Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts (also called phasefronts) whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.

A curved phasefront may be synthesized from an infinite number of these "natural modes" i.e., from plane wave phasefronts oriented in different directions in space. Far from its sources, an expanding spherical wave is locally tangent to a planar phase front (a single plane wave out of the infinite spectrum), which is transverse to the radial direction of propagation. In this case, a Fraunhofer diffraction pattern is created, which emanates from a single spherical wave phase center. In the near field, no single well-defined spherical wave phase center exists, so the wavefront isn't locally tangent to a spherical ball. In this case, a Fresnel diffraction pattern would be created, which emanates from an extended source, consisting of a distribution of (physically identifiable) spherical wave sources in space. In the near field, a full spectrum of plane waves is necessary to represent the Fresnel near-field wave, even locally. A "wide" wave moving forward (like an expanding ocean wave coming toward the shore) can be regarded as an infinite number of "plane wave modes", all of which could (when they collide with something such as a rock in the way) scatter independently of one other. These mathematical simplifications and calculations are the realm of Fourier analysis and synthesis – together, they can describe what happens when light passes through various slits, lenses or mirrors that are curved one way or the other, or is fully or partially reflected.

Fourier optics forms much of the theory behind image processing techniques, as well as applications where information needs to be extracted from optical sources such as in quantum optics. To put it in a slightly complex way, similar to the concept of frequency and time used in traditional Fourier transform theory, Fourier optics makes use of the spatial frequency domain (kx, ky) as the conjugate of the spatial (x, y) domain. Terms and concepts such as transform theory, spectrum, bandwidth, window functions and sampling from one-dimensional signal processing are commonly used.

Fourier optics plays an important role for high-precision optical applications such as photolithography in which a pattern on a reticle to be imaged on wafers for semiconductor chip production is so dense such that light (e.g., DUV or EUV) emanated from the reticle is diffracted and each diffracted light may correspond to a different spatial frequency (kx, ky). Due to generally non-uniform patterns on reticles, a simple diffraction grating analysis may not provide the details of how light is diffracted from each reticle.

Propagation of light in homogeneous, source-free media

Light can be described as a waveform propagating through a free space (vacuum) or a material medium (such as air or glass). Mathematically, a real-valued component of a vector field describing a wave is represented by a scalar wave function u that depends on both space and time:

where

represents a position in a three dimensional space (in the Cartesian coordinate system here), and t represents time.

The wave equation

Fourier optics begins with the homogeneous, scalar wave equation (valid in source-free regions):

where  is the speed of light and u(r,t) is a real-valued Cartesian component of an electromagnetic wave propagating through a free space (e.g.,  for  where Ei is the i-axis component of an electric field E in the Cartesian coordinate system).

Sinusoidal steady state

If light of a fixed frequency in time/wavelength/color (as from a single-mode laser) is assumed, then, based on the engineering time convention, which assumes an  time dependence in wave solutions at the angular frequency  with  where  is a time period of the waves, the time-harmonic form of the optical field is given as
where  is the imaginary unit,  is the operator taking the real part of ,
 is the angular frequency (in radians per unit time) of light waves, and
 is, in general, a complex quantity, with separate amplitude  in non-negative real number and phase .

The Helmholtz equation

Substituting this expression into the scalar wave equation above yields the time-independent form of the wave equation,

where

with the wavelength  in vacuum, is the wave number (also called propagation constant),  is the spatial part of a complex-valued Cartesian component of an electromagnetic wave. Note that the propagation constant  and the angular frequency  are linearly related to one another, a typical characteristic of transverse electromagnetic (TEM) waves in homogeneous media.

Since the originally desired real-valued solution  of the scalar wave equation can be simply obtained by taking the real part of , solving the following equation, known as the Helmholtz equation, is mostly concerned as treating a complex-valued function is often much easier than treating the corresponding real-valued function.

Solving the Helmholtz equation

Solutions to the Helmholtz equation in the Cartesian coordinate system may readily be found via the principle of separation of variables for partial differential equations. This principle says that in separable orthogonal coordinates, an elementary product solution to this wave equation may be constructed of the following form:

i.e., as the product of a function of x, times a function of y, times a function of z. If this elementary product solution is substituted into the wave equation, using the scalar Laplacian in the Cartesian coordinates system

then the following equation for the 3 individual functions is obtained

which is readily rearranged into the form:

It may now be argued that each quotient in the equation above must, of necessity, be constant. To justify this, let's say that the first quotient is not a constant, and is a function of x. Since none of the other terms in the equation has any dependence on the variable x, so the first term also must not have any x-dependence; it must be a constant. (If the first term is a function of x, then there is no way to make the left hand side of this equation be zero.) This constant is denoted as -kx2. Reasoning in a similar way for the y and z quotients, three ordinary differential equations are obtained for the fx, fy and fz, along with one separation condition:

Each of these 3 differential equations has the same solution form: sines, cosines or complex exponentials. We'll go with the complex exponential as  to be a complex function. As a result, the elementary product solution  is

with a generally complex number . This solution is the spatial part of a complex-valued Cartesian component (e.g., , , or  as the electric field component along each axis in the Cartesian coordinate system) of a propagating plane wave.  (, , or ) is a real number here since waves in a source-free medium has been assumed so each plane wave is not decayed or amplified as it propagates in the medium. The negative sign of  (, , or ) in a wave vector  (where ) means that the wave propagation direction vector has a positive  (, , or )-component, while the positive sign of  means a negative  (, , or )-component of that vector.

Product solutions to the Helmholtz equation are also readily obtained in cylindrical and spherical coordinates, yielding cylindrical and spherical harmonics (with the remaining separable coordinate systems being used much less frequently).

The complete solution: the superposition integral

A general solution to the homogeneous electromagnetic wave equation at a fixed time frequency  in the Cartesian coordinate system may be formed as a weighted superposition of all possible elementary plane wave solutions as

with the constraints of , each  as a real number, and  where . In this superposition,  is the weight factor or the amplitude of the plane wave component with the wave vector  where  is determined in terms of  and  by the mentioned constraint.

Next, let
Then:

The plane wave spectrum representation of a general electromagnetic field (e.g., a spherical wave) in the equation () is the basic foundation of Fourier optics (this point cannot be emphasized strongly enough), because at z = 0, the equation simply becomes a Fourier transform (FT) relationship between the field and its plane wave contents (hence the name, Fourier optics).

Thus:

and

All spatial dependence of each plane wave component is described explicitly by an exponential function. The coefficient of the exponential is a function of only two components of the wave vector for each plane wave (since other remained component can be determined via the above mentioned constraints), for example  and , just as in ordinary Fourier analysis and Fourier transforms.

Connection between Fourier optics and imaging resolution
Let's consider an imaging system where the z-axis is the optical axis of the system and the object plane (to be imaged on the image plane of the system) is the plane at . On the object plane, the spatial part of a complex-valued Cartesian component of a wave is, as shown above,  with the constraints of , each  as a real number, and  where . The imaging is the reconstruction of a wave on the object plane (having information about a pattern on the object plane to be imaged) on the image plane via the proper wave propagation from the object to the image planes, (E.g., think about the imaging of an image in an aerial space.) and the wave on the object plane, that fully follows the pattern to be imaged, is in principle, described by the unconstrained inverse Fourier transform  where  takes an infinite range of real numbers. It means that, for a given light frequency, only a part of the full feature of the pattern can be imaged because of the above-mentioned constraints on ; (1) a fine feature which representation in the inverse Fourier transform requires spatial frequencies , where  are transverse wave numbers satisfying , can not be fully imaged since waves with such  do not exist for the given light of  (This phenomenon is known as the diffraction limit.), and (2) spatial frequencies with  but close to  so higher wave outgoing angles with respect to the optical axis, requires a high NA (Numerical Aperture) imaging system that is expensive and difficult to build. For (1), even if complex-valued longitudinal wavenumbers  are allowed (by an unknown interaction between light and the object plane pattern that is usually a solid material),  give rise to light decay along the  axis (Light amplification along the  axis does not physically make sense if there is no amplification material between the object and image planes, and this is an usual case.) so waves with such  may not reach the image plane that is usually sufficiently far way from the object plane.

In connection with photolithography of electronic components, these (1) and (2) are the reasons why light of a higher frequency (smaller wavelength, thus larger magnitude of ) or a higher NA imaging system is required to image finer features of integrated circuits on a photoresist on a wafer. As a result, machines realizing such an optical lithography have become more and more complex and expensive, significantly increasing the cost of the electronic component production.

