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Difference between revisions of "Electromagnetic wave"
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Paul Wormer (Talk | contribs)
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\mathbf{E}(\mathbf{r}, t) = \mathbf{e}_x E_0 \sin\big[k(z-ct)\big]\quad\hbox{with}\quad k \equiv \frac{2\pi}{\lambda}.
\mathbf{E}(\mathbf{r}, t) = \mathbf{e}_x E_0 \sin\big[k(z-ct)\big]\quad\hbox{with}\quad k \equiv \frac{2\pi}{\lambda}.
The snapshot is taken at <math>\
The snapshot is taken at <math> \t = 2\pi n /(ck)</math> for some arbitrary integer ''n''. We assumed here that the direction of '''E''' defines the direction of the ''x''-axis with unit vector '''e'''''x'' along this axis. The quantity ''E''0 is the [[amplitude]] of the wave. Insertion of this expression in in the left hand side of the wave equation for '''E''' gives
\boldsymbol{\nabla}^2 \mathbf{E} = \mathbf{e}_x E_0 \frac{\partial^2 \sin\big[k(z-ct)\big]}{\partial z^2} = - k^2 \mathbf{e}_x E_0 \sin\big[k(z-ct)\big].
\boldsymbol{\nabla}^2 \mathbf{E} = \mathbf{e}_x E_0 \frac{\partial^2 \sin\big[k(z-ct)\big]}{\partial z^2} = - k^2 \mathbf{e}_x E_0 \sin\big[k(z-ct)\big].
Revision as of 14:10, 14 January 2009
In physics, an electromagnetic wave is a change, periodic in space and time, of an electric field E(r,t) and a magnetic field B(r,t). A stream of electromagnetic waves is referred to as electromagnetic radiation. Because an electric as well as a magnetic field is involved, the term electromagnetic (EM) is used, a contamination of electric and magnetic. Examples of EM waves in increasing wavelength are: gamma rays, X-rays, ultraviolet light, visible light, infrared, microwaves, and radio waves. All these waves propagate in vacuum with the same speed c, the speed of light. The speed of light in air (at standard temperature and pressure) is very close to the speed of light in vacuum (the refractive index of air, n, is 1.0002926, meaning that the speed of electromagnetic waves in air is c/n ≈ c).
Classically (non-quantum mechanically), EM radiation is produced by accelerating charges, for instance, by the oscillating charge in a radio antenna. Quantum mechanically, EM radiation is emitted whenever a system in an energetically high state (of energy E2) makes a transition to a state of lower energy (E1); during this transition a photon (light quantum) of energy proportional to the difference E2 - E1 > 0 is emitted. This is what happens in a fluorescent tube: mercury atoms are brought into an energetically high state by collisions with electrons, and upon subsequent falling down to their lowest energy state they emit photons.
Electromagnetic waves were predicted on theoretical grounds by James Clerk Maxwell in 1861 and first emitted and received in the laboratory by Heinrich Hertz a quarter century later. The first to see the applicability for communication purposes was the inventor of radiotelegraphy Guglielmo Marconi (around 1900). Later applications are radio, television, radar, cellular phones, gps, and all kinds of wireless applications, from remote control of television sets to internet access in public places.
In figure 1 we see a snapshot (i.e., a picture at a certain point in time) of the magnetic and electric fields in adjacent points of space. In each point, the vector E is perpendicular to the vector B. The wave propagates to the right, along an axis which we conveniently refer to as z-axis. Both E and B are perpendicular to the propagation direction, which is expressed by stating that an electromagnetic wave is a transverse wave, in contrast to sound waves, which are longitudinal waves (i.e., air molecules vibrate parallel to the propagation direction of the sound).
Assume that the snapshot in figure 1 is taken at time t, then at a certain point z we see an arrow of certain length representing E(z,t) and also a vector B(z,t). At a point in time Δt later, the same values of E and B (same arrows) are seen at z + c Δt. The arrows seem to have propagated to the right with a speed c.
The time t is fixed and the position z varies in figure 1. Conversely, we can keep the position fixed and imagine what happens if time changes. Focus on a fixed point z, then in progressing time the two vectors E(z,t) and B(z,t) in the point z, grow to a maximum value, then shrink to zero, become negative, go to a minimum value, and grow again, passing through zero on their way to the same maximum value. This cycle is repeated indefinitely. When we now plot E and B in the fixed point z as a function of time t, we see the same type (sine-type) function as in figure 1. The number of times per second that the vectors go through a full cycle is the frequency of the electromagnetic wave.
Periodicity in space means that the EM wave is repeated after a certain distance. This distance, the wavelength is traditionally designated by λ, see figure 1. If we go at a fixed time a distance λ to the right or to the left we encounter the very same fields E and B.
Basically, the only property distinguishing different kinds of EM waves, is their wavelength, see figure 2. Note the enormous span in wavelengths, from one trillionth of a millimeter for gamma-rays (radioactive rays) up to the VLF (very low frequency) radio waves of about 100 kilometer.
Frequency of electromagnetic waves
Often EM waves are characterized by their frequency ν, instead of their wavelength λ. If the EM field goes through ν full cycles in a second, where ν is a positive integral number, the field has a frequency of ν Hz (hertz). The speed of propagation of the EM waves being c, in 1/ν seconds the wave propagates a distance c × (1/ν) meter (according to the formula: distance traveled is speed times traveling time). The distance covered in 1/ν seconds is by definition the wavelength λ:
- <math>
\lambda = \frac{c}{\nu} \quad \Longrightarrow\quad \nu = \frac{c}{\lambda} </math> If we express c in m/s then λ is obtained in m. To convert quickly from wavelength to frequency we can approximate c by 3·108 m/s.
As was pointed out above, the wavelengths of the various parts of the EM spectrum differ many orders of magnitude. Furthermore, the sources of the radiations, the interactions with matter, and the detectors employed differ widely, too. So, it is not surprising that in the past different parts of the spectrum were discovered at different times and that electromagnetic radiation of different wavelengths is called by different names. In the table and in figure 2 some illustrative values are given for several kinds of EM waves, together with their names.
Some typical values of: wavelength (λ), frequency (ν = c/λ), photon energy (hν), cycle time (T = 1/ν), and inverse wavelength [1/(100⋅λ)].
|EM wave
|λ (m)
|ν (1/s)
|hν (J)
|T (s)
|1/λ (cm−1)
Monochromatic linearly polarized waves
The wave depicted in figure 1 is monochromatic, i.e., it is characterized by a single wavelength (monochromatic means "of one color". In the visible region, different wavelengths correspond to light of different colors). It is known that EM waves can be linearly superimposed, which is due to the fact that they are solutions of a linear partial differential equation, the wave equation (see next section). A linear superposition of waves is a solution of the same wave equation as the waves themselves. Such a superposition is also an electromagnetic wave (a propagating periodic EM field). If waves of different wavelengths are superimposed, then a non-monochromatic wave is obtained (the term multi-chromatic wave would be apt, but is never used). By means of Fourier analysis a non-monochromatic wave can be decomposed into its monochromatic components.
The electric field vectors in figure 1 are all in one plane, this is the plane of polarization, and a wave with one fixed polarization plane, is called linearly polarized.
The radiation of many lasers is monochromatic and linearly polarized (at least to a very good approximation).
Relation to Maxwell's equations
In this section it will be shown that the electromagnetic wave depicted in figure 1 is a solution of the Maxwell equations in the vacuum.
We assume that at some distance away from the source of EM waves (a radio transmitter, a laser, gamma radiating nuclei, etc.), there is no charge density ρ and no current density J. For that region of space, the microscopic (vacuum) Maxwell equations become (in SI units):
- <math>
\boldsymbol{\nabla} \cdot \mathbf{B} = 0, \qquad \boldsymbol{\nabla} \cdot \mathbf{E} = 0, </math> and
- <math>
\boldsymbol{\nabla} \times \mathbf{B}= \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}, \qquad \boldsymbol{\nabla} \times \mathbf{E}= -\frac{\partial \mathbf{B}}{\partial t}. </math> Apply to the last Maxwell equation the following relation, known from vector analysis and valid for any (differentiable) vector field,
- <math>
\boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \mathbf{E}) = \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{E}) - \boldsymbol{\nabla}^2 \mathbf{E} </math> and use that ∇ · E = 0, then E satisfies the wave equation,
- <math>
\boldsymbol{\nabla}^2 \mathbf{E} = \frac{\partial(\boldsymbol{\nabla}\times \mathbf{B})}{\partial t} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2}. </math> Note that the displacement current (time derivative of E) is essential in this equation, if it were absent (zero), the field E would be a static, time-independent, electric field, and there would be no waves.
In the very same way we derive a wave equation for B,
- <math>
\boldsymbol{\nabla}^2 \mathbf{B} = \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2}. </math> Observe that E and B are related by the third and fourth Maxwell equation, which express the fact that a displacement current causes a magnetic field, and a changing magnetic field causes an electric field (Faraday's law of induction), respectively. So a time-dependent electric field that is not associated with a time-dependent magnetic field cannot exist, and conversely. Indeed, in special relativity E and c B can be transformed into one another by a Lorentz transformation of the electromagnetic field tensor, which shows their close relationship.
The wave equation is without doubt the most widely studied differential equation in mathematical physics. In figure 1 the electric field depicts a particular solution, with special initial and boundary conditions. The snapshot that is depicted has the analytic form
- <math>
\mathbf{E}(\mathbf{r}, t) = \mathbf{e}_x E_0 \sin\big[k(z-ct)\big]\quad\hbox{with}\quad k \equiv \frac{2\pi}{\lambda}. </math> The snapshot is taken at <math> \, t = 2\pi n /(ck)</math> for some arbitrary integer n. We assumed here that the direction of E defines the direction of the x-axis with unit vector ex along this axis. The quantity E0 is the amplitude of the wave. Insertion of this expression in in the left hand side of the wave equation for E gives
- <math>
\boldsymbol{\nabla}^2 \mathbf{E} = \mathbf{e}_x E_0 \frac{\partial^2 \sin\big[k(z-ct)\big]}{\partial z^2} = - k^2 \mathbf{e}_x E_0 \sin\big[k(z-ct)\big]. </math> Insertion of this expression in in the right hand side of the wave equation for E gives
- <math>
\frac{\mathbf{e}_x E_0}{c^2} \frac{\partial^2 \sin\big[k(z-ct)\big]}{\partial t^2} = - \frac{c^2\,k^2}{c^2} \mathbf{e}_x E_0 \sin\big[k(z-ct)\big] = - k^2 \mathbf{e}_x E_0 \sin\big[k(z-ct)\big], </math> so that it follows that the special solution, depicted in figure 1, is indeed a solution of the wave equation for E.
We could now proceed in the very same way and solve the wave equation for B, but then we could easily overlook the relation between the two fields. So we rather substitute the solution for E into the fourth Maxwell equation and use the definition of curl as a determinant,
- <math>
\begin{vmatrix} \mathbf{e}_x &\quad \mathbf{e}_y & \mathbf{e}_z \\ \frac{\partial}{\partial x}&\quad \frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ E_0 \sin\big[k(z-ct)\big] &\quad 0 & \quad 0 \\ \end{vmatrix} = k \mathbf{e}_y E_0 \cos\big[k(z-ct)\big] = - \frac{\partial \mathbf{B}}{\partial t}. </math> It is easy to see that
- <math>
\mathbf{B}(\mathbf{r},t) = \mathbf{e}_y B_0 \sin\big[k(z-ct)\big] \quad \hbox{with}\quad B_0 \equiv \frac{E_0}{c} </math> is a solution of this equation. It follows that E and B are perpendicular (along the x-axis and y-axis, respectively) and are in phase. That is, E and B are simultaneously zero and attain simultaneously their maximum and minimum. The fact that (in vacuum) the amplitude of B is a factor c smaller than that of E is due to the use of SI units, in which the amplitudes have different dimensions. In Gaussian units this is not the case and E0 = B0.
Energy and energy flow in electromagnetic field
Relation between power densities
In this section the following balance of power (energy per unit time) densities will be discussed:
- <math>
\boldsymbol{\nabla}\cdot \mathbf{S} + \frac{\partial \mathcal{E}_\textrm{Field}}{\partial t} = - \mathcal{E}_\textrm{Joule} \qquad\qquad\qquad(1) </math> where
- <math>
\begin{align} \mathbf{S} &= \mathbf{E} \times \mathbf{H}\equiv \mathbf{E} \times \frac{1}{\mu_0}\mathbf{B} \qquad\mathrm{(the\,\, Poynting\,\,vector)} \\ \mathcal{E}_\textrm{Field} &= \frac{1}{2} ( \mathbf{E}\cdot\mathbf{D}+ \mathbf{B}\cdot\mathbf{H}) \equiv \frac{1}{2} ( \epsilon_0 \mathbf{E}\cdot\mathbf{E}+ \frac{1}{\mu_0}\mathbf{B}\cdot\mathbf{B}) \\ \mathcal{E}_\textrm{Joule} &= \mathbf{E}\cdot \mathbf{J} \\ \end{align} </math> The terms in equation (1) have dimension W/m3 (watt per volume). The quantity <math>\scriptstyle -\mathcal{E}_\textrm{Joule}\, >\, 0</math> represents the rate at which energy is produced per unit volume by ordinary Joulean (resistance) heating. The quantity <math>\scriptstyle \mathcal{E}_\textrm{Field}</math> is the energy density of the EM field and the time derivative is the rate of increase (power density). The vector S, Poynting's vector, is the power flux, the amount of energy crossing unit area perpendicular to the vector, per unit time.
Multiplying the terms in equation (1) by a volume ΔV and a time span Δt, both small enough that the terms in the equation may assumed to be constant over the volume and the time span, the equation represents the conservation of energy: the energy generated in ΔVΔt (the right hand side) is equal to the increase in energy in ΔV plus the net flow of energy leaving ΔV. Hence, when this equation is multiplied by ΔVΔt it is an equation of continuity for energy.
The intensity I of an electromagnetic wave is by definition the modulus of the Poynting vector, the amount of energy carried by the wave across a unit surface in unit time [dimension: volt×ampere/meter2 = joule/(second×m2) = W/m2],
- <math>
I \equiv |\mathbf{E} \times \mathbf{H}| = \Big[ (\mathbf{E}\cdot\mathbf{E})\,(\mathbf{H}\cdot\mathbf{H})- (\mathbf{E}\cdot\mathbf{H})\,(\mathbf{E}\cdot\mathbf{H})\Big]^{1/2}. </math> Use E ⋅ H = 0 and that in vacuum and SI units
- <math>
\mathbf{H}(\mathbf{r},t) = c\,\epsilon_0\ \mathbf{E}(\mathbf{r},t), </math> then
- <math>
I = c\,\epsilon_0\, |\mathbf{E}(\mathbf{r},t)|^2. </math> Often I is time-averaged over a complete cycle. Since the integral of cos2 and sin2 over a cycle is 1/2, we get for the time-averaged intensity:
- <math>
\bar{I} = \frac{1}{2} c\,\epsilon_0\, E_0^2, </math> where E0 is the amplitude of the electric field.
Clearly, the cycle averaged intensity of a wave going through vacuum is constant, independent of the direction of propagation, z. This is because in vacuum the wave does not lose energy to the medium. In a medium this may be different, <math>\,\bar{I}\,</math> may decrease because of energy loss. The Lambert-Beer law states in that case
- <math>
\frac{d\bar{I}(z)}{dz} = - k \bar{I}(z) \quad\Longrightarrow\quad \bar{I}(z) = \bar{I}_0 e^{-kz}. </math> Hence the electromagnetic wave is damped by the factor exp[− kz/2]. This damping gives rise to an imaginary component of the index of refraction.
Derivation of relation between power densities
Equation (1), the balance of power, will be proved. Recall from elementary electricity theory the laws of Joule and Ohm. They state that the amount of energy W per unit time, produced by a conduction current I, is equal to
- <math>
W = I^2\, R = I\,V, \quad\hbox{with}\quad R,\, W > 0 , </math> where R is the resistance and V a voltage difference.
Assuming that the current flows along z, we introduce the current density Jz, and using
- <math>
V = - E_z \Delta z \quad\hbox{and}\quad I = J_z \Delta x \Delta y </math> we obtain
- <math>
W = - E_z J_z\, \Delta x\Delta y\Delta z . </math> We could continue discussing the system with the small volume <math> \Delta x\Delta y\Delta z</math>. However, because all terms in the equations would be multiplied by the same volume, it is more convenient to consider densities and to divide out the volume. Nevertheless, we still refer to the system. Thus, we define
- <math>
\mathcal{E}_\textrm{Joule} \equiv -\frac{W}{ \Delta x\Delta y\Delta z} = \mathbf{E}\cdot \mathbf{J} < 0. </math> The negative quantity <math>\scriptstyle \mathcal{E}_\textrm{Joule}</math> is the loss of energy of the system per unit time and per unit volume (according to Joule's and Ohm's laws). One may look upon the quantity <math>\scriptstyle -\mathcal{E}_\textrm{Joule}</math> as the work (per unit time and unit volume) done by the Lorentz force <math>\scriptstyle q (\mathbf{v}\times \mathbf{B} + \mathbf{E})</math> on the moving particles constituting the current density J. Since <math>\scriptstyle \mathbf{v}\cdot (\mathbf{v}\times \mathbf{B}) = 0 </math>, this work depends only on the electric field E.
Apply one of Maxwell's equations:
- <math>
\mathbf{E}\cdot \mathbf{J} = \mathbf{E}\cdot\left( - \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} + \frac{1}{\mu_0} \boldsymbol{\nabla}\times \mathbf{B} \right) </math> Use a rule known from vector analysis and apply another one of Maxwell's equations,
- <math>
\begin{align} \mathbf{E}\cdot(\boldsymbol{\nabla}\times \mathbf{B}) =& \mathbf{B}\cdot(\boldsymbol{\nabla}\times \mathbf{E}) - \boldsymbol{\nabla}\cdot(\mathbf{E} \times \mathbf{B}) \\ =& -\mathbf{B}\cdot\left(\frac{\partial \mathbf{B}}{\partial t}\right) - \boldsymbol{\nabla}\cdot(\mathbf{E} \times \mathbf{B}) \end{align} </math> Define
- <math>
\mathcal{E}_\textrm{Field} \equiv \frac{1}{2}\left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) = \frac{1}{2}\left( \mathbf{E}\cdot\mathbf{D} + \mathbf{B}\cdot\mathbf{H} \right) </math> with
- <math>
\mathbf{D}\equiv \epsilon_0\mathbf{E}\quad\hbox{and}\quad\mathbf{B} \equiv \mu_0\mathbf{H}, </math> where ε0 is the electric constant and μ0 the magnetic constant of the vacuum. Define also
- <math>
\mathbf{S} \equiv \frac{1}{\mu_0} \mathbf{E}\times\mathbf{B} = \mathbf{E}\times\mathbf{H} </math> where S is the Poynting vector called after John Henry Poynting. This vector is perpendicular to the plane of E and B and by the right-hand rule, it points in the direction of propagation of the EM wave. The divergence of the Poynting vector is the energy flow associated with the electromagnetic wave, i.e., with the pair E(r,t) and B(r,t). By definition ∇·S gives the flow leaving the system and − ∇·S gives the flow entering the system. The total energy balance becomes
- <math>
-\mathcal{E}_\textrm{Joule} = \frac{\partial \mathcal{E}_\textrm{Field}}{\partial t} + \boldsymbol{\nabla}\cdot \mathbf{S}. </math>
Here we have found an example of the conservation of energy, known as Poynting's theorem: The energy produced per unit time according to Joule's law <math>\scriptstyle -\mathcal{E}_\textrm{Joule}</math> is equal to the rate in increase of the electromagnetic energy of the system <math>\scriptstyle \mathcal{E}_\textrm{Field}</math>, plus the flow of EM radiation ∇·S leaving the system.
If there is no current, J = 0, then
- <math>
\frac{\partial \mathcal{E}_\textrm{Field}}{\partial t} = - \boldsymbol{\nabla}\cdot \mathbf{S}, </math> which is the continuity equation. The increase of field energy per unit time is the flow of radiation energy into the system.
As an example we give an order-of-magnitude-calculation of an electromagnetic energy density. Consider to that end a radio station with a signal of strength P kW. We compute the energy density at a distance R from the station. First we must assume what the shape is of the waves emitted by the antenna, are they spherical or cylindrical? We choose the latter and call the cylinder height z. Further it is assumed that power density is homogeneous and that all power crosses the cylindrical walls, that is, power crossing the top and bottom of the cylinder is assumed to be zero. Also no absorption by the atmosphere or the Earth will occur. When a steady state is reached (some time after the beginning of the transmission), the time derivative of <math>\scriptstyle \mathcal{E}_\mathrm{Field}</math> vanishes. The energy density at a distance R becomes constant in time,
- <math>
\mathcal{E}_\mathrm{Field}(R) = \frac{P} {2\pi R z c} \quad \mathrm{J/m^3}, </math> where c is the speed of propagation of the radio signal (is speed of ligth ≈ 3·108 m/s).[1] To give a numerical example: P = 100 kW, R = 5 km (about 3 miles), z = 50 m, then <math>\scriptstyle \mathcal{E}_\mathrm{Field}</math> = 2.1·10−10 J/m3.
Fourier expansions of the fields
The quantization of the EM fields leads to photons, light quanta of well-defined energy and momentum. In field quantization a key role is played by the Fourier expansion of the different vector fields. Hence, we will now discuss the Fourier expansions of the fields E, B, and A. It will be seen that the expansion of the vector potential A yields immediately the expansions of the fields E and B.
Fourier expansion of a vector field
For an arbitrary real vector field F its Fourier expansion is the following:
- <math>
\mathbf{F}(\mathbf{r}, t) = \sum_\mathbf{k} \left( \mathbf{f}_k(t) e^{i\mathbf{k}\cdot\mathbf{r}} + \bar{\mathbf{f}}_k(t) e^{-i\mathbf{k}\cdot\mathbf{r}} \right) </math> where the bar indicates complex conjugation. Such an expansion, labeled by a discrete (countable) set of vectors k, is always possible when F satisfies periodic boundary conditions, i.e., F(r + p,t) = F(r,t) for some finite vector p. To impose such boundary conditions, it is common to consider EM waves as if they are in a virtual box of finite volume V. Waves on opposite walls of the box are enforced to have the same value (usually zero). Note that the waves are not restricted to the box: the box is replicated an infinite number of times in x, y, and z direction.