The paraxial approximation

Paraxial wave propagation (optic axis assumed as z axis)

A solution to the Helmholtz equation as the spatial part of a complex-valued Cartesian component of a single frequency wave is assumed to take the form:

where  is the wave vector, and

and

is the wave number. Next, use the paraxial approximation, that is a small-angle approximation such that

so, up to the second order approximation of trigonometric functions (that is, taking only up to the second term in the Taylor series expansion of each trigonometric function),

where  is the angle (in radian) between the wave vector k and the z-axis as the optical axis of an optical system under discussion.

As a result,

and

The paraxial wave equation

Substituting this expression into the Helmholtz equation, the paraxial wave equation is derived:

where

is the transverse Laplace operator in the Cartesian coordinates system. In the derivation of the paraxial wave equation, the following approximations are used.

  is small () so a term with  is ignored.
 Terms with  and  are much smaller than a term with  (or ) so these two terms are ignored.
  so a term with  is ignored. It is the slowly varying envelope approximation, means that the amplitude or envelope of a wave  is slowly varying compared with the major period of the wave .

The far field approximation

The equation () above may be evaluated asymptotically in the far field (using the stationary phase method) to show that the field at a distant point  is indeed due solely to the plane wave component with the wave vector  which propagates parallel to the vector , and whose plane is tangent to the phasefront at . The mathematical details of this process may be found in Scott [1998] or Scott [1990]. The result of performing a stationary phase integration on the expression above is the following expression,

which clearly indicates that the field at  is directly proportional to the spectral component in the direction of , where,

and

Stated another way, the radiation pattern of any planar field distribution is the FT (Fourier Transform) of that source distribution (see Huygens–Fresnel principle, wherein the same equation is developed using a Green's function approach). Note that this is NOT a plane wave. The  radial dependence is a spherical wave - both in magnitude and phase - whose local amplitude is the FT of the source plane distribution at that far field angle. A plane wave spectrum does not necessarily mean that the field as the superposition of the plane wave components in that spectrum behaves something like a plane wave at far distances.

Spatial versus angular bandwidth

The equation () above is critical to making the connection between spatial bandwidth (on the one hand) and angular bandwidth (on the other), in the far field. Note that the term "far field" usually means we're talking about a converging or diverging spherical wave with a pretty well defined phase center. The connection between spatial and angular bandwidth in the far field is essential in understanding the low pass filtering property of thin lenses. See the section 6.1.3 for the condition defining the far field region.

Once the concept of angular bandwidth is understood, the optical scientist can "jump back and forth" between the spatial and spectral domains to quickly gain insights which would ordinarily not be so readily available just through spatial domain or ray optics considerations alone. For example, any source bandwidth which lies past the edge angle to the first lens (This edge angle sets the bandwidth of the optical system.) will not be captured by the system to be processed.

As a side note, electromagnetics scientists have devised an alternative means to calculate an electric field in a far zone which does not involve stationary phase integration. They have devised a concept known as "fictitious magnetic currents" usually denoted by M, and defined as

In this equation, it is assumed that the unit vector in the z-direction points into the half-space where the far field calculations will be made. These equivalent magnetic currents are obtained using equivalence principles which, in the case of an infinite planar interface, allow any electric currents J to be "imaged away" while the fictitious magnetic currents are obtained from twice the aperture electric field (see Scott [1998]). Then the radiated electric field is calculated from the magnetic currents using an equation similar to the equation for the magnetic field radiated by an electric current. In this way, a vector equation is obtained for the radiated electric field in terms of the aperture electric field, and the derivation requires no use of stationary phase ideas.

The plane wave spectrum: the foundation of Fourier optics

Fourier optics is somewhat different from ordinary ray optics typically used in the analysis and design of focused imaging systems such as cameras, telescopes and microscopes.  Ray optics is the very first type of optics that most of us encounter in our lives; it's simple to conceptualize and understand, and works very well in gaining a baseline understanding of common optical devices. Unfortunately, ray optics does not explain the operation of Fourier optical systems, which are in general not focused systems. Ray optics is a subset of wave optics (In the jargon, it is "the asymptotic zero-wavelength limit" of wave optics.) and therefore has limited applicability. We have to know when it is valid and when it is not - and this is one of those times when it is not. For our current task, we must expand our understanding of optical phenomena to encompass wave optics, in which the optical field is seen as a solution to Maxwell's equations. This more general wave optics accurately explains the operation of Fourier optics devices.

In this section, we won't go all the way back to Maxwell's equations, but will start instead with the homogeneous Helmholtz equation (valid in source-free media), which is one level of refinement up from Maxwell's equations (Scott [1998]). From this equation, we'll show how infinite uniform plane waves comprise one field solution (out of many possible) in free space. These uniform plane waves form the basis for understanding Fourier optics.

The plane wave spectrum concept is the basic foundation of Fourier Optics. The plane wave spectrum is a continuous spectrum of uniform plane waves, and there is one plane wave component in the spectrum for every tangent point on the far-field phase front. The amplitude of that plane wave component would be the amplitude of the optical field at that tangent point. Again, this is true only in the far field, roughly defined as the range beyond  where  is the maximum linear extent of the optical sources and  is the wavelength (Scott [1998]). The plane wave spectrum is often regarded as being discrete for certain types of periodic gratings, though in reality, the spectra from gratings are continuous as well, since no physical device can have the infinite extent required to produce a true line spectrum.

Likely to electrical signals, bandwidth in optics is a measure of how finely detailed an image is; the finer the detail, the greater the bandwidth required to represent it. A DC (Direct Current) electrical signal is constant and has no oscillations; a plane wave propagating parallel to the optic () axis has constant value in any x-y plane, and therefore is analogous to the (constant) DC component of an electrical signal. Bandwidth in electrical signals relates to the difference between the highest and lowest frequencies present in the spectrum of a signal, practically with a criterion to cut off high and low frequency edges of the spectrum to represent bandwidth in a number. For optical systems, bandwidth also relates to spatial frequency content (spatial bandwidth), but it also has a secondary meaning. It also measures how far from the optic axis the corresponding plane waves are tilted, and so this type of bandwidth is often referred to also as angular bandwidth. It takes more frequency bandwidth to produce a short pulse in an electrical circuit, and more angular (or, spatial frequency) bandwidth to produce a sharp spot in an optical system (see discussion related to Point spread function).

The plane wave spectrum arises naturally as the eigenfunction or "natural mode" solution to the homogeneous electromagnetic wave equation in rectangular coordinates (see also Electromagnetic radiation, which derives the wave equation from Maxwell's equations in source-free media, or Scott [1998]). In the frequency domain, with an assumed time convention of , the homogeneous electromagnetic wave equation becomes what is known as the Helmholtz equation and takes the form

where  and  is the wavenumber of the medium.

Eigenfunction (natural mode) solutions: background and overview

In the case of differential equations, as in the case of matrix equations, whenever the right-hand side of an equation is zero (For example, a forcing function, forcing vector, or the source of a force is zero.), the equation may still admit a non-trivial solution, known in applied mathematics as an eigenfunction solution, in physics as a "natural mode" solution, and in electrical circuit theory as the "zero-input response." This is a concept that spans a wide range of physical disciplines. Common physical examples of resonant natural modes would include the resonant vibrational modes of stringed instruments (1D), percussion instruments (2D) or the former Tacoma Narrows Bridge (3D). Examples of propagating natural modes would include waveguide modes, optical fiber modes, solitons and Bloch waves. an Infinite homogeneous media admits the rectangular, circular and spherical harmonic solutions to the Helmholtz equation, depending on the coordinate system under consideration. The propagating plane waves that we'll study in this article are perhaps the simplest type of propagating waves found in any type of media.

There is a striking similarity between the Helmholtz equation () above, which may be written

and the usual equation form for the eigenvalues / eigenvectors of a square matrix A,

particularly since both the scalar Laplacian  and the matrix A are linear operators on their respective functions / vector spaces. (The minus sign in this matrix equation is, for all intents and purposes, immaterial. However, the plus sign in the Helmholtz equation is significant.) It is perhaps worthwhile to note that the eigenfunction solutions / eigenvector solutions to the Helmholtz equation / the matrix equation, often yield an orthogonal set of the eigenfunctions / the eigenvectors which span (i.e., form a basis set for) the function space / vector space under consideration. The interested reader may investigate other functional linear operators (so for different equations than the Helmholtz equation) which give rise to different kinds of orthogonal eigenfunctions such as Legendre polynomials, Chebyshev polynomials and Hermite polynomials.