Vector potential and its expansion
The magnetic field B is a transverse field and hence can be written as
- <math>
\mathbf{B}(\mathbf{r}, t) = \boldsymbol{\nabla}\times \mathbf{A}(\mathbf{r}, t), </math> in which the vector potential A is introduced. Also the electric field E is transverse, because earlier we assumed absence of charge distributions. The electric field E also follows from A,
- <math>
\mathbf{E}(\mathbf{r}, t) = - \frac{\partial \mathbf{A}(\mathbf{r}, t)}{\partial t}. </math> The fact that E can be written this way is due to the choice of Coulomb gauge for A:
- <math>
\boldsymbol{\nabla}\cdot\mathbf{A}(\mathbf{r}, t) = 0. </math> By definition, a choice of gauge does not affect any measurable properties (the best known example of a choice of gauge is the fixing of the zero of an electric potential, for instance at infinity). The Coulomb gauge makes A transverse as well, and clearly A is in the same plane as E. (The time differentiation does not affect direction.) So, the vector fields A, B, and E are all in a plane perpendicular to the propagation direction and can be written in terms of ex and ey (in the definition of figure 1). It is more convenient to choose complex unit vectors:
- <math>
\mathbf{e}^{(1)} \equiv \frac{-1}{\sqrt{2}}(\mathbf{e}_x + i \mathbf{e}_y)\quad\hbox{and}\quad\mathbf{e}^{(-1)} \equiv \frac{1}{\sqrt{2}}(\mathbf{e}_x - i \mathbf{e}_y) </math> which are orthonormal,
- <math>
\mathbf{e}^{(\mu)}\cdot\bar{\mathbf{e}}^{(\mu')} = \delta_{\mu,\mu'}\quad\hbox{with}\quad\mu,\mu'= 1,\, -1.
</math> The Fourier expansion of the vector potential reads
- <math>
\mathbf{A}(\mathbf{r}, t) = \sum_\mathbf{k}\sum_{\mu=-1,1} \left( \mathbf{e}^{(\mu)}(\mathbf{k}) a^{(\mu)}_\mathbf{k}(t) \, e^{i\mathbf{k}\cdot\mathbf{r}} + \bar{\mathbf{e}}^{(\mu)}(\mathbf{k}) \bar{a}^{(\mu)}_\mathbf{k}(t) \, e^{-i\mathbf{k}\cdot\mathbf{r}} \right). </math> The vector potential obeys the wave equation. The substitution of the Fourier series of A into the wave equation yields for the individual terms,
- <math>
-k^2 a^{(\mu)}_\mathbf{k}(t) = \frac{1}{c^2} \frac{\partial^2 a^{(\mu)}_\mathbf{k}(t)}{\partial t^2} \quad \Longrightarrow \quad a^{(\mu)}_\mathbf{k}(t) \propto e^{-i\omega t}\quad\hbox{with}\quad\omega = kc. </math> It is now an easy matter to construct the corresponding Fourier expansions for E and B from the expansion of the vector potential A. For instance, the expansion for E follows from differentiation with respect to time,
- <math>
\mathbf{E}(\mathbf{r}, t) = i\sum_\mathbf{k}\sum_{\mu=-1,1} \omega \left( \mathbf{e}^{(\mu)}(\mathbf{k}) a^{(\mu)}_\mathbf{k}(t) \, e^{i\mathbf{k}\cdot\mathbf{r}} - \bar{\mathbf{e}}^{(\mu)}(\mathbf{k}) \bar{a}^{(\mu)}_\mathbf{k}(t) \, e^{-i\mathbf{k}\cdot\mathbf{r}} \right) </math>
Fourier-expanded energy
The electromagnetic energy density <math>\scriptstyle \mathcal{E}_\mathrm{Field}</math>, defined earlier in this article, can be expressed in terms of the Fourier coefficients. We define the total energy (classical Hamiltonian) of a finite volume V by
- <math>
H = \iiint_V \mathcal{E}_\mathrm{Field}(\mathbf{r},t) \mathrm{d}^3\mathbf{r}. </math>
The classical Hamiltonian in terms of Fourier coefficients takes the form
- <math>
H = V\epsilon_0 \sum_\mathbf{k}\sum_{\mu=1,-1} \omega^2 \big(\bar{a}^{(\mu)}_\mathbf{k}(t)a^{(\mu)}_\mathbf{k}(t)+ a^{(\mu)}_\mathbf{k}(t)\bar{a}^{(\mu)}_\mathbf{k}(t)\big) </math> with
- <math>
\omega \equiv |\mathbf{k}|c = 2\pi \nu = 2\pi \frac{c}{\lambda} </math> and ε0 is the electric constant. The two terms in the summand of H are identical (factors commute) and may be summed. However, after quantization (interpretation of the expansion coefficients as operators) the factors do no longer commute and according to quantum mechanical rules one must depart from the symmetrized classical Hamiltonian.
Fourier-expanded momentum
The electromagnetic momentum, PEM, of EM radiation enclosed by a volume V is proportional to an integral of the Poynting vector (see above). In SI units:
- <math>
\mathbf{P}_\textrm{EM} \equiv \frac{1}{c^2} \iiint_V \mathbf{S}\, \textrm{d}^3\mathbf{r} = \epsilon_0 \iiint_V \mathbf{E}(\mathbf{r},t)\times \mathbf{B}(\mathbf{r},t)\, \textrm{d}^3\mathbf{r}. </math>
Quantization of the electromagnetic field
Einstein postulated in 1905 that an electromagnetic field consists of energy parcels (light quanta, later called photons) of energy hν, where h is Planck's constant. In 1927 Paul A. M. Dirac was able to fit the photon concept into the framework of the new quantum mechanics. He applied a technique which is now generally called second quantization,[2] although this term is somewhat of a misnomer for EM fields, because they are, after all, solutions of the classical Maxwell equations, and it is the first time that they are quantized.
Second quantization
Second quantization starts with an expansion of a field in a basis consisting of a complete set of functions. The coefficients multiplying the basis functions are then interpreted as operators and (anti)commutation relations between these new operators are imposed, commutation relations for bosons and anticommutation relations for fermions (nothing happens to the basis functions themselves). By doing this, the expanded field is converted into a fermion or boson operator field. The expansion coefficients have become creation and annihilation operators: a creation operator creates a particle in the corresponding basis function and an annihilation operator annihilates a particle in this function.
In the case of EM fields the required expansion of the field is the Fourier expansion.
Quantization of EM field
The best known example of quantization is the replacement of the time-dependent linear momentum of a particle by the rule
- <math>\mathbf{p}(t) \rightarrow -i\hbar\boldsymbol{\nabla}</math>.
Note that Planck's constant is introduced here and that time disappears (in the so-called Schrödinger picture).
For the EM field we do something similar and apply the quantization rules:
- <math>
\begin{align} a^{(\mu)}_\mathbf{k}(t)\, &\rightarrow\, \sqrt{\frac{\hbar}{2 \omega V\epsilon_0}}\, a^{(\mu)}(\mathbf{k}) \\ \bar{a}^{(\mu)}_\mathbf{k}(t)\, &\rightarrow\, \sqrt{\frac{\hbar}{2 \omega V\epsilon_0}}\, {a^\dagger}^{(\mu)}(\mathbf{k}) \\ \end{align} </math> subject to the boson commutation relations
- <math>
\begin{align} \big[ a^{(\mu)}(\mathbf{k}),\, a^{(\mu')}(\mathbf{k}') \big] & = 0 \\ \big[{a^\dagger}^{(\mu)}(\mathbf{k}),\, {a^\dagger}^{(\mu')}(\mathbf{k}')\big] &=0 \\ \big[a^{(\mu)}(\mathbf{k}),\,{a^\dagger}^{(\mu')}(\mathbf{k}')\big]&= \delta_{\mathbf{k},\mathbf{k}'} \delta_{\mu,\mu'}. \end{align} </math> The square brackets indicate a commutator, defined by
- <math>
\big[ A, B\big] \equiv AB - BA </math> for any two quantum mechanical operators A and B.
The quantized fields (operator fields) are the following
- <math>
\begin{align} \mathbf{A}(\mathbf{r}) &= \sum_{\mathbf{k},\mu} \sqrt{\frac{\hbar}{2 \omega V\epsilon_0}} \left(\mathbf{e}^{(\mu)} a^{(\mu)}(\mathbf{k}) e^{i\mathbf{k}\cdot\mathbf{r}} + \bar{\mathbf{e}}^{(\mu)} {a^\dagger}^{(\mu)}(\mathbf{k}) e^{-i\mathbf{k}\cdot\mathbf{r}} \right) \\ \mathbf{E}(\mathbf{r}) &= i\sum_{\mathbf{k},\mu} \sqrt{\frac{\hbar\omega}{2 V\epsilon_0}} \left(\mathbf{e}^{(\mu)} a^{(\mu)}(\mathbf{k}) e^{i\mathbf{k}\cdot\mathbf{r}} - \bar{\mathbf{e}}^{(\mu)} {a^\dagger}^{(\mu)}(\mathbf{k}) e^{-i\mathbf{k}\cdot\mathbf{r}} \right) \\ \mathbf{B}(\mathbf{r}) &= i\sum_{\mathbf{k},\mu} \sqrt{\frac{\hbar}{2 \omega V\epsilon_0}} \left((\mathbf{k}\times\mathbf{e}^{(\mu)}) a^{(\mu)}(\mathbf{k}) e^{i\mathbf{k}\cdot\mathbf{r}} - (\mathbf{k}\times\bar{\mathbf{e}}^{(\mu)}) {a^\dagger}^{(\mu)}(\mathbf{k}) e^{-i\mathbf{k}\cdot\mathbf{r}} \right), \\ \end{align} </math> where ω = c |k| = ck.
Hamiltonian of the field
Substitution of the operators into the classical Hamiltonian gives the Hamilton operator of the EM field
- <math>
\begin{align} H &= \frac{1}{2}\sum_{\mathbf{k},\mu=-1,1} \hbar \omega \Big({a^\dagger}^{(\mu)}(\mathbf{k})\,a^{(\mu)}(\mathbf{k}) + a^{(\mu)}(\mathbf{k})\,{a^\dagger}^{(\mu)}(\mathbf{k})\Big) \\ &= \sum_{\mathbf{k},\mu} \hbar \omega \Big({a^\dagger}^{(\mu)}(\mathbf{k})a^{(\mu)}(\mathbf{k}) + \frac{1}{2}\Big) \end{align} </math> By the use of the commutation relations the second line follows from the first. Note that <math> \hbar\omega = h\nu =\hbar c |\mathbf{k}|</math> , which is the well-known Einstein expression for photon energy. Remember that ω depends on k, even though it is not explicit in the notation. The notation ω(k) could have been introduced, but is not common.
Digression: harmonic oscillator
The second quantized treatment of the one-dimensional quantum harmonic oscillator is a well-known topic in quantum mechanical courses. We digress and say a few words about it. The harmonic oscillator Hamiltonian has the form
- <math>
H = \hbar \omega \big( a^\dagger a + \frac{1}{2} \big) </math> where ω ≡ 2πν is the fundamental frequency of the oscillator. The ground state of the oscillator is designated by | 0 > and is referred to as vacuum state. It can be shown that <math>\scriptstyle a^\dagger</math> is an excitation operator, it excites from an n fold excited state to an n+1 fold excited state:
- <math>
a^\dagger |n \rangle = |n+1 \rangle \sqrt{n+1} \quad\hbox{in particular}\quad a^\dagger |0 \rangle = |1 \rangle \quad\hbox{and}\quad (a^\dagger)^n |0\rangle \propto |n\rangle. </math> Since harmonic oscillator energies are equidistant, the n-fold excited state | n > can be looked upon as a single state containing n particles (sometimes called vibrons) all of energy hν. These particles are bosons. For obvious reason the excitation operator <math> a^\dagger</math> is called a creation operator.
From the commutation relation follows that the Hermitian adjoint <math>\scriptstyle a</math> de-excites:
- <math>
a |n \rangle = |n-1 \rangle \sqrt{n} \quad\hbox{in particular}\quad a |0 \rangle \propto 0 \rightarrow a |0 \rangle = 0, </math> because a function times the number 0 is the zero function. For obvious reason the de-excitation operator <math> \,a</math> is called an annihilation operator.
Suppose now we have a number of non-interacting (independent) one-dimensional harmonic oscillators, each with its own fundamental frequency ωi. Because the oscillators are independent, the Hamiltonian is a simple sum:
- <math>
H = \sum_i \hbar\omega_i \Big(a^\dagger(i) a(i) +\frac{1}{2} \Big). </math>
Making the substitution
- <math>
i \rightarrow (\mathbf{k}, \mu) </math> we see that the Hamiltonian of the EM field can be looked upon as a Hamiltonian of independent oscillators of energy ω = |k| c and oscillating along direction e(μ) with μ=1,−1.
Photon energy
The quantized EM field has a vacuum (no photons) state | 0 >. The application of, say,
- <math>
\big( {a^\dagger}^{(\mu)}(\mathbf{k}) \big)^m \, \big( {a^\dagger}^{(\mu')}(\mathbf{k}') \big)^n \, |0\rangle \propto \Big| m^{(\mu)}(\mathbf{k}), \, n^{(\mu')}(\mathbf{k}') \, \Big\rangle, </math> gives a quantum state of m photons in mode (μ, k) and n photons in mode (μ', k'). We use the proportionality symbol because the state on the right-hand is not normalized to unity.
We can shift the zero of energy and rewrite the Hamiltonian as
- <math>
H= \sum_{\mathbf{k},\mu} \hbar \omega N^{(\mu)}(\mathbf{k}) \quad\hbox{with}\quad N^{(\mu)}(\mathbf{k}) \equiv {a^\dagger}^{(\mu)}(\mathbf{k})a^{(\mu)}(\mathbf{k}) </math> The operator <math> N^{(\mu)}(\mathbf{k})</math> is the number operator. When acting on a quantum mechanical photon state, it returns the number of photons in mode (μ, k). Such a photon state is an eigenstate of the number operator. This is why the formalism described here, is often referred to as the occupation number representation. The effect of H on a single-photon state is
- <math>
H \left({a^\dagger}^{(\mu)}(\mathbf{k}) \,|0\rangle\right) = \hbar\omega \left( {a^\dagger}^{(\mu)}(\mathbf{k}) \,|0\rangle\right). </math> Apparently, the single-photon state is an eigenstate of H and <math>\hbar \omega = h \nu </math> is the corresponding energy.
Example photon density
In an earlier example we introduced a radio station and calculated the electromagnetic energy density that the station creates in its environment; at 5 km from the station this was 2.1·10−10 J/m3. Let us now see if we need quantum mechanics to describe the broadcasting of this station.
The classical approximation to EM radiation is good when the number of photons is much larger than unity in the volume
- <math>
\left(\frac{\lambda}{2\pi}\right)^3 , </math> where λ is the length of the radio waves. In that case quantum fluctuations are negligible and cannot be heard.
Suppose the radio station broadcasts at ν = 100 MHz, then it is sending out photons with an energy content of νh = 1·108× 6.6·10−34 = 6.6·10−26 J, where h is Planck's constant. The wavelength of the station is λ = c/ν = 3 m, so that λ/(2π) = 48 cm and the volume is 0.111 m3. The energy content of this volume element is 2.1·10−10 × 0.111 = 2.3 ·10−11 J, which amounts to
- 3.5 ·1012 photons per <math>\left (\frac{\lambda}{2\pi}\right)^3</math>
Obviously, 3.5 ·1012 is much larger than one and hence quantum effects do not play a role; the waves emitted by this station are well into the classical limit, even when it plays non-classical music, for instance of Led Zepplin.
Photon momentum
Introducing the operator expansions for E and B into the classical form
- <math>
\mathbf{P}_\textrm{EM} = \epsilon_0 \iiint_V \mathbf{E}(\mathbf{r},t)\times \mathbf{B}(\mathbf{r},t)\, \textrm{d}^3\mathbf{r}, </math> yields
- <math>
\mathbf{P}_\textrm{EM} = \sum_{\mathbf{k},\mu} \hbar \mathbf{k} \Big({a^\dagger}^{(\mu)}(\mathbf{k})a^{(\mu)}(\mathbf{k}) + \frac{1}{2}\Big) = \sum_{\mathbf{k},\mu} \hbar \mathbf{k} N^{(\mu)}(\mathbf{k}). </math> The 1/2 that appears can be dropped because when we sum over the allowed k, k cancels with −k. The effect of PEM on a single-photon state is
- <math>
\mathbf{P}_\textrm{EM} \left({a^\dagger}^{(\mu)}(\mathbf{k}) \,|0\rangle \right) = \hbar\mathbf{k} \left( {a^\dagger}^{(\mu)}(\mathbf{k}) \,|0\rangle\right). </math> Apparently, the single-photon state is an eigenstate of the momentum operator, and <math>\hbar \mathbf{k}</math> is the eigenvalue (the momentum of a single photon).
Photon mass
The photon having non-zero linear momentum, one could imagine that it has a non-vanishing rest mass m0, which is its mass at zero speed. However, we will now show that this is not the case: m0 = 0.
Since the photon propagates with the speed of light, special relativity is called for. The relativistic expressions for energy and momentum squared are,
- <math>
E^2 = \frac{m_0^2 c^4}{1-v^2/c^2}, \quad p^2 = \frac{m_0^2 v^2}{1-v^2/c^2}. </math> From p2/E2,
- <math>
\frac{v^2}{c^2} = \frac{c^2p^2}{E^2} \quad\Longrightarrow\quad E^2= \frac{m_0^2c^4}{1 - c^2p^2/E^2} \quad\Longrightarrow\quad m_0^2 c^4 = E^2 - c^2p^2. </math> Use
- <math>
E^2 = \hbar^2 \omega^2\quad\mathrm{and}\quad p^2 = \hbar^2 k^2 = \frac{\hbar^2 \omega^2}{c^2} </math> and it follows that
- <math>
m_0^2 c^4 = E^2 - c^2p^2 = \hbar^2 \omega^2 - c^2 \frac{\hbar^2 \omega^2}{c^2} = 0, </math> so that m0 = 0.
Photon spin
The photon can be assigned a triplet spin with spin quantum number S = 1. This is similar to, say, the nuclear spin of the 14N isotope, but with the important difference that the state with MS = 0 is zero, only the states with MS = ±1 are non-zero.
We define spin operators:
- <math>
S_z \equiv -i\hbar\Big( \mathbf{e}_{x} \mathbf{e}_{y} - \mathbf{e}_{y} \mathbf{e}_{x}\Big) \quad\hbox{and cyclically}\quad x\rightarrow y \rightarrow z \rightarrow x. </math> The products between the unit vectors on the right-hand side are dyadic products. The unit vectors are perpendicular to the propagation direction k (the direction of the z axis, which is the spin quantization axis).
The spin operators satisfy the usual angular momentum commutation relations
- <math>
[S_x, \, S_y] = i \hbar S_z \quad\hbox{and cyclically}\quad x\rightarrow y \rightarrow z \rightarrow x. </math> Indeed,
- <math>
[S_x, \, S_y] = -\hbar^2 \Big( \mathbf{e}_{y} \mathbf{e}_{z} - \mathbf{e}_{z} \mathbf{e}_{y}\Big) \cdot \Big( \mathbf{e}_{z} \mathbf{e}_{x} - \mathbf{e}_{x} \mathbf{e}_{z}\Big) + \hbar^2 \Big( \mathbf{e}_{z} \mathbf{e}_{x} - \mathbf{e}_{x} \mathbf{e}_{z}\Big) \cdot \Big( \mathbf{e}_{y} \mathbf{e}_{z} - \mathbf{e}_{z} \mathbf{e}_{y}\Big) = i\hbar \Big[ -i\hbar \big(\mathbf{e}_{x} \mathbf{e}_{y} - \mathbf{e}_{y} \mathbf{e}_{x}\big)\Big] =i\hbar S_z. </math> Define states
- <math>
| \mathbf{k}, \mu \rangle \equiv {a^\dagger}^{(\mu)}(\mathbf{k}) \,|0\rangle \leftrightarrow \mathbf{e}^{(\mu)} e^{i\mathbf{k}\cdot \mathbf{r}}. </math> By inspection it follows that
- <math>
-i\hbar\Big( \mathbf{e}_{x} \mathbf{e}_{y} - \mathbf{e}_{y} \mathbf{e}_{x}\Big)\cdot \mathbf{e^{(\mu)}} = \mu \mathbf{e}^{(\mu)}, \quad \mu=1,-1, </math> and correspondingly we see that μ labels the photon spin,
- <math>
S_z | \mathbf{k}, \mu \rangle = \mu | \mathbf{k}, \mu \rangle,\quad \mu=1,-1. </math> Because the vector potential A is a transverse field, the photon has no forward (μ = 0) spin component.
- ↑ P is the integral of <math>\scriptstyle -\mathcal{E}_\textrm{Joule}</math> = −E⋅J over the volume of the cylinder; <math>\scriptstyle (2\pi R z c)\,\mathcal{E}_\mathrm{Field}</math> is the integral of the Poynting vector S over the surface of the cylinder.
- ↑ The name derives from the second quantization of quantum mechanical wave functions. Such a wave function is a scalar field: the "Schrödinger field" and can be quantized in the very same way as EM fields. Since a wave function is derived from a "first" quantized Hamiltonian, the quantization of the Schrödinger field is the second time quantization is performed, hence the name.


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Gnarly Adventurer
Joined: Jun 2010
Location: Deep-ish South E Point GA
Oddometer: 261
Day 5 big bend to creel
Odometer: 20409
Nature called around 330 AM, so I got up and went off to answer the call... and as always when you get away from the city, was enthralled by the stars above... so many, so brilliant... I stared and pondered for a moment, looking at them all, wondering as men have for ages, if we are all alone... my mind contemplated if maybe, just maybe, we aren't alone, but maybe we are nothing more interesting than an Ant Farm of and to the Gods???? ;) Look how fun we would be to watch: wars, civilizations grown and burned, now flight and automobiles, space travel! Dudes and dudettes riding little two wheeled combustion devices and falling over a lot in canyons? I'd subscribe! :)
Arose at 7am, made hot tea and had some fruit and Cliff bar. Packed up camp and headed out... My morning blast off song? East down and bound loaded up and truckin' we're gonna do what they say can't be done... we've got a long way to go and short time to get there... watch ol' Bandit run............................... You're welcome!
The road out was the Old Maverick road. 14 miles of gravel/dirt/curves/washes... vast openness. Brilliant.
The sun was just coming up and it was getting warmer, so riding was enjoyable to say the least... I had gone half way, and seen no one. No sign of anything man made (road excepted)...
I got an overwhelming sense of just having the whole world to myself. The light on the hills, the scrub, the trees, the wind, the sun, the air. All mine. Even moving along on the bike, I felt a very comforting sense of peace and belonging, as I moved through the landscape... that feeling doesn't come often. Old Maverick Road was the best ride of my life..... so far.
"Can't stop stupid, can't fix crazy."
Gnarly Adventurer
Joined: Jun 2010
Location: Deep-ish South E Point GA
Oddometer: 261
Big Bend to Creel Part 2
DAY 5 BIG BEND TO CREEL... Part 2
Out of big bend, you head in a NW direction along the river. Plenty of twisties and scrub, high dessert areas. Very pretty, and not much in the way of development. I was zipping along, feeling the cold. I had had an issue with my heated grips, and now the throttle side wasn't working. I arrived at a look out over the Pecos River and decided to correct that issue. A bit of finagling, along with some electrical tape and I had warm hands again! The heated grips were a pain to install, but have definitely paid for themselves... $20 on sale at Cyclegear.