In the matrix equation case in which A is a square matrix, eigenvalues  may be found by setting the determinant of the matrix equal to zero, i.e. finding where the matrix has no inverse. (Such a square matrix is said to be singular.) Finite matrices have only a finite number of eigenvalues/eigenvectors, whereas linear operators can have a countably infinite number of eigenvalues/eigenfunctions (in confined regions) or uncountably infinite (continuous) spectra of solutions, as in unbounded regions.

In certain physics applications such as in the computation of bands in a periodic volume, it is often a case that the elements of a matrix will be very complicated functions of frequency and wavenumber, and the matrix will be non-singular (I.e., it has the inverse matrix.) for most combinations of frequency and wavenumber, but will also be singular (I.e., it does not have the inverse matrix.) for certain specific combinations. By finding which combinations of frequency and wavenumber drive the determinant of the matrix to zero, the propagation characteristics of the medium may be determined. Relations of this type, between frequency and wavenumber, are known as dispersion relations and some physical systems may admit many different kinds of dispersion relations. An example from electromagnetics is an ordinary waveguide, which may admit numerous dispersion relations, each associated with a unique propagation mode of the waveguide. Each propagation mode of the waveguide is known as an eigenfunction solution (or eigenmode solution) to Maxwell's equations in the waveguide. Free space also admits eigenmode (natural mode) solutions (known more commonly as plane waves), but with the distinction that for any given frequency, free space admits a continuous modal spectrum, whereas waveguides have a discrete mode spectrum. In this case, the dispersion relation is linear, as in section 1.3.

K-space

For a given  such as  for a homogeneous vacuum space, the separation condition,

which is identical to the equation for the Euclidean metric in a three-dimensional configuration space, suggests the notion of a k-vector in a three-dimensional "k-space", defined (for propagating plane waves) in rectangular coordinates as:

and in the spherical coordinate system as

Use will be made of these spherical coordinate system relations in the next section.

The notion of k-space is central to many disciplines in engineering and physics, especially in the study of periodic volumes, such as in crystallography and the band theory of semiconductor materials.

The two-dimensional Fourier transform

A spectrum analysis equation (calculating the spectrum of a function ):

A synthesis equation (reconstructing the function  from its spectrum):

The normalizing factor of  is present whenever angular frequency (radians) is used, but not when ordinary frequency (cycles) is used.

Optical systems: general overview and analogy with electrical signal processing systems

In a high level overview, an optical system consists of three parts; an input plane, and output plane, and a set of components between these planes that transform an image f formed in the input plane into a different image g formed in the output plane. The optical system output image g is related to the input image f by convolving the input image with the optical impulse response function of the optical system, h (known as the point-spread function, for focused optical systems). The impulse response function uniquely defines the input-output behavior of the optical system. By convention, the optical axis of the system is taken as the z-axis. As a result, the two images and the impulse response function are all functions of the transverse coordinates, x and y.

The impulse response of an optical imaging system is the output plane field which is produced when an ideal mathematical optical field point source of light, that is an impulse input to the system, is placed in the input plane (usually on-axis, i.e., on the optical axis). In practice, it is not necessary to have an ideal point source in order to determine an exact impulse response. This is because any source bandwidth which lies outside the bandwidth of the optical system under consideration won't matter anyway (since it cannot even be captured by the optical system), so therefore it's not necessary in determining the impulse response. The source only needs to have at least as much (angular) bandwidth as the optical system.

Optical systems typically fall into one of two different categories. The first is ordinary focused optical imaging systems (e.g., cameras), wherein the input plane is called the object plane and the output plane is called the image plane. An optical field in the image plane (the output plane of the imaging system) is desired to be a high-quality reproduction of an optical field in the object plane (the input plane of the imaging system). The impulse response function of an optical imaging system is desired to approximate a 2D delta function, at the location (or a linearly scaled location) in the output plane corresponding to the location of the impulse (an ideal point source) in the input plane. The actual impulse response function of an imaging system typically resembles an Airy function, whose radius is on the order of the wavelength of the light used. The impulse response function in this case is typically referred to as a point spread function, since the mathematical point of light in the object plane has been spread out into an Airy function in the image plane.

The second type is optical image processing systems, in which a significant feature in the input plane optical field is to be located and isolated. In this case, the impulse response of such a system is desired to be a close replica (picture) of that feature which is being searched for in the input plane field, so that a convolution of the impulse response (an image of the desired feature) against the input plane field will produce a bright spot at the feature location in the output plane. It is this latter type of optical image processing system that is the subject of this section. The section 6.2 presents one hardware implementation of the optical image processing operations described in this section.

Input plane

The input plane is defined as the locus of all points such that z = 0. The input image f is therefore

Output plane

The output plane is defined as the locus of all points such that z = d. The output image g is therefore

The 2D convolution of input function against the impulse response function

i.e.,

The alert reader will note that the integral above tacitly assumes that the impulse response is NOT a function of the position (x',y') of the impulse of light in the input plane (if this were not the case, this type of convolution would not be possible). This property is known as shift invariance (Scott [1998]). No optical system is perfectly shift invariant: as the ideal, mathematical point of light is scanned away from the optic axis, aberrations will eventually degrade the impulse response (known as a coma in focused imaging systems). However, high quality optical systems are often "shift invariant enough" over certain regions of the input plane that we may regard the impulse response as being a function of only the difference between input and output plane coordinates, and thereby use the equation above with impunity.

Also, this equation assumes unit magnification. If magnification is present, then eqn. () becomes

which basically translates the impulse response function, hM(), from x′ to x = Mx′. In eqn. (), hM will be a magnified version of the impulse response function h of a similar, unmagnified system, so that hM(x,y) = h(x/M,y/M).

Derivation of the convolution equation
The extension to two dimensions is trivial, except for the difference that causality exists in the time domain, but not in the spatial domain. Causality means that the impulse response h(t − t′) of an electrical system, due to an impulse applied at time t', must of necessity be zero for all times t such that t − t′ < 0.

Obtaining the convolution representation of the system response requires representing the input signal as a weighted superposition over a train of impulse functions by using the sifting property of Dirac delta functions.

It is then presumed that the system under consideration is linear, that is to say that the output of the system due to two different inputs (possibly at two different times) is the sum of the individual outputs of the system to the two inputs, when introduced individually. Thus the optical system may contain no nonlinear materials nor active devices (except possibly, extremely linear active devices). The output of the system, for a single delta function input is defined as the impulse response of the system, h(t − t′). And, by our linearity assumption (i.e., that the output of system to a pulse train input is the sum of the outputs due to each individual pulse), we can now say that the general input function f(t) produces the output:

where h(t − t′) is the (impulse) response of the linear system to the delta function input δ(t − t′), applied at time t. This is where the convolution equation above comes from. The convolution equation is useful because it is often much easier to find the response of a system to a delta function input - and then perform the convolution above to find the response to an arbitrary input - than it is to try to find the response to the arbitrary input directly. Also, the impulse response (in either time or frequency domains) usually yields insight to relevant figures of merit of the system. In the case of most lenses, the point spread function (PSF) is a pretty common figure of merit for evaluation purposes.

The same logic is used in connection with the Huygens–Fresnel principle, or Stratton-Chu formulation, wherein the "impulse response" is referred to as the Green's function of the system. So the spatial domain operation of a linear optical system is analogous in this way to the Huygens–Fresnel principle.

 System transfer function
If the last equation above is Fourier transformed, it becomes:

where
  is the spectrum of the output signal
  is the system transfer function
  is the spectrum of the input signal

In like fashion, eqn. () may be Fourier transformed to yield:

The system transfer function, . In optical imaging this function is better known as the optical transfer function (Goodman).

Once again it may be noted from the discussion on the Abbe sine condition, that this equation assumes unit magnification.

This equation takes on its real meaning when the Fourier transform,  is associated with the coefficient of the plane wave whose transverse wavenumbers are . Thus, the input-plane plane wave spectrum is transformed into the output-plane plane wave spectrum through the multiplicative action of the system transfer function. It is at this stage of understanding that the previous background on the plane wave spectrum becomes invaluable to the conceptualization of Fourier optical systems.

Applications of Fourier optics principles
Fourier optics is used in the field of optical information processing, the staple of which is the classical 4F processor.

The Fourier transform properties of a lens provide numerous applications in optical signal processing such as spatial filtering, optical correlation and computer generated holograms.

Fourier optical theory is used in interferometry, optical tweezers, atom traps, and quantum computing. Concepts of Fourier optics are used to reconstruct the phase of light intensity in the spatial frequency plane (see adaptive-additive algorithm).

Fourier transforming property of lenses

If a transmissive object is placed at one focal length in front of a lens, then its Fourier transform will be formed at one focal length behind the lens. Consider the figure to the right (click to enlarge)

In this figure, a plane wave incident from the left is assumed. The transmittance function in the front focal plane (i.e., Plane 1) spatially modulates the incident plane wave in magnitude and phase, like on the left-hand side of eqn. () (specified to z = 0), and in so doing, produces a spectrum of plane waves corresponding to the FT of the transmittance function, like on the right-hand side of eqn. () (for z > 0). The various plane wave components propagate at different tilt angles with respect to the optic axis of the lens (i.e., the horizontal axis). The finer the features in the transparency, the broader the angular bandwidth of the plane wave spectrum. We'll consider one such plane wave component, propagating at angle θ with respect to the optic axis. It is assumed that θ is small (paraxial approximation), so that

and

and

In the figure, the plane wave phase, moving horizontally from the front focal plane to the lens plane, is

and the spherical wave phase from the lens to the spot in the back focal plane is:

and the sum of the two path lengths is f (1 + θ2/2 + 1 − θ2/2) = 2f; i.e., it is a constant value, independent of tilt angle, θ, for paraxial plane waves. Each paraxial plane wave component of the field in the front focal plane appears as a point spread function spot in the back focal plane, with an intensity and phase equal to the intensity and phase of the original plane wave component in the front focal plane. In other words, the field in the back focal plane is the Fourier transform of the field in the front focal plane.

All FT components are computed simultaneously - in parallel - at the speed of light. As an example, light travels at a speed of roughly  per nanosecond, so if a lens has a  focal length, an entire 2D FT can be computed in about 2 ns (2 × 10−9 seconds). If the focal length is 1 in, then the time is under 200 ps. No electronic computer can compete with these kinds of numbers or perhaps ever hope to, although supercomputers may actually prove faster than optics, as improbable as that may seem. However, their speed is obtained by combining numerous computers which, individually, are still slower than optics. The disadvantage of the optical FT is that, as the derivation shows, the FT relationship only holds for paraxial plane waves, so this FT "computer" is inherently bandlimited. On the other hand, since the wavelength of visible light is so minute in relation to even the smallest visible feature dimensions in the image i.e.,

(for all kx, ky within the spatial bandwidth of the image, so that kz is nearly equal to k), the paraxial approximation is not terribly limiting in practice. And, of course, this is an analog - not a digital - computer, so precision is limited. Also, phase can be challenging to extract; often it is inferred interferometrically.

Optical processing is especially useful in real time applications where rapid processing of massive amounts of 2D data is required, particularly in relation to pattern recognition.

Object truncation and Gibbs phenomenon

The spatially modulated electric field, shown on the left-hand side of eqn. (), typically only occupies a finite (usually rectangular) aperture in the x,y plane. The rectangular aperture function acts like a 2D square-top filter, where the field is assumed to be zero outside this 2D rectangle. The spatial domain integrals for calculating the FT coefficients on the right-hand side of eqn. () are truncated at the boundary of this aperture. This step truncation can introduce inaccuracies in both theoretical calculations and measured values of the plane wave coefficients on the RHS of eqn. ().

Whenever a function is discontinuously truncated in one FT domain, broadening and rippling are introduced in the other FT domain. A perfect example from optics is in connection with the point spread function, which for on-axis plane wave illumination of a quadratic lens (with circular aperture), is an Airy function, J1(x)/x. Literally, the point source has been "spread out" (with ripples added), to form the Airy point spread function (as the result of truncation of the plane wave spectrum by the finite aperture of the lens). This source of error is known as Gibbs phenomenon and it may be mitigated by simply ensuring that all significant content lies near the center of the transparency, or through the use of window functions which smoothly taper the field to zero at the frame boundaries. By the convolution theorem, the FT of an arbitrary transparency function - multiplied (or truncated) by an aperture function - is equal to the FT of the non-truncated transparency function convolved against the FT of the aperture function, which in this case becomes a type of "Greens function" or "impulse response function" in the spectral domain. Therefore, the image of a circular lens is equal to the object plane function convolved against the Airy function (the FT of a circular aperture function is J1(x)/x and the FT of a rectangular aperture function is a product of sinc functions, sinx/x).

 Fourier analysis and functional decomposition

Even though the input transparency only occupies a finite portion of the x-y plane (Plane 1), the uniform plane waves comprising the plane wave spectrum occupy the entire x-y plane, which is why (for this purpose) only the longitudinal plane wave phase (in the z-direction, from Plane 1 to Plane 2) must be considered, and not the phase transverse to the z-direction. It is of course, very tempting to think that if a plane wave emanating from the finite aperture of the transparency is tilted too far from horizontal, it will somehow "miss" the lens altogether but again, since the uniform plane wave extends infinitely far in all directions in the transverse (x-y) plane, the planar wave components cannot miss the lens.

This issue brings up perhaps the predominant difficulty with Fourier analysis, namely that the input-plane function, defined over a finite support (i.e., over its own finite aperture), is being approximated with other functions (sinusoids) which have infinite support (i.e., they are defined over the entire infinite x-y plane). This is unbelievably inefficient computationally, and is the principal reason why wavelets were conceived, that is to represent a function (defined on a finite interval or area) in terms of oscillatory functions which are also defined over finite intervals or areas. Thus, instead of getting the frequency content of the entire image all at once (along with the frequency content of the entire rest of the x-y plane, over which the image has zero value), the result is instead the frequency content of different parts of the image, which is usually much simpler. Unfortunately, wavelets in the x-y plane don't correspond to any known type of propagating wave function, in the same way that Fourier's sinusoids (in the x-y plane) correspond to plane wave functions in three dimensions. However, the FTs of most wavelets are well known and could possibly be shown to be equivalent to some useful type of propagating field.

On the other hand, sinc functions and Airy functions - which are not only the point spread functions of rectangular and circular apertures, respectively, but are also cardinal functions commonly used for functional decomposition in interpolation/sampling theory [Scott 1990] - do''' correspond to converging or diverging spherical waves, and therefore could potentially be implemented as a whole new functional decomposition of the object plane function, thereby leading to another point of view similar in nature to Fourier optics. This would basically be the same as conventional ray optics, but with diffraction effects included. In this case, each point spread function would be a type of "smooth pixel," in much the same way that a soliton on a fiber is a "smooth pulse."

Perhaps a lens figure-of-merit in this "point spread function" viewpoint would be to ask how well a lens transforms an Airy function in the object plane into an Airy function in the image plane, as a function of radial distance from the optic axis, or as a function of the size of the object plane Airy function. This is somewhat like the point spread function, except now we're really looking at it as a kind of input-to-output plane transfer function (like MTF), and not so much in absolute terms, relative to a perfect point. Similarly, Gaussian wavelets, which would correspond to the waist of a propagating Gaussian beam, could also potentially be used in still another functional decomposition of the object plane field.

 Far-field range and the 2D2 / λ criterion

In the figure above, illustrating the Fourier transforming property of lenses, the lens is in the near field of the object plane transparency, therefore the object plane field at the lens may be regarded as a superposition of plane waves, each one of which propagates at some angle with respect to the z-axis. In this regard, the far-field criterion is loosely defined as: Range = 2D2/λ where D is the maximum linear extent of the optical sources and λ is the wavelength (Scott [1998]). The D of the transparency is on the order of cm (10−2 m) and the wavelength of light is on the order of 10−6 m, therefore D/λ for the whole transparency is on the order of 104. This times D is on the order of 102 m, or hundreds of meters. On the other hand, the far field distance from a PSF spot is on the order of λ. This is because D for the spot is on the order of λ, so that D/λ is on the order of unity; this times D (i.e., λ) is on the order of λ (10−6 m).

Since the lens is in the far field of any PSF spot, the field incident on the lens from the spot may be regarded as being a spherical wave, as in eqn. (), not as a plane wave spectrum, as in eqn. (). On the other hand, the lens is in the near field of the entire input plane transparency, therefore eqn. () - the full plane wave spectrum - accurately represents the field incident on the lens from that larger, extended source.

 Lens as a low-pass filter

A lens is basically a low-pass plane wave filter (see Low-pass filter). Consider a "small" light source located on-axis in the object plane of the lens. It is assumed that the source is small enough that, by the far-field criterion, the lens is in the far field of the "small" source. Then, the field radiated by the small source is a spherical wave which is modulated by the FT of the source distribution, as in eqn. (), Then, the lens passes - from the object plane over onto the image plane - only that portion of the radiated spherical wave which lies inside the edge angle of the lens. In this far-field case, truncation of the radiated spherical wave is equivalent to truncation of the plane wave spectrum of the small source. So, the plane wave components in this far-field spherical wave, which lie beyond the edge angle of the lens, are not captured by the lens and are not transferred over to the image plane. Note: this logic is valid only for small sources, such that the lens is in the far field region of the source, according to the 2D2/λ criterion mentioned previously. If an object plane transparency is imagined as a summation over small sources (as in the Whittaker–Shannon interpolation formula, Scott [1990]), each of which has its spectrum truncated in this fashion, then every point of the entire object plane transparency suffers the same effects of this low pass filtering.