Apparently the bridge was built to help move materials for the World War II war effort. It replaced trails, tunnels and a ferry that you had previously needed to get across. The roads are still there, and you can take a boat trip along the river when it isn't the Winter season if you want... and I can tell you you want!
As usual, amazing views at the river and along the plains.
Did you find the train?
I stopped in a small store for some lunch, and from everyone's garb, you could tell hunting season was in full swing... these girls were totally cute, though I don't know about handing them a gun... the Grandfather was bilingual, but English was toughish, you could tell... Son was good at both, Girls defaulted to English but switched fluidly between the two. It was fun listening to them.
I finally get to Presidio and go to the local Presidio bank to change dollars to pesos... sorry sir, we don't do that here.... WTF? Fine, to the border we go.
I don't take pictures as I dont feel like starting out on the wrong foot going in... I got lucky going across... Immigration was professional and helpful, the teller at the bank (pay import fee, pay copy fee, pay bank fee, agree to pay more for not exporting bike, etc) was fast and efficient. I was through and gone in 30 minutes!
Finally MEXICO!!!!
I found a local cambio and changed some cash to pesos... should, as usual, have changed more, then headed towards Chihuahua city. Aaaaayyy, Chihuahua!
Rolling towards Ciudad Chihuahua (it is also the name of the state I am in), you hit a series of flat plains, followed by ridge climbs up the mountains... I crossed a whole series of these, easily 7-8.
I round the curve on one, and woops, there is a Aduana/Customs check point 50 km from the border... sneaky.
Young customs guy comes up and immediately asks about the motorcycle... what does it cost, what did you do to it, where are you going, I have a Hayabusa and a Rebel 200, etc and finally... ok, let me look in there. (points to hard case) Fine, Amigo, no problema! Has a quick look, then back to motorcycles bonding time... I escape only when another truck comes, and I almost thought he was going to wave them through... ok, you can go. Gracias 'cause I gotta make some time! :)
Alas, even though I am making good time, it is not enough and night closes in... the wind starts gusting... I am going West, the are going North and things get cold and scary...
I stop at a rest stop and a random dude in Mexgas truck asks Where are you going amigo? I say Creel and he stops and says, Amigo, that is dangerous.
Me: Er, why, because of animals on the road?
Him: No amigo, because of the mafioso. The boys will be out tonight...
Enough said. I was cold and tired and knew I shouldn't be on the road anyway...
Half an hour later it is Bienvenidos a Hotel del Parque, in Cuauhtemoc!http://www.hoteldelparque.org/
Creel, however lies 2 hours down the road... I'm not gonna meet up with Boulderguy and sjeproductions....
"Can't stop stupid, can't fix crazy."
Gnarly Adventurer
Joined: Jun 2010
Location: Deep-ish South E Point GA
Oddometer: 261
Day 6 Cuauhtemoc to Creel...
Day 6 Cuauhtemoc to Creel...
I'm jumpy, so I wake up at 3 am in the nite... surf the web, write some stuff... plot, plan and basically turn in circles... the usual when stuff is bugging you... I nod off to sleep again eventually... and awake at 530 and start packing... I had to wait till 7am to get my deposit back... and it turns out I was in mountain time, so the wait was longer than I expected as my watch and computer were in Central time... woops.
I had the receptionist call the hotel in Creel where Boulderguy and sjeproductions were staying.... Ok. I know some Spanish, but fluent, not even close. Apparently you need verbs for that, who knew. I could tell something wasn't right, but not enough to know exactly what... she told him the basics then just said Si too many times... no me gusta.
I got on the road knowing I had done all I could.
Here is what the road looked like from my perspective... not good.
I rolled towards Creel, and things improved... kinda.
Snow? Really? Really. I knew it was cold but still....
Motored through some small villages... kinda twistie. Went through one where the early morning smoke layer reminded me of pics of LA back in the smogy 70's. Everything smelled nicely woody. I rode on.
I get to Creel eventually. I am at the hotel discussed (I think) when the manager finds me.
Conversation, with loose interpretation a given:
Him: You need a room?
Me: No. Looking for some amigoes on motorcycles, you seen any?
Him: I will show you.
Me: Eh? Ok, good.
Him: Follow me.
Me: Bueno.
We go into the hotel, he opens a door to a room.
Him: Here is your room.
Me: Aaaaaah, looking for my buddies... one on a motorcycle, nero (black), the other on one, naranja (orange).
Him: Ok, here is your room. How many nights? They stay too?
Me: Gotta go.....
OK, there were communication issues. However he was helpful and got me to their OTHER hotel... where I stumbled onto this:
Followed closely by this:
Got a lot of electrical widgets you like to run? Better have a battery monitor! At least we know what to get him for Christmas.
That was how I met The Professionals.
Boulderguy chatted with me and explained the route and what to expect... very good overview...
We blasted off after I warned them of my skill level... or lack there of. :) Hey, somebody has to provide the entertainment!
Joke, but no joke. Boulderguy and SJEproductions know THEIR stuff. Been to Copper Canyon, done that. Doin' something different this time. IF they offer up another trip, jump at it. They know their sh*t. Respect the people, respect the land... chill. It will be majestic... really. Plus, you will laugh. A lot.
We got moving and amazing beauty and views were at every turn.
We arrived at the train stop at Copper Canyon... amazing overview.
Bridge built of railroad ties... glad it is not wet.
Very large roadside Memorial for a loved one...
Very typical trail/road for the area, smoother than most, actually.
and million dollar views everywhere else!
"Can't stop stupid, can't fix crazy."
Gnarly Adventurer
Joined: Dec 2010
Location: Atlanta, GA
Oddometer: 442
Awesome RR Gentri, maps of Copper Canyon and surrounding areas please. So stoked for you, can't wait to hear about it when you get back.
I don't need loud pipes, I've got a Thumper
Gnarly Adventurer
Joined: Jan 2008
Location: Boulder, CO
Oddometer: 271
Lucky for you....
I managed to capture a few ummmm, let's just call them memorable moments from our ride together which should really help round out this portion of your trip report quite nicely!!!
Let me be the first to say, you road the sh!t out of that DR650!!! I look forward to heading out on future adventures with you mi amigo!! Look for pics tonight.....
Beastly Adventurer
Joined: Jun 2008
Location: Louisville, KY...really too far from the hills!
Oddometer: 2,006
DUDE! You are living a DREAM!
Joined: May 2008
Location: Atlanta
Oddometer: 281
What an amazing trip Gentri. Once you get settled, we need to ride together.
I don't know! I don't know why I did it, I don't know why I enjoyed it, and I don't know why I'll do it again!
Bart Simpson
Gnarly Adventurer
Joined: Jan 2008
Location: Boulder, CO
Oddometer: 271
it was our density...
A little back to the future reference for those paying attention...but when I think of all the emails back-and-forth with Gentri before this trip started and his ill-fated first cold start, I didn't know if meeting up would really happen. BoulderGuy and I left CO as planned and headed for the border. After two solid days of cold, cold, cold weather riding (I'll save that for another ride report) we hit Creel on schedule and after a long night of sleep, I awoke to find an email from Gentri telling me he was only about two hours behind us. Hot damn, he was on his way!!
He asked that we leave the name of our next hotel at the front desk of our current hotel and would try to either meet us in Creel or catch us that evening. I honestly didn't think it was going to happen, but BoulderGuy and I decided we'd leave the note and see what happens. As we loaded up the bikes, I heard a voice in the courtyard that was distinctly not...ummm, how should I say this...local. I stepped out of my room into the courtyard and got my first glimpse of Gentri!! Holy crap, what time did you get started?? 7:15am!! God damn, I liked him already!!
So, as Gentri noted, BoulderGuy had a bit of technical difficulty in the AM. So with a bit of pushing and a hotel on a downhill run, we got the 950 fired up and ready to go. I asked Gentri quite clearly "are you sure you want to saddle up to this trainwreck?" He never hesitated.
So, the good news about the technical delay was the temps had been slowly creeping up and BoulderGuy showed his thanks...
So, after changing layers, we set out from Creel with Urique as our destination. We'd see some miles of sweet twisty tarmac before pavement turned to dirt, which would be our home for the next three days....Gentri wasted no time heading out to the front.
After warming up for a bit, we stopped for some scenic picture taking, route discussion and some proper dirt preparation on the DR...
BoulderGuy offering some thoughts on what exactly, I'm not sure.
Properly prepared, we hit dirt and Gentri (as I noted above) stuck to our wheels like glue...
Snacks always help...
With some food in the system, we set out for the second half of day one, but I'll save those details for later and let Gentri take it from here!!!
Gnarly Adventurer
Joined: Jun 2010
Location: Deep-ish South E Point GA
Oddometer: 261
The plot thickens....
sjeproductions is just waiting to pounce!
Should get up 2nd installment tonight.... awesome supplement!
"Can't stop stupid, can't fix crazy."
ADV Demo Rider
Joined: Jun 2009
Location: Woodstock, GA
Oddometer: 1,486
I'm sorry
I was blinded by the CHROME
Ride on and stay warm man----- Skype worked great while down in Mexico. Highly recommend to everyone traveling to at least have an account.
Yes, I'd love to try
What ever happens to be in the stable at the current time.
H-ADV in fabrication
out ridin'....
Joined: Jan 2008
Location: Boulder, Co or lost in Mexico....again
Oddometer: 433
Someone is up early and already posting at 4:10 AM. More trip report please!
Hegelian Scum
Joined: Jul 2005
Location: Asheville NC
Oddometer: 4,954
I like it.
It's great to see a guy so honest about his year and then so open about his ride.
So there I was walking into a biker bar in Chattanooga with liberty spikes on my head and a drag queen beside me... - bcurvin
Joined: May 2008
Location: Atlanta
Oddometer: 281
Boulderguy and SJEproductions know THEIR stuff. Been to Copper Canyon, done that. Doin' something different this time. IF they offer up another trip, jump at it. They know their sh*t. Respect the people, respect the land... chill. It will be majestic... really. Plus, you will laugh. A lot.
I like the quote = Respect the people, respect the land...chill it will be majestic.
I don't know! I don't know why I did it, I don't know why I enjoyed it, and I don't know why I'll do it again!
Bart Simpson
Gnarly Adventurer
Joined: Jan 2008
Location: Boulder, CO
Oddometer: 271
Copper Canyon sort of speaks for itself....
Gentri and me as taken by BoulderGuy....
Beastly Adventurer
Joined: Jun 2008
Location: Louisville, KY...really too far from the hills!
Oddometer: 2,006
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Medvedev: I would recommend that Mr. Gorbachev take a closer look at Putin's words. He merely said that if Vladimir Putin and Dmitry Medvedev are still attractive to the people as political figures at the time of the next presidential election, then we will sit down and decide which of us will stand for election -- so that we don't impede each other. He did not maintain that we would decide among ourselves who would be the next president. That would of course be ridiculous.
SPIEGEL: In the meantime, there are enough political challenges. On Dec. 5 the START treaty on arms control will expire. US President Barack Obama is already dreaming of a world without nuclear weapons. Did he speak with you about how he wants to achieve this objective?
Medvedev: Who else should do it, if not us? The greatest nuclear potential is currently in the hands of Russia and the US. If we don't address this, there will be no disarmament. We have recently moved at quite a brisk pace, also because the new administration in Washington has made this issue a top priority -- in contrast to its predecessor, which appeared to be totally uninterested in strategic disarmament. Now we have every opportunity to agree on lower thresholds and define monitoring measures. At the end of the year, we could sign a legally binding document.
SPIEGEL: The ground-breaking role of Moscow and Washington won't be enough.
Medvedev: Correct, other nuclear powers show no signs of such an attitude. Even our close European partners don't share this attitude -- you can imagine who I mean.
SPIEGEL: This could only be France and the UK.