Loss of the high (spatial) frequency content causes blurring and loss of sharpness (see discussion related to point spread function). Bandwidth truncation causes a (fictitious, mathematical, ideal) point source in the object plane to be blurred (or, spread out) in the image plane, giving rise to the term, "point spread function." Whenever bandwidth is expanded or contracted, image size is typically contracted or expanded accordingly, in such a way that the space-bandwidth product remains constant, by Heisenberg's principle (Scott [1998] and Abbe sine condition).

 Coherence and Fourier transforming

While working in the frequency domain, with an assumed ejωt (engineering) time dependence, coherent (laser) light is implicitly assumed, which has a delta function dependence in the frequency domain. Light at different (delta function) frequencies will "spray" the plane wave spectrum out at different angles, and as a result these plane wave components will be focused at different places in the output plane. The Fourier transforming property of lenses works best with coherent light, unless there is some special reason to combine light of different frequencies, to achieve some special purpose.

Hardware implementation of the system transfer function: The 4F correlator

The theory on optical transfer functions presented in the section 5 is somewhat abstract. However, there is one very well known device which implements the system transfer function H in hardware using only 2 identical lenses and a transparency plate - the 4F correlator. Although one important application of this device would certainly be to implement the mathematical operations of cross-correlation and convolution, this device - 4 focal lengths long - actually serves a wide variety of image processing operations that go well beyond what its name implies. A diagram of a typical 4F correlator is shown in the figure below (click to enlarge). This device may be readily understood by combining the plane wave spectrum representation of the electric field (section 1.5) with the Fourier transforming property of quadratic lenses (section 6.1) to yield the optical image processing operations described in the section 5.

The 4F correlator is based on the convolution theorem from Fourier transform theory, which states that convolution in the spatial (x,y) domain is equivalent to direct multiplication in the spatial frequency (kx, ky) domain (aka: spectral domain). Once again, a plane wave is assumed incident from the left and a transparency containing one 2D function, f(x,y), is placed in the input plane of the correlator, located one focal length in front of the first lens. The transparency spatially modulates the incident plane wave in magnitude and phase, like on the left-hand side of eqn. (), and in so doing, produces a spectrum of plane waves corresponding to the FT of the transmittance function, like on the right-hand side of eqn. (). That spectrum is then formed as an "image" one focal length behind the first lens, as shown. A transmission mask containing the FT of the second function, g(x,y), is placed in this same plane, one focal length behind the first lens, causing the transmission through the mask to be equal to the product, F(kx,ky) × G(kx,ky). This product now lies in the "input plane" of the second lens (one focal length in front), so that the FT of this product (i.e., the convolution of f(x,y) and g(x,y)), is formed in the back focal plane of the second lens.

If an ideal, mathematical point source of light is placed on-axis in the input plane of the first lens, then there will be a uniform, collimated field produced in the output plane of the first lens. When this uniform, collimated field is multiplied by the FT plane mask, and then Fourier transformed by the second lens, the output plane field (which in this case is the impulse response of the correlator) is just our correlating function, g(x,y). In practical applications, g(x,y) will be some type of feature which must be identified and located within the input plane field (see Scott [1998]). In military applications, this feature may be a tank, ship or airplane which must be quickly identified within some more complex scene.

The 4F correlator is an excellent device for illustrating the "systems" aspects of optical instruments, alluded to in the section 5 above. The FT plane mask function, G(kx,ky) is the system transfer function of the correlator, which we'd in general denote as H(kx,ky), and it is the FT of the impulse response function of the correlator, h(x,y) which is just our correlating function g(x,y). And, as mentioned above, the impulse response of the correlator is just a picture of the feature we're trying to find in the input image. In the 4F correlator, the system transfer function H(kx,ky) is directly multiplied against the spectrum F(kx,ky) of the input function, to produce the spectrum of the output function. This is how electrical signal processing systems operate on 1D temporal signals.

 Image Restoration
Image blurring by a point spread function is studied extensively in optical information processing, one way to alleviate the blurring is to adopt Wiener Filter. For example, assume that  is the intensity distribution from an incoherent object,  is the intensity distribution of its image which is blurred by a space-invariant point-spread function  and a noise  introduced in the detection process:

The goal of image restoration is to find a linear restoration filter that minimize the mean-squared error between the true distribution and the estimation . That is, to minimize

The solution of this optimization problem is Wiener filter:

where , ,  are the power spectral densities of the point-spread function, the object and the noise.

Ragnarsson proposed a method to realize Wiener restoration filters optically by holographic technique like setup shown in the figure. The derivation of the function of the setup is described as follows.

Assume there is a transparency as the recording plane and an impulse emitted from a point source S. The wave of impulse is collimated by lens L1, forming a distribution equal to the impulse response . Then the distribution  is then split into two parts:

 The upper portion is first focused (i.e., Fourier transformed) by a lens L2 to a spot in the front focal plan of lens L3, forming a virtual point source generating a spherical wave. The wave is then collimated by lens L3 and produces a tilted plane wave with the form  at the recording plane.
 The lower portion is directly collimated by lens L3'', yielding an amplitude distribution .

Therefore, the total intensity distribution is

Assume  has an amplitude distribution  and a phase distribution  such that

then we can rewrite intensity as follows:

Note that for the point at the origin of the film plane (), the recorded wave from the lower portion should be much stronger than that from the upper portion because the wave passing through the lower path is focused, which leads to the relationship .

In Ragnarsson' s work, this method is based on the following postulates:

 Assume there is a transparency, with its amplitude transmittance  proportional to , that has recorded the known impulse response of the blurred system.
 The maximum phase  shift introduced by the filter is much smaller than  radians so that .
 The phase shift of the transparency after bleaching is linearly proportional to the silver density  present before bleaching.
 The density is linearly proportional to the logarithm of exposure
 The average exposure  is much stronger than varying exposure

By these postulates, we have the following relationship:

Finally, we get a amplitude transmittance with the form of a Wiener filter:

Afterword: Plane wave spectrum within the broader context of functional decomposition

Electrical fields can be represented mathematically in many different ways. In the Huygens–Fresnel or Stratton-Chu viewpoints, the electric field is represented as a superposition of point sources, each one of which gives rise to a Green's function field. The total field is then the weighted sum of all of the individual Green's function fields. That seems to be the most natural way of viewing the electric field for most people - no doubt because most of us have, at one time or another, drawn out the circles with protractor and paper, much the same way Thomas Young did in his classic paper on the double-slit experiment. However, it is by no means the only way to represent the electric field, which may also be represented as a spectrum of sinusoidally varying plane waves. In addition, Frits Zernike proposed still another functional decomposition based on his Zernike polynomials, defined on the unit disc. The third-order (and lower) Zernike polynomials correspond to the normal lens aberrations. And still another functional decomposition could be made in terms of Sinc functions and Airy functions, as in the Whittaker–Shannon interpolation formula and the Nyquist–Shannon sampling theorem. All of these functional decompositions have utility in different circumstances. The optical scientist having access to these various representational forms has available a richer insight to the nature of these marvelous fields and their properties. These different ways of looking at the field are not conflicting or contradictory, rather, by exploring their connections, one can often gain deeper insight into the nature of wave fields.

Functional decomposition and eigenfunctions

The twin subjects of eigenfunction expansions and functional decomposition, both briefly alluded to here, are not completely independent. The eigenfunction expansions to certain linear operators defined over a given domain, will often yield a countably infinite set of orthogonal functions which will span that domain. Depending on the operator and the dimensionality (and shape, and boundary conditions) of its domain, many different types of functional decompositions are, in principle, possible.

See also

 Angular spectrum method
 Abbe sine condition
 Adaptive-additive algorithm
 Huygens–Fresnel principle
 Point spread function
 Phase contrast microscopy
 Fraunhofer diffraction
 Fresnel diffraction
 Geometrical optics
 Hilbert space
 Optical correlator
 Optical Hartley transform
 Wave field synthesis

References

 Intro to Fourier Optics and the 4F correlator

External links

Physical optics
Fourier analysis

Fourier optics


The Académie Mallarmé is a French literary academy of writers and poets, founded in 1937.  Since 1976, the Académie has awarded the Prix Mallarmé literary prize at the Brive book fair.

Founding members include Paul Valéry, Édouard Dujardin, André Fontainas, Charles Vildrac, Maurice Maeterlinck, Ferdinand Hérold, Albert Mockel, Léon-Paul Fargue, Francis Viélé-Griffin, Paul Fort, Saint-Pol-Roux, Valéry Larbaud, and Jean Ajalbert. When asked, André Gide, Francis Jammes and Paul Claudel declined the invitation to become founding members.

Later, Jean Cocteau, Gérard d'Houville, Henry Charpentier, Jacques Audiberti, and Henri Mondor became members.

In 2008, Academy members include Jean L'Anselme, Jean-Michel Maulpoix, Robert Sabatier, Sylvestre Clancier, Abdellatif Laâbi, Venus Ghoury-Ghata, Andrée Chedid, Anise Koltz, Philippe Jones, Jean Portante, Robert Marteau, François Montmaneix, Jean Orizet, Claude Beausoleil, Dominique Grandmont, Marie-Claire Bancquart, Claude Vigée, Richard Rognet, Charles Dobzynski, Pierre Oster, Lionel Ray, Édouard Maunick, Jean Pérol, Bernard Mazo, Philippe Delaveau, Pierre Dhainaut, Henri Meschonnic, Michel Deguy, Alexandre Voisard, Max Alhau, Claudine Helft...

French writers' organizations

Académie Mallarmé


Abdul Aziz Zakari (born September 2, 1976) is a Ghanaian athlete specializing in the 100 metres. He was born in Accra, Ghana.

Participating in the 2000 Summer Olympics, he made it to the final of the 100 metres, but failed to finish after becoming injured at about the 35m mark.

Also participating in the 2004 Summer Olympics, he achieved second place in his 100 metres heat, thus making it through to the second round. Heading into the second round, he was victorious in a tough sprint, before achieving qualification from his semi-final. This good form was not able to continue, as he failed to finish in the final, staged on August 22, renowned as possibly the fastest collective 100 metre race in history, where six of the seven finishing athletes completed the race in ten seconds or less.

He finally, for the first time, achieved a sub-10 second run on June 14, 2005, in Athens when he ran 9.99 seconds in Asafa Powell's world record breaking race. The Ghanaian record currently belongs to Leonard Myles-Mills with 9.98 seconds.

In July 2006, Zakari was reported to have failed a drugs test for the banned substance stanozolol. The test was conducted In-Competition on 29 April 2006 at the IAAF Grand Prix Meeting in Dakar. On 25 September 2006 the IAAF suspended him for two years. After his suspension Zakari participated at the 2008 Summer Olympics in which he ran the fourth time in his 100 metres heat, behind Richard Thompson, Martial Mbandjock and Simone Collio. His time of 10.34 was the third losing time after the 10.25 of Nobuharu Asahara, advancing him to the second round. There he improved his time to 10.24, but finished in fifth place of his heat, causing his elimination from the competition.

Personal bests

Outdoor

Indoor

See also
 List of sportspeople sanctioned for doping offences

References

External links

1976 births
Living people
Ghanaian male sprinters
Athletes (track and field) at the 1996 Summer Olympics
Athletes (track and field) at the 2000 Summer Olympics
Athletes (track and field) at the 2004 Summer Olympics
Athletes (track and field) at the 2008 Summer Olympics
Olympic athletes for Ghana
Athletes (track and field) at the 1998 Commonwealth Games
Athletes (track and field) at the 2002 Commonwealth Games
Athletes (track and field) at the 2006 Commonwealth Games
Athletes (track and field) at the 2010 Commonwealth Games
Doping cases in athletics
Ghanaian sportspeople in doping cases
Sportspeople from Accra
Commonwealth Games competitors for Ghana
World Athletics Championships athletes for Ghana
African Games gold medalists for Ghana
African Games medalists in athletics (track and field)
African Games bronze medalists for Ghana
Athletes (track and field) at the 2003 All-Africa Games

Aziz Zakari


Michael John Seaton  (16 January 1923 – 29 May 2007) was an influential British mathematician, atomic physicist, and astronomer.

He was born in Bristol, and educated at Wallington County Grammar School (WCGS), a grammar school in Surrey, where he won prizes for his achievements in chemistry.

From 1941 to 1946 he served in the wartime Royal Air Force as a Flight Lieutenant. In this capacity he served first in RAF Bomber Command, navigating Avro Lancasters, and later in one of the elite Pathfinder squadrons, such was his capacity to apply his understanding of mathematics to the task before him.

After demobilisation, he returned to his studies, and the start of a long career at University College London. Gaining a First Class BSc in physics just two years later, he continued, obtaining his PhD on Quantal Calculations of certain reaction rates with applications to Astrophysical and Geophysical problems in 1951. He later did important work on the Quantum Defect Theory.

With a break as Chargé de Recherché at the Institut d'astrophysique de Paris from 1954 to 1955, he rose through the ranks at the Department of Physics at UCL, becoming a Reader in 1959, and Professor of Physics in 1963. He was made a Fellow of the College in 1972, the year in which the Departments of Physics and Astronomy merged. He held the status of Professor Emeritus and Honorary Research Fellow from 1988 until his death.

In 1964 he became Fellow-Adjoint at the Joint Institute for Laboratory Astrophysics (JILA) in Boulder, Colorado, a combined venture between the American National Institute of Standards and Technology and the University of Colorado.

In 1967 he was elected Fellow of the Royal Society. He held Honorary Membership of the American Astronomical Society, awarded in 1983, and was made Foreign Associate of the United States National Academy of Sciences in 1986.

Seaton held the Presidency of the Royal Astronomical Society [RAS] between 1979 and 1981, and was awarded its Gold Medal in 1983. This was followed by the Guthrie Medal and Prize, from the Institute of Physics in 1984, and the Hughes Medal of the Royal Society in 1992.

Seaton served as Senior Fellow to the Science and Engineering Research Council EPSRC, 1984–88.

Other honours include: Honorary Doctorate, Observatoire de Paris, 1976; Honorary DSc QUB, 1982.

References

External links
Obituary, The Times, 7 June 2007
Obituary, The Guardian, 31 August 2007
Obituary, UCL, September 2007
Obituary, Bulletin of the American Astronomical Society, 31 December 2007
Biographical Notes, Harvard-Smithsonian Centre for Astrophysics, August 2006
 Atomic Physics & Astrophysics Group - "Research B - Theoretical" section in: From Lardner to Massey, by JW Fox
 Atoms and Astrophysics: Mike Seaton's legacy - Research meeting held at UCL, April 2008

1923 births
2007 deaths
20th-century English mathematicians
Royal Air Force officers
Royal Air Force personnel of World War II
Alumni of University College London
Academics of University College London
Fellows of the Royal Society
Scientists from Bristol
Recipients of the Gold Medal of the Royal Astronomical Society
People educated at Wallington County Grammar School
Foreign associates of the National Academy of Sciences
English nuclear physicists
20th-century British astronomers
Presidents of the Royal Astronomical Society

M. J. Seaton


A Good Man in Africa is William Boyd's first novel, published in 1981. It won both the Whitbread Book Award for a first novel and the Somerset Maugham Award that year.

Plot introduction
Morgan Leafy is First Secretary to the British Deputy High Commission in Nkongsamba in the fictional West African country of Kinjanja. Leafy's life is becoming increasingly problematic: he is being blackmailed by a local politician, his plan to fix the forthcoming elections has come unstuck, and a coup is looming. In his personal life he has contracted gonorrhea from Hazel, his black mistress who is cheating on him, while Priscilla, his boss's daughter on whom Leafy has lustful ambitions, has just got engaged to his hated underling. To complete his woes Priscilla's father is threatening to dismiss him unless he can dispose of a corpse that has been left to rot in the sun in accordance with tribal laws.

Publication
Morgan Leafy also appears in two short stories, "Next Boat from Douala" and "The Coup" which concern his departure from Africa. The stories appear in the collection On the Yankee Station, published later in 1981, but as Boyd explained in an interview the collection was actually written before the novel, though Boyd claimed he had written both when he sent the collection to potential publishers. Hamish Hamilton agreed to publish the novel (as yet unwritten) and collection in that order, Boyd admits "So I said to my new editor, Christopher Sinclair-Stevenson, 'Look, the manuscript is in a shocking state, I just need a couple of months to knock into shape', and I sat down and wrote A Good Man in Africa in a white heat of dynamic endeavour in three months at my kitchen table.