Medvedev: Emerging nuclear powers show even less understanding, particularly those that illegally attempt to acquire nuclear technologies. And then there are countries that won't admit that they have nuclear weapons, but won't deny it, either. We have to think of ways that we can convince everyone to renounce nuclear weapons.
SPIEGEL: You are familiar with the West's fears of a nuclear-armed Iran. How will Russia act in this case? To what extent do you intend to accommodate Tehran with arms deliveries and the pursuit of its nuclear energy program?
Medvedev: Iran has the right to the peaceful use of nuclear energy, under the oversight of the International Atomic Energy Agency -- there is no objection to that. The country must only respect the applicable regulations; it cannot attempt to conceal any facilities. The discovery of the new plant at Qom is alarming. It is also surprising that this information has only now been made public. Should the negotiations on enriching Iran's uranium for peaceful purposes prove successful, we would be happy to take part in this program.
SPIEGEL: And if not?
Medvedev: Theoretically, all options would still be on the table. I have spoken with Obama in New York about this. I don't want it all to end with sanctions. But if things don't move forward, such a scenario cannot be ruled out.
SPIEGEL: What about the Russian arms deliveries?
Medvedev: We will only deliver arms that serve defensive purposes, no offensive weapons.
SPIEGEL: Do you see a danger that the West in Afghanistan could suffer the same fate as the Soviet Union, which, after nine years of war and 15,000 military casualties, withdrew its troops from the Hindu Kush in 1989?


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1882 T$1 PR66 Cameo PCGS. CAC....
2013 April 24 - 28 CSNS US Coin Signature Auction - Chicago #1184
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Renaissance Schaumburg Convention Center Hotel
1551 North Thoreau Drive
Schaumburg, IL 60173
|Printable auction results for all items in the 2013 April 24 - 28 CSNS US Coin Signature Auction - Chicago.
1882 Trade Dollar, PR66 Cameo1882 T$1 PR66 Cameo PCGS. CAC. The 1882 Trade dollar was a proof-only issue, as the series was rapidly winding down, and would see its last official mintage in 1883. The 1882 mintage was limited to 1,097 pieces, and few survivors can match the quality or visual appeal of this spectacular Premium Gem. The design elements are sharply detailed throughout, and the frosty devices contrast boldly with the deeply mirrored fields to produce a dramatic cameo effect. Both sides are blanketed in a patina of vivid cobalt-blue and champagne-gold hues, with exceptional eye appeal. Only the most insignificant signs of contact are evident on either side. Population: 7 in 66 Cameo, 3 finer (3/13).(Registry values: N4719) (PCGS# 87062)
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Book your room at Residence Inn by Marriott Silicon Valley Sunnyvale II
Studio, 1 Queen Bed with SofabedMax Occupancy: 3 guests (up to 2 children, 2 infants)
Executive Penthouse, 2 Queen BedsMax Occupancy: 4 guests (up to 3 children, 3 infants)
Get an overview of this hotelFamily-friendly hotel with free breakfast and outdoor pool
This family-friendly Sunnyvale hotel is located near the airport, within 3 mi (5 km) of Our Lady of Peace Church and Shrine and California's Great America. Golfland USA and Lace Museum and Guild are also within 3 mi (5 km).
Hotel Features
This smoke-free hotel features an outdoor pool, a 24-hour fitness center, and a 24-hour business center. Free buffet breakfast is provided, as well as free WiFi in public areas, free self parking, and a free manager's reception. Other amenities include coffee/tea in the lobby, dry cleaning, and laundry facilities.
Room Amenities
All 247 rooms feature kitchens with refrigerators, stovetops, microwaves, and dishwashers. Other amenities available to guests include free WiFi, free wired Internet, and sofa beds.Information missing or incorrect? Tell us!
Hotel Amenities
Hotel Amenities
Residence Inn by Marriott Silicon Valley Sunnyvale II features an outdoor pool and a 24-hour fitness center. High-speed wireless Internet access is complimentary. A 24-hour business center is on site at this 3-star property. A complimentary breakfast and manager's reception are offered (may not be available daily). This family-friendly hotel also offers barbecue grills, a picnic area, and coffee/tea in the lobby. Complimentary self parking is available on site.
Residence Inn by Marriott Silicon Valley Sunnyvale II is a smoke-free property.
- Free breakfast
- Free WiFi
- 24-hour fitness facilities
- Year Built 1985
- 24-hour business center
- Free reception
- Free newspapers in lobby
- Express check-in
- 24-hour front desk
- Express check-out
- Dry cleaning/laundry service
- Fireplace in lobby
- Barbecue grill(s)
- Coffee/tea in lobby
- Number of floors - 2
- Laundry facilities
- Total number of rooms - 247
- Picnic area
- Safe-deposit box at front desk
- Free self parking
- Smoke-free property
- Outdoor pool
Family Friendly Amenities
- Family Hotel
- Outdoor pool
- Free Wi-Fi
- Kitchen
- Laundry facilities
- Sofa bed
- Refrigerator
- Microwave
- Cribs/infant beds (complimentary)
Available in all rooms: Free WiFi , Free wired high-speed Internet
Available in some public areas: Free WiFi
Free self parking
Room Amenities
- Dishwasher
- Stovetop
- Free cribs/infant beds
- Cookware, dishware, and utensils
- Free wired high-speed Internet
- Desk
- Premium TV channels
- Pay movies
- Video-game console
- Separate sitting area
- Access via exterior corridors
- Climate control
- Air conditioning
- Phone
- Refrigerator
- Coffee/tea maker
- Kitchen
- Microwave
- Free toiletries
- Hair dryer
- Shower/tub combination
- Iron/ironing board
- Daily housekeeping
- Cable TV service
- Sofa bed
- Free WiFi
Where to Eat
Guests are offered a complimentary buffet breakfast. A complimentary manager's reception is offered on select days.
Nearby Things to Do
Recreational amenities at the hotel include an outdoor pool and a 24-hour fitness center.
The recreational activities listed below are available either on site or nearby; fees may apply.
- Golfing nearby
- Basketball on site
- Volleyball on site
To make requests for specific accessibility needs, complete the "Special or Accessibility Requests" section of "Trip Preferences" when you book your hotel.
- Accessible path of travel
- Accessible bathroom
- Roll-in shower
- Accessible parking
- In-room accessibility
- Accessibility equipment for the deaf
- Braille or raised signage
Hotel Policies
Check-in time starts at 3 PM
Check-out time is noon
Payment types
Children and extra beds
- Children are welcome.
- Free cribs (infant beds)!
Pets allowed, Pet maximum weight in lb is 100
You need to know
Extra-person charges may apply and vary depending on hotel policy.
Government-issued photo identification and a credit card or cash deposit are required at check-in for incidental charges.
Special requests are subject to availability upon check-in and may incur additional charges. Special requests cannot be guaranteed.
- Pet fee: USD 100 per pet, per stay


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Naidu reaction to Pawan speech July 08, 2015 09:07
Andhra Pradesh Chief Minister Chandrababu Naidu warned the Seemandhra MPs not to move against Pawan Kalyan as he is one of the reasons for the party to win in the 2014 elections.On Monday, reacting to the current situations in the dual Telugu states, Pawan Kalyan spoke to the media. One among the various topics he spoke was about Andhra Pradesh special status. Speaking on the issue, Pawan said that, "Seemandhra MPs are concentrating more on their personal businesses and omitting to fight for the special status. Specially the MPs like Kesineni Nani were very active during the time of getting the tickets and are now completely silent on politics."When the MPs took the issue to the CM, Naidu said that, "Pawan is our friend, who has supported our party during the elections and is one of the reasons for the party's victory in the elections. So instead of reacting against him, explain the struggle Seemandhra MPs experienced and not more."TDP MPs against Naidu? It seems, TDP MP Nani was personally hurt, as Pawan took out his name. Regarding this, he joined hands with Bandar MP Konakalla Narayana and arranged a called for a press meet yesterday. Knowing this, the TDP office head asked to cancel the press meet, but as the meeting was already scheduled, the MPs went forward, with the promise of not to react against Pawan, but finally busted on him in the press meet.This issue was taken seriously by the TDP party officials and it is heard that one of the senior leaders warned Nani about his comments on Pawan.However, it seems, Pawan could finally managed to balance his comments and compliments on both state parties, so that no leader is completely able to come out and speak against Pawan.By PhaniRead More
Recipe: Healthy Broccoli Tikki July 07, 2015 18:53
Broccoli is one of the healthy leafy vegetable. It is loaded with dietary fiber, minerals, vitamins, anti-oxidants and phyto-chemicals. If you are interested to take healthy snack during tea time, try this Broccoli Tikki at your home and enjoy its taste. Ingredients:Broccoli - 200 gms, finely choppedPotatoes - 125 gms Plain Flour (Maida) - 100 gmsCorn Flour - 1/4 cupButter - 2 to 3 tblspGinger Garlic Paste - 2 tspOil for fryingRed Chilli Flakes - 2 tsp (adjust as required)Salt as per tasteBlack Pepper Powder as per tasteMethod of preparation: Boil, peel and mash the potatoes well.Heat butter in a pan over medium flame.Saute the broccolis for a minute or two.Combine the mashed potatoes, corn flour, maida, broccolis, ginger garlic paste, red chilli flakes, salt and black pepper powder.Mix well and shape the mixture into small or medium tikkis.Heat oil in a deep frying pan.Fry the prepared tikkis until golden brown.Remove and drain excess oil.Serve hot with chutney or sauce of choice.By LizithaRead More
RGV responds to Pawan's speech July 07, 2015 18:28
From the film industry, it was Ram Gopal Varma, who first called Pawan to react on the cash for vote scam and finally, when Pawan reacted to the issue yesterday, RGV has tweeted on the issue.Immediately after the Pawan's speech, the common people waited for the RGV's response one the issue, but to the shock, though RGV tweeted, saying that he has just seen Pawan's video and ended his tweet there itself. With this, there are various kinds of buzzes that are going on among the people saying that, RGV might be speechless after watching the video, he might have thoroughly inspired by Pawan's speech and his tweet might mean that he is symbolically saying that he could not put it into the words. On the other side, the critics also opine that, RGV got confused after watching the speech and might be depressed with the way Pawan expressed his views. However, a clarity on the issue would be found, when RGV re-tweets on the issue.Kona Venkat responds on Pawan speechOn the other side, Kona Venkat, the writer cum producer said that, his respect on Pawan has increased ten times more after knowing the way, he reacted to the hot issues of the state. He even said that, "a real leader would come out from the common people and Pawan is one such leader."By PhaniRead More
Girl thrashes an eve teaser in police station July 07, 2015 18:16
Girls, especially in India need to learn Karate, King Fu, kalaripayattu etc; instead of 'Yoga'. Know why? Then read.He is a roadside Romeos who roams around and harass the girls in the area. His notoriousness carried out well until this girl decided to stand against him. Sanjana, a Class 11 student from Uttar Pradesh's Pilibhit district, notices that a boy was following her. Without any delay, she walked towards the police station, dragged him inside and thrashed him with her sandals right in front of officers, until he apologized and fell at her feet. At one point, she twists his arm.The video clip of the same is doing the rounds in social media. A lesson he is unlikely to forget soon. - ManoharRead More
No Aadhar - No Marriage July 07, 2015 17:42
The government is considering a plan of linking the Aadhar card to the matrimonial accounts.Be it women or men, more than half of the accounts in the matrimonial sites are identified as fake. Due to this, the people who are completely dependent on these sites are going forward basing on the details specified and finally realizing that, they are cheated. Especially, the women are being cheated in these cases, and there are instances, where they even committed suicide. So to avoid these kinds of situations in the future, central minister Menaka Gandhi came out with an innovative proposal. She said that, when anyone decides to give an advertisement on any kind of matrimonial platforms, they should mention their Aadhar number too, which is to be noted by the concerned matrimonial sites and only then the advertisement will be valid. Speaking on the issue, the minister said that, when the above procedure is followed, it would be easier to find out the people with the fake accounts by their Aadhaar card number. The proposal was unofficially granted by the government and regarding this, the department of Electronics and Information Technology is conducting a meeting with the marriage bureau managements on Wednesday. The decision on the issue would be taken by considering the ideas given by the marriage bureau managements in the meeting.Hope the new proposal comes into existence and save the women, who are completely being cheated by the fake accounts on the sites.By PhaniRead More