Inspiration
William Boyd grew up in Western Africa, living in both Ghana and Nigeria. He explains that the setting for the novel "is completely set in Ibadan in Western Nigeria even though I changed the names, but everybody in it is made up. It's rooted in my autobiography in terms of its colour, texture and smells but the story is – and that's something that's always been the case with me – invented. There is an autobiographical element in that the character of Dr Murray is very much a two-dimensional portrait of my father."

Reception
Michiko Kakutani in The New York Times praises the novel, likening it to the work of Evelyn Waugh and Kingsley Amis: "it is as though Lucky Jim had been suddenly transported to the mythical kingdom of Azania in Black Mischief." She concludes, "There are, of course, things a reader might quarrel with: Mr. Boyd's penchant for broad humor and narrative pratfalls makes, at times, for an irritating glibness; and his technical mastery of the novel form obscures the fact that he has yet to develop a voice that is truly his own - the echoes of his predecessors haunt the achievement of this book. Still, this remains a precocious debut indeed, and I eagerly await Mr. Boyd's next novel".
Kirkus Reviews concludes "Boyd can lapse from credible black-comedy into cheap farce. Still, if the worst of this energetic novel is reminiscent of crude sit-corns, the best recalls Waugh and Amis – in a dark yet cheerful nightmare that's juiced along by humiliation, fury, and a highly unsentimental view of post-colonial Africa."

Adaptations
In 1985 BBC Radio 4 broadcast an audio adaptation starring Alan Rickman as Leafy. It was repeated on BBC Radio 4 Extra in 2022.

In 1994 the novel was made into a film of the same name, with a script written by Boyd.

References

1981 British novels
Scottish novels
Costa Book Award-winning works
Novels set in Nigeria
British novels adapted into films
Postcolonial novels
Novels by William Boyd (writer)
Hamish Hamilton books
Novels set in Ibadan
1981 debut novels

A Good Man in Africa (novel)


Isopropenyl acetate is an organic compound, which is the acetate ester of the enol tautomer of acetone.  This colorless liquid is significant commercially as the principal precursor to acetylacetone.  In organic synthesis, it is used to prepare enol acetates of ketones and acetonides from diols.

Preparation and reactions
Isopropenyl acetate is prepared by treating acetone with ketene. Upon heating over a metal surface, isopropenyl acetate rearranges to acetylacetone.

Isopropenyl acetate is used to prepare other isopropenyl esters by transesterification.

References

Acetate esters
Isopropenyl esters

Isopropenyl acetate


Upper Mesa Falls is a waterfall on the Henrys Fork in the Caribou-Targhee National Forest. Upstream from Lower Mesa Falls, it is roughly  away from Ashton, Idaho.

Upper Mesa Falls is roughly  high and  wide.

Formation

Mesa Falls Tuff, which is the rock over which Upper Mesa Falls cascades, was formed 1.3 million years ago.  A cycle of rhyolitic volcanism from the Henrys Fork caldera deposited a thick layer of rock and ash across the area.  This layer compressed and hardened over time.

Between 200,000 and 600,000 years ago, the river eroded a wide canyon which was subsequently partly filled with basalt lava flows. The Henrys Fork of the Snake River then carved the channel through the basalt; which is the inner canyon seen today.

See also

 List of waterfalls in Idaho

References

External links
 Mesa Falls Visitor Center (USDA Forest Service - Caribou-Targhee National Forest)

Landforms of Fremont County, Idaho
Waterfalls of Idaho
Tourist attractions in Fremont County, Idaho
Block waterfalls
Articles containing video clips

Upper Mesa Falls


Paramin is a village located on one of the highest points of western area of the Northern Range in Trinidad, Trinidad and Tobago, which forms part of the Maraval area.  It is a sprawling, steep and mountainous village whose residents have traditionally been farmers, producing herbs like chives, thyme and parsley, as well as vegetables like tomatoes and yams.

Population

Many of the original residents of the area are descendants of the French Creole migrants to the island following the Cedula of Population (1783)  and the surnames reflect this heritage:  Constantine, Boisson, Fournillier (Fournier), Isidore, Laurent (Lawrence), Pascal, Romany, etc.   Following the Cedula of Population, Paramin was peopled by immigrants from Martinique, Guadeloupe, Grenada (La Grenade), Dominica (La Dominique), St. Lucia (Ste. Lucie), etc.

Paramin  was later also populated by Cocoa Panyols  from Venezuela in South America who aided in establishing the cocoa industry.  The Cocoa Panyols  are an Hispanic (Latino) ethnic mixture of mainly European (Spanish and/or French), Amerindian and African ancestry.

Some of the early settlers in the village of Paramin were runaway slaves from the Maraval (Negre Marron Caves) and Diego Martin Valleys (Camp Marron or Cameron).

Names of places within Paramin
Within Paramin many locations have French names including Finette Road  Saut D'Eau, Morne Espoire, Morne Roche, Beau Pres, Morne Rene

Music
The area is also home to two specific music genres—Crèche and Parang  Both are specifically played at Christmas time.  They reflect the French and Spanish influences of the island mixed with Afro/Caribbean beats.  Parang has seen a large resurgence in recent years with mixes of both Soca and even Chutney music with lyrics in English.  The Maraval Folk Choir, known for both Crèche and Parang, produced a Crèche album in the 1970s, and the Paramin Folk Choir produced one in 2004.

Catholicism
The area is predominantly Roman Catholic with many of the residents being practicing Catholics.  Within the village of Paramin are a Roman Catholic primary school and a Catholic church (Our Lady of Guadalupe) that serve the area. The churches within Paramin and Maraval are linked in their record keeping of the Sacraments.

Activity

French Creole or Patois mass on Dimanche Gras
The church still offers a well-known French Creole or Patois mass on Dimanche Gras, the Sunday before Carnival, exploring the steep Paramin hills. Persons from areas outside of Maraval and Paramin, including foreign dignitaries of French origin and French Creole origin such as Louisiana attend the Mass either to experience the culture of the area or to participate in an activity that falls outside of the Carnival setting.

Most of the Mass is said in Creole or Patois. There is a choir which leads the singing of the hymns at the Mass.

Members of the choir and the congregation at the Mass are  dressed in colorful plaids, in the style of La Reine Rive, which is a white cotton top and skirt, with an overskirt in plaid, which drapes to the sides of the front of the skirt and leaves the front of the white skirt visible. A head "tie" also in a matching plaid completes the outfit.

In recent times, given the timing of the Patois Mass in the morning lunch and other snacks are available for sale after the Mass.

In recent times, the Patois mass has been videotaped.

Caliente Monday
Caliente Monday refers to a Monday evening celebration in Paramin of Parang and food. This event has been held for more than thirty years. The event used to be open to the public or those brave enough to go up to Paramin by jeep earlier in an evening when the sun sets earlier in the day and the air gets cold earlier in the evening, just to hear the Parang Bands from across the country sing either their compositions or renditions of popular songs. At the event, residents of Paramin sell provisions and pork, and various types of pastelles.

Over the last few years, the event has been held at the Country Club on Long Circular Road in Maraval.

Hiking
Over the last fifteen to twenty years, persons have hiked to the top of Saut D'Eau (Hill and Waterfall) to experience the ruggedness of the mountain trek.

There are also tours or hikes to the Cave in Paramin. According to Sokah2Soca "it should also be noted that the area is also famous for its bat caves with eye catching limestone formations. It is reported that these caves are some of the most easily accessible caves in the island".

References

Villages in Trinidad and Tobago

Paramin


Microphotography is a writing style that emerged in the early 1990s in science journalism. The style is named after micrographs and is distinctive for its highly detailed, worm's eye or microscopic view of the macroscopic world.

One of the flagship works in this style was David Bodanis's commercially prominent The Secret House: 24 hours in the strange & wonderful world in which we spend our nights and days. This book followed daily life in a typical enclosed human habitat in minute detail, featuring detailed physical and biological explanations.

Notes

External links
Academic Essay Writing

Science writing

Microphotography (literature)


Volney Vallencourt "V.V." Ashford (1844 – 21 March 1900) was an American soldier and involved in 19th-century rebellion in Hawaii.

Military career
Ashford joined the Union Army in 1863 after leaving his home in Port Hope, Ontario. He was a British citizen since Canada was a territory of the United Kingdom. He became a Quartermaster Sergeant in the Grizwold Light Cavalry during the American Civil War. After the Civil War, Ashford, became a surveyor for the Beaverton railway and joined the Prince of Wales' Canadian Dragoons where he made Captain. He was widowed twice in addition to the death of his daughter.  In 1884 his brother and future Attorney General of Hawaii, Clarence W. Ashford, invited him to visit Honolulu. After moving to Hawaii he joined the Honolulu Rifles, a militia. Ashford became responsible for the Rifles' growth from 100 men in 1886 to a battalion of three companies in 1887 and recognition as a paramilitary force. He rapidly moved up the ranks entering with his Canadian rank of Captain, he rose through Major and Lieutenant Colonel becoming Colonel in 1887, the highest rank and commander of the Rifles itself.