Try to solve this Hong Kong's first grader puzzle July 07, 2015 17:22
Do you love solving the puzzles? Are you interested in logical and reasoning? Then this is definitely for you. Get a pen and paper out there. Here is the head scratcher for you:From the above picture, try to figure out the number of spot, in which the car was parked.The puzzle was given to the first graders in Hong Kong and the time frame given is 20 seconds. They solved it also. Provided that you are older than them, it shouldn't take more than that, if you are intellectual enough.So get cracking on it and share your answers in the comments section below. - ManoharRead More
Best home made scrubs to try during monsoon July 07, 2015 17:14
Monsoon season brings pleasurable experiences to all of us. But taking care of your skin is the most important thing that you must consider. As exfoliation is the common problem during this season, here are some home made scrubs that will help you in eliminating the dead cells your skin. Oat meal scrub: Take crushed oatmeal and mix it with a small amount of milk. Make it into a paste and use it to exfoliate your skin. Apply this paste on your skin and scrub it with dry finger tips for two to three minutes and leave it on for 10 minutes. Then rinse it off.Sandalwood scrub: Make a paste by mixing sandalwood powder in orange peel and Fuller's Earth (multani mitti). The paste will help in exfoliating dead skin cells, and bring a glow to the skin.Baking soda scrub: Mix two table spoons of baking soda with one spoon of cinnamon powder, half a lemon juice and five spoons of honey. Apply the paste and rinse off after five minutes. Natural Grams scrub: Mix equal amounts of green gram powder and Bengal gram powder and half the quantity of fenugreek seeds. Prepare a paste by mixing all these in rosewater and apply the paste generously on the entire body. The method deeply cleanses and refreshes the body.Multani scrub: Mix together Fuller's Earth, gram flour and sandalwood powder in equal quantities. Store in an air tight container. Mix one tablespoon of the powder in water to make a paste. Apply evenly on the face and leave it on to dry then rinse with cold water.BY LizithaRead More
Hyderabad theatres gamble with Baahubali tickets July 07, 2015 16:46
As the prestigious Indian epic Baahubali is getting ready to release on July 10th, the entire cinema lovers across the world are very eager to watch the visual wonder on the first day itself. But the ticket rates are making them to think upon.As per the reports, the Baahubali is scheduled to release in 4000 screens across the world on the first day itself. Regarding this, the benefit show ticket rates are going high day by day, as the movie is nearing its release. Also read: Baahubali release date trailer talkAlong with the benefit show ticket rates, the normal shows ticket rates are also showing the sky. Though the advance booking is supposed to take place before the release and sell the tickets, the theatres in Hyderabad are already showing houseful boards for the entire first week. The reason behind this is, as Baahubali is the hot cake now, the audience is ready to spend any amount for the tickets and this effect would be more on the first day of release. Keeping that situation in mind, the theaters might be holding on the tickets to sell them on the release day. By PhaniRead More
New app to detect eye diseases early July 07, 2015 16:22
A team from the Medical and Surgical Centre for Retina in Mexico have developed an application that can detect eye diseases like diabetic macular edema. It occurs when fluid and protein deposits collect on or under the macula of the eye and causes it to thicken and swell. The swelling may distort a person's central vision."The idea is to detect and prevent diseases in general practice. We are not replacing the specialist, we want to know which patients have a disease and make an early detection," said Dr. Juan Carlos Altamirano Vallejo, medical director of the Medical and Surgical Centre for Retina in Mexico. He said that the technology is designed for general physicians, "who support the health system in Mexico and, even without in-depth knowledge of ophthalmology, can, with this tool, detect certain abnormalities and send the patient to the specialist." This new app helps in prevention of eye diseases and will also cut down on costs. It is developed in collaboration with biomedical engineers from the Spain-based the Monterrey Institute of Technology and Higher Education (ITESM) has been satisfactory and is expected to soon be marketed and incorporated the basic health system.By LizithaRead More
First look talk: Varun Tej 'Kanche' July 07, 2015 16:07
Mega hero Varun Tej has released his new look in his upcoming movie 'Kanche'. As per the reports, the shooting of the film is completed. Director Krish maintained utter secrecy about the project. In fact, except the title, nothing about the movie was revealed into the media. Finally, Varun himself posted the picture on the social networking site. Director Krish is identified for his thought provoking films like Gamyam, Vedam etc. After watching the Kanche poster, it can be understood that he is still not out of his identity. Also read: It is Puri for Varun's next!Varun is spotted with a mission gun in the fenced area. If a wild guess is to be made, he might be the soldier protecting the country at the border. If that is true, Krish has already proven that, he is the master in dealing such subjects and so Varun is guaranteed a super hit this time. It seems the mega hero is concentrating on different films, rather than on mass films, where the mega heroes are generally known for. So whether the audience accepts the message oriented, lighter mode films from a mega hero or not is yet to be known. By PhaniRead More


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Free Callen Seminar - Kitchen & Bathroom Planning 101: How To Save Money Planning Your New Designer Space
Kitchens and Baths by Callen, specializing in kitchen, bathroom, lower level, attics, additions, and other interior living space renovations, has scheduled a free seminar titled "Kitchen & Bathroom Planning 101: How to Save Money Planning Your New Designer Space" at the company showroom, S63 W13131 Janesville Rd. in Muskego.
The seminar will be presented on Wed., Oct. 17 at 6:00 p.m. and Sat., Oct. 20 at 10:00 a.m. Callen design professionals will discuss the solutions to creating a more functional and usable space, especially those found in kitchens and baths. Information will include identifying the functional needs and materials to renovate an outdated or uniquely shaped kitchen or bathroom to create a more useful and comfortable environment.
Refreshments will be served. Space is limited and reservations are required. E-mail firstname.lastname@example.com or call 414-529-5509 to reserve your space.
Callen is a family-owned and award-winning home improvement and remodeling contractor operating since 1986. Callen's Muskego showroom contains a variety of samples, including full kitchen and bath displays featuring Kohler and Starmark Cabinetry products, countertops, and Infinity® Fiberglass replacement windows from Marvin.
Callen is a member of several home improvement and area associations, including the Milwaukee/NARI Home Improvement Council, National Kitchen & Bath Association, the Metropolitan Builders Association Remodelers Council, Better Business Bureau, Angie's List, the Muskego Area Chamber of Commerce, the Brookfield Chamber of Commerce, and the Hales Corners Chamber of Commerce.
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You can now invite anyone to meet at your favorite Starbucks.
Limited-time feature
30 year old man | Somerville, MA, USA | Seeking women 23-29 within 15 miles
Seeking hazardous adventures.
Vitals: I'm an easy-going guy who loves being active, the outdoors, and traveling, and I aim to make those things a focus rather than something I do in my spare time. I'm looking for someone amazing to have adventures of all shapes and sizes with. I'm driven by curiosity and a desire to discover new things and develop new skills, and to experience as much as possible first hand. I'm always interested in gaining a better understanding of how things work. I try to consistently find ways to improve myself. I love to create and do and interact, rather than simply consume and observe. I experiment with food and cooking; I enjoy learning to play and sing songs that I love on the guitar; I try to capture beautiful moments with my camera; when I read about an interesting place or event, I want to experience it myself. For me, a lot of the pleasure comes from being part of the process and gaining deeper understanding through it. My professional expertise is in web development. The code itself has a beauty and art to it, and developing elegant, creative tools provides me with immense satisfaction. Job-wise, I set out a few years ago with the goal of building a location-independent business that allows me the flexibility to travel pretty much wherever, whenever, and I'm very fortunate to be able to say that that's finally become a reality. I spent most of November traveling through Thailand and Cambodia, which was amazing. Peru, Patagonia, Eastern Europe, the American Southwest, and Barcelona are high on my list of places to visit in 2013 - but I'll go anywhere I have the opportunity. My family is awesome, both immediate and extended. They're close enough that I get to see them with some frequency, and far enough that it's never an intrusion. Family holidays tend to be highly entertaining affairs - in the best possible way. Stuff that makes me happy Making and designing things. Adventures. Reaching places you can't get to by road. Getting lost. Spontaneous road trips. Wandering. Discovering new places. Encountering smells that recall lost memories. Singing. Campfires. Snowy nights. Playing board games with friends. Winter cabins. Telling stories. Surprising friends. Red wine. Self sufficiency and personal responsibility. Simplicity. Planning grand schemes that inevitably disintegrate into something even better. Integrity. Authenticity. Teaching. Movie nights. Sharp wit. Passionate discourse. Finely wrought sentences. The smell of crisp fall days. Mountains. Walking barefoot. The sea. Stuff that doesn't make me happy Clubs or bars where it's so loud you can't have a conversation. Self-serving rationalizations. Bad grammar. People who don't think for themselves. Catching myself in any of the above. Who I'm looking for I'm hoping to find someone who shares some or most of my passions - traveling with some frequency (anything from backpacking to 5 star hotels), being outdoors (especially hiking), music (I play guitar and sing with a modestly competent amount of enthusiasm), food (cooking or eating, either works), photography (well, perhaps a healthy tolerance if not a personal passion), great storytelling (films, novels, television series), and a desire to explore in all senses of the word. I'm also looking for someone who will challenge me and introduce me to new things, both physically and intellectually; a best friend; someone highly intelligent, athletic, confident in themself and excited about life; someone who wants to do fantastic things together and be equal partners in a relationship. And I'm serious about finding someone special - I'm not looking for a short-term relationship without the possibility of long-term potential. I'm excited/nervous/hopeful to find someone amazing to share it all with!
Camping , Coffee and conversation , Cooking , Dining out , Movies/Videos , Music and concerts , Exploring new areas , Travel/Sightseeing
Basketball, Cycling, Other types of exercise, Skiing, Soccer, Tennis / Racquet sports, VolleyBall, Walking / Hiking, Weights / Machines
I exercise 5 or more times per week.
I like Cats, Dogs
Middle of the Road
Cape Cod; Franconia Notch, NH; Sydney, Australia; Koh Lipe, Thailand; Santorini, Greece (I was there when I was young and I'd love to go back)
Duke University, Durham, NC
Movies: Inception, The Usual Suspects, Love Actually, Good Will Hunting, Casablanca, North by Northwest, Amelie, Up, The Princess Bride Music: Foo Fighters, Third Eye Blind, Incubus, Mumford and Sons, KT Tunstall, Florence and The Machine
Traveling, hiking, photography, guitar, cooking, tennis, soccer, kayaking, biking, snowboarding,
Bossypants by Tina Fey (that was the hardest I've laughed in a while), The Fault in Our Stars by John Green, Ready Player One by Ernest Cline.
Z{{Cta}} {{#if HasMoreInfo}} ?{{CtaSecondary}} {{/if}}
|What he is looking for
|6'1" (185cms)
|5'4" (162cms) to 6'0" (182cms)
|Body type:
|Athletic and toned
|Slender, Athletic and toned
|No preference
|Dark blonde
|No preference
|No Way
|No Way
|Social Drinker
|No preference
|Self-Employed / Entrepreneur
|No preference
I develop and sell digital goods online that help people build websites. I love creating new products, and I'm fortunate that the business has really taken off.
|$100,001 to $150,000
|No preference
|Never Married
|Never Married
|Have kids:
|Wants kids:
|No preference
|White / Caucasian
|No preference
|I'll tell you later
|No preference
|Bachelors degree
|Bachelors degree, Graduate degree, PhD / Post Doctoral
Degree in Computer Science and English from Duke


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A "Big" thank you to the Homer Foundation's Youth Advisory Committee and the Sheldon Youth to Youth fund for their donation to Big Brothers Big Sisters and our matches. Through their generosity, our program was able to purchase several swim passes for Bigs and Littles. This grant not only supp...Thu, 06/05/2014 - 9:44am
My friend Mike Heimbuch offered his opinion recently that it may have been a mistake for the legislature to overturn the oil tax law known as ACES in favor of Senate Bill 21. Mike and I have some common ground in that we are both offspring of Alaska homesteaders and were enjoying Alaska long b...Thu, 05/29/2014 - 9:44am
I'm struck by the passages we go through in our lives. There are many passages we make by choice, from graduation and promotions, to enlistments, marriages and becoming parents. For me, however, the most important passages are the "natural passages," those we go through, but not by choice. We ...Thu, 05/29/2014 - 9:40am
Sprout Imagination Library has been thrilled to see our whole community getting involved in reading to kids. In collaboration with the Dolly Parton Imagination Library, we provide books monthly to all children ages birth to 5 years old throughout the Southern Kenai Peninsula. These engag...Thu, 05/29/2014 - 9:30am
A big shoutout goes to all the volunteers, staff members, parents and students at Voznesenka School for making another Marine Safety season a great success....Thu, 05/29/2014 - 9:30am
I just recently visited my mother at the Homer hospital. I have to say that the lunch that she was served looked so good that I went down to the cafeteria and bought some. It was delicious. I had eaten there a few years ago and the food was, well, hospital food. But they have improved exponentially....Thu, 05/29/2014 - 9:29am
Paul Banks Elementary was fortunate to have Jimmy Riordan, an Alaska artist and bookmaker in our school this spring. We learned techniques and skills for new ways of presenting information and ideas through paper projects and book binding. We also got to have fun with printmaking. We appreciate the ...Thu, 05/29/2014 - 9:28am
Every year, the City of Homer allocates general operating funds to the non-profit sector through an application process managed by the Homer Foundation. The board of directors and staff of Homer Council on the Arts would like to thank the city for its support of the non-profit sector and particularl...Thu, 05/29/2014 - 9:28am
To Kellie Blue and the Bay Club staff: Thank you for hosting our elementary folks for Learn-to-Swim. This opportunity could not have been possible without a generous contribution from migrant education; KPBSD Federal Programs Department. ...Thu, 05/29/2014 - 9:28am
I would like to offer a big thank you to the Drew Scalzi family and the Homer Foundation for honoring me by awarding me the Drew Scalzi Memorial Maritime Scholarship. This award will go a long way toward offsetting the cost of the information technology program at AVTEC which I will be attending thi...Thu, 05/29/2014 - 9:27am
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The lights are on
You can get three trophies easily, kick 50 dogs, blind 100 enemies and set 100 enemies on fire. The numbers you have to do may be different but those are the 3 trophies you can get all at one part. You will get to a part in GOW where you have to save posiediens daughter, and you have to kick dogs through a portal onto a platform to weight it down, well the dogs will keep spawning, so get all these trophies here, you only have a max of 2 dogs, so it is easy to just sit here and kick all the dogs you want, than blind them and than set them on fire witih apollo's bow. If you happen to kick the dogs onto the platform well go kill them and go back down to where the spawn, this was a easy way to get all these trophies.
Also the Labyrinth part, there is a trophy that you need to make it without dying or failing, and the labryinth is crazy hard, well I suggest to quit if you die and re-load last save, the last save should be at each checkpoint and if you do this, you will make it without dying. But trust me, you will most likely die more than a few times through the Labyrinth. One part is with spikes, make sure to grab a harpie right before the spikes all go up, the room rotates but if you grab the harpy to quick you let go onto the spikes, so make sure to time it right, and you will hold on until the room rotates. There is 2 times you will do this.


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Jucarii - 4
Tali Zorah DLC Mission
This is a nice parody of Mass Effect. See these heroes from many angles having sex.Vizualizări: 77201
Another Late Night at the Office 2
Orgies of office-life continues. In this episode Gerry tries to seduce Angela (the main heroine of this game with nice ass). Will she sleep with her boss? How everybody else will react after that? I'm sure You'll find out!Vizualizări: 117872
Another Late Night at the Office
Everyone has stayed at the office for a few extra hours. But how will it work when you're a hot secretary and every other colleague would love to fuck you so badly. Angela will be in this situation today.Vizualizări: 123155
Strip Poker with Lauren
Ready for another great video poker game? This time you play against sexy brunette named Lauren. Guess what?! She has a dildo in her pocket. So beat her and she'll show you how to use that pink dildo.Vizualizări: 98636
Rear Guard
This is funny advertisement for your personal sexual security. In this short sex movie you'll find all necessary information how to protect your asshole from different intruders.Vizualizări: 78921
Nursery Fertilization Experiment
Enjoy this big compilation of nice made illustrations about a girl who gets penetrated and fucked in any possible way. All this is a huge experiment - it means that there's a scientific explanation for this :)Vizualizări: 160526
Passion Fruit
Looks like our heroes of this small sex animation are at some market buying vegetables. She is on her knees and her ass is ready for penetration with those vegetables and some black meat bone :)Vizualizări: 72498
Strip Shifumi with Ellie
Another simple adult game where you have to play rock-paper-scissors. Just imagine - you and some random girl playing this game and the looser takes of some clothes :)Vizualizări: 118784
MrPinku: Dungeons And Morons
This is a funny adventure game where you'll have to fight against various creatures and girls to get some new items and progress the game. Just pick the right weapon combination for each girl and you'll get lucky. As the first path opens fully, heal yourself regularly.Vizualizări: 156515
The first thing you do when you become a girl 2: Lesbian
Story about the girl who lately became adult continues. This time she discovers how it is to make love with a girl. They are gentle to each other and later they use nice toys to please each other. Of course, some surprise later will come in their room.Vizualizări: 105686
Inspector J Episode 4
Inspector J series are back with new clues and now you have to find the link between the two first girls who are missing. And it seems that they both know Michelle. Fuck her and maybe you'll get some new information.Vizualizări: 194601
BDSM Dungeon Slave: the Beginning
Maybe you remember 2 our previous games with this heroine! Now we made a story how did this started. Her name is Veronica. She's going to USA from Ukraine to find some job and make money! But everything isn't so bright as she expected. This game has various endings (6). It depends on your answers during the game. Make sure you reach all endings! A lot of sex scenes are waiting for you!Vizualizări: 222087
King of Porn City: March 2014 update
Another great update for King of Porn City game. As always new upgrades brings you new poses, models, new stages, upgrades and many more! Earn money, buy new locations and spread your seed around the city!Vizualizări: 103169
WAIT UNTIL New Game button appears (all files need to be preloaded). This adult adventure game is for those who love hard adult games. Plan your actions and improve your skills. There is a good tutorial inside the game in English. Read everything to learn how to fuck all girls: Asuka, Haruhi, Horo, Lala, Mari, Menace, Miko, Mikuru, Mio, Mugi, Orihime, Princess, Ranka, Rena, Saber, Sakura, Sheryl, Yoko, Yoruichi.Vizualizări: 514905
Playful 3D Cat
Enjoy this 3D animation where some kind of cat plays with big flashlight dildo. Watch the movie and click some times to proceed further in animation.Vizualizări: 112028
Geo Quiz with Lucy and Amanda
Two busty blondes are here for you. In this adult game you have to show your knowledge in Geography once again. Pick the right capitals for specific countries to collect enough right answers and get the reward.Vizualizări: 85053
Sex Slave
An old guy from BDSM club? Seriously? Yes, this old perv is going to fuck some chained slut and cum all over her. Select the action from the bottom of the game.Vizualizări: 90428
KGB Training
In this game you can play with hot girl from the front and behind. First of all try to touch her, then remove all her clothes. After that you'll be able to use all available sex toys. Do not fill the red bar. Your task is to fill the green one.Vizualizări: 98093
Robotic Puzzle
Assemble the image with the help of industrial manipulator. Assemble 9 pieces of image in natural order, to see the picture. Click on the plate that you want to move, and then click where you want to put it. Click Next button when done. Last level will be video puzzle. After that you'll be able to create your puzzle from any picture on the internet.Vizualizări: 61540
Have you been a good girl? Asks Santa to large breasted girl Unohana. It's a Xmas game for all the lovers of Bleach series. Use different tools and techniques to strip her and fill pleasure bar. After that you'll be able to fuck her sweet ass.Vizualizări: 193812


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Gear Review
Neil Provo
Ultralight shell for those unexpected storms
The size of an apple and the weight of a fly box, the Helium2 jacket will fit into every outdoorsman kit out there! You never know when that perfect blue day in the mountains will turn into a dreadful wet and cold retreat. Weather your skiing/snowboarding, biking, fishing, camping, or climbing, with packable gear this light there's no reason to get stuck without proper weather protection when sky decides to unleash. Toss it in your pack and never look back!
View More Details on Outdoor Research Helium II Jacket - Men's


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Follow This Seminar on Twitter #avtax
Conklin & de Decker Associates, Inc. and the National Air Transportation Association (NATA) continue to offer the only tax seminar for commercial operators. The Commercial Operators Tax Seminar ("Tax Seminar") deals with taxes that you, the commercial operator, face every day in your operations.
Are you concerned with the recent IRS Chief Counsel Advisory relating to the application of the commercial FET to owner flights and management fees? This seminar will provide insight on this and other IRS opinions and how to prepare and survive an IRS audit.
A comprehensive review of the Federal Aviation Regulations, Ops Spec A008 and language that should be included in charter/management and management agreements will be provided on the first day.
who should attend:
This Tax Seminar is a must attend for anyone involved with aircraft charter, whether you are an aircraft owner, operator, certificate holder, management company, charter-broker, insurance broker, consultant or attorney.
Agenda & Seminar Outline:
View the 2015 Commercial Operators Tax Seminar Agenda
Registration Information:
CPE/CLE Credit:
This seminar has not been submitted for CPE or CLE approval to any state provider. However, upon written request, Conklin & de Decker will provide individual attorneys and CPA's with an attendance confirmation letter to support their application for CPE or CLE credit in their respective states. Based on a 50-minute hour, the Commercial Operators Tax Seminar should be eligible for a maximum of 12.5 CPE/CLE credits for the full seminar.
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Westfields Marriott
14750 Conference Center Dr.
Chantilly, VA 20151
Guaranteed Rate: $160, mention Conklin & de Decker
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Sponsorship of a Conklin & de Decker Seminar entitles your company to have promotional materials at the Seminar, your logo included in any printed materials and on the Conklin & de Decker website. Depending on the level of sponsorship you may be entitled to a free registration or a registration at a reduced cost. If you are interested in being a sponsor of this Seminar or any other Conklin & de Decker Seminar, please contact us via email at firstname.lastname@example.com or call us at 928.443.8676. Click Here to view the sponsorship opportunities.


text: This week should have been nothing but hype with Apex 2015 quickly approaching, EVO2015 game announcement being held this week. All of this coming on the tail end of 2 very exciting tournaments. The scene for Smash is anything but grass roots at this point and it is insane to think that just 2 years ago it was struggling to survive. However times have changed and things are definitely only getting better.