Ashford and Annexation of Hawaii

Shortly after becoming Colonel, a group of conspirators within the Missionary Party wanted to annex Hawaii to the United States. Ashford believed the monarchy was corrupt and was recruited to what would be known as the Hawaiian League. The Hawaiian League used the Rifles in the Rebellion of 1887 a bloodless revolution to enact the Bayonet Constitution. In 1889 the Rifles did battle with forces of Robert Wilcox in the Wilcox Rebellion. Upon Wilcox's surrender, he offered Ashford his sabre, he accepted the surrender but turned down the sabre. Ashford began to regret his support for the Hawaiian League. He came to believe that the League was more corrupt than the monarchy at advancing their own interests rather than improving Hawaii. He joined the Liberal Party, a third party that supported neither the Monarchy nor the Missionaries. Ashford was arrested in a plot to overthrown both the Monarchy and Bayonet Constitution known as the Burlesque Conspiracy and went into exile. His brother took his place in his absence as commander of the Honolulu Rifles. After the Overthrow of the Hawaiian monarchy, Ashford returned to Hawaii and retired from his position as commander of the Rifles. He was a strong supporter of annexation, but with the establishment of the Republic of Hawaii he gave little support for the new government which he believed was corrupt and later revealed at his military tribunal, disapproved of the mistreatment of Hawaiians by the government.  Ashford was approached by counter-revolutionaries as a consultant to restore the monarchy.

Exile and illness

The counter-revolution ended in failure in 1895 and Ashford was charged with misprision of treason and received one year in prison and a $1000 fine. Due to his ill health he was given the option of the one-year sentence or become exiled (if he returned he would receive his original sentence). After accepting exile he moved to California, his brother moved in with him under similar circumstances of exile. Ashford died on March 21, 1900, in Oakland, California.

References

On a Bayonet Throne by Neil B. Dukas

People associated with the overthrow of the Hawaiian Kingdom
Prisoners and detainees of the Republic of Hawaii
People from Northumberland County, Ontario
Hawaiian insurgents and supporters
1844 births
1900 deaths
National Liberal Party (Hawaii) politicians
Exiles from Hawaii
Hawaiian Kingdom military officers
Union Army soldiers
Canadian Militia officers
Canadian emigrants
Immigrants to the Hawaiian Kingdom
Canadian expatriates in the United States

Volney V. Ashford


Pakistan is the second largest Muslim country in terms of population (behind Indonesia). Pakistan is the only Muslim country known to have nuclear weapons. Pakistan is also an important member of the Organization of the Islamic Conference (OIC).

Pakistan is an active member of the United Nations. Pakistan had some problems with its Foreign relations. Pakistan's relations with its neighbor, India is ok, but mostly not good. Both countries have been fighting over the disputed territory of Kashmir,over which they have fought two to three wars. Pakistan has also had difficult relations with neighbors Afghanistan and Iran. Pakistan has had a long and good relationship with the China. Pakistan has interest over the Persian Gulf and has wide-ranging bilateral relations with the United States and other Western countries.

Pakistan distrusted the Soviet Union during much of the Cold War. Because of this Pakistan had strong relations with both the United States of America and the People's Republic of China.

Pakistan was a member of the American-sponsored CENTO and SEATO military alliances. Pakistans' alliance with the United States was especially close after the Soviets invaded the neighboring country of Afghanistan. In 1964, Pakistan signed the Regional Cooperation for Development (RCD) Pact with Turkey and Iran, when all three countries were closely allied with the U.S., and as neighbors of the Soviet Union, wary of perceived Soviet expansionism. To this day, Pakistan has had a close relationship with Turkey. RCD became defunct after the Iranian Revolution, and a Pakistani-Turkish initiative led to the founding of the Economic Cooperation Organization (ECO) in 1985. For several years, Pakistan's relations with the Republic of India had been gradually improving, which opened up Pakistan's foreign policy to issues beyond security.

Foreign relations of Pakistan

Foreign relations of Pakistan


Almaty Region (; ) is a region in Kazakhstan, located in the southeastern part of the country. Its capital, from 1997 to 2022 was the city of Taldyqorğan. But with the creation of the new Jetysu Region in 2022, Taldyqorğan was chosen to be its capital and the capital of Almaty region was moved to the city of Kunayev.

Geography
Almaty Region surrounds the city of Almaty. The region borders Kyrgyzstan and Xinjiang in the People's Republic of China. The region also touches three other regions of Kazakhstan: Jambyl Region to the west, Karaganda Region to the northwest, and East Kazakhstan Region to the north. Almaty Region has an area of 224,000 square kilometres.

Much of the northwestern border of the region runs along  Lake Balkhash, whose main affluent, the  Ili River, is the most significant river of the region. In the region's northeast, it shares the four lakes of the Alakol Depression (Lakes Alakol, Sasykkol, Koshkarkol, and Zhalanashkol) with East Kazakhstan Region. Other lakes within the region include Bartogay Lake and Kapshagay lakes, Lake Issyk, Kaindy and Kolsai lakes.

The Trans-Ili Alatau branch of the Tian Shan Mountains extends from China and Kyrgyzstan into Almaty Region, along its southern border with Kyrgyzstan. Farther to the northeast, the Dzungarian Alatau runs along the region's border with China.
The Charyn Canyon is one of the region's famous scenic spots.

History
Alma-Ata Region,  the predecessor of today's Almaty Region, was created from the historical region of Zhetysu on March 10, 1932. Its capital was Alma-Ata (Almaty). Several times during the Soviet period, the north-eastern part of the region, centered on Taldyqorğan, was separated from Alma-Ata Region, forming a separate Taldy-Kurgan Region, only to be merged into Alma-Ata Region again several years later.

In April 2001 the administrative center of Almaty Region was moved from Almaty to Taldyqorğan. Thus Taldyqorğan has become a regional capital again - this time without splitting the region.
Almaty region was formed on March 10, 1932 with a center in the city of Almaty. The region included 18 districts.
At the end of 1939, part of the districts was transferred to the Dzhambul and Semipalatinsk regions, 23 districts remained in the Alma-Ata region, including the newly formed 7.
In March 1944, Taldy-Kurgan Region was allocated from its structure, to which 11 districts were transferred.

Heads of the region
 Baiken  Ashymov  1968—1970
 Bimende Sadvokasov	1970—1972
 Aripbay  Alybaev 1972—1978
 Sakan  Kussainov 1978—1982
 Abubakir  Tynynybaev 1982—1986
 Vladislav  Anufriev 1986—1988
 Anatoly Zhigulin 1988—1990
 Saginbek  Tursunov 1990—1993
 Umirzak Uzbekov (1994-1996)
 Serik  Ahimbekov 1996—1997
 Zamanbek Nurkadilov (December 1997 - May 2001)
 Shalbai Kulmakhanov (May 2001 - August 11, 2005)
 Serik  Umbetov (2005-2011)
 Anzar Musakhanov (from April 13, 2011 to August 20, 2014)
 Amandyk Batalov (from August 20, 2014)

Awards
On October 28, 1966, Alma-Ata region was awarded by Order of Lenin for its success in increasing the production and procurement of grain, meat, milk and other agricultural products.
On December 3, 1970, Alma-Ata region was awarded by the second Order of Lenin for the great successes achieved by the working people of the region in fulfilling the five-year plan for the development of industrial and agricultural production.

Administrative divisions
The region is administratively divided into seventeen districts and the cities of Konaev, Taldyqorğan, and Tekeli.
 Aksu District, administrative center is Zhansugirov;
 Alakol District, the administrative center is Üşaral;
 Balkhash District,  the administrative center is Bakanas;
 Enbekshikazakh District, the administrative center is  Esik;
 Eskeldi District, the administrative center is Karabulak;
 Ile District,  the administrative center is  Otegen Batyr;
 Karasay District, the administrative center is Kaskelen;
 Karatal District, the administrative center is Ushtobe;
 Kerbulak District, the administrative center is Sary-Ozek;
 Koksu District, the administrative center is Balpyk Bi;
 Kegen District, the administrative center is Kegen;
 Panfilov District, the administrative center is Zharkent;
 Raiymbek District, the administrative center is Narynkol;
 Sarkand District, the administrative center is Sarkand;
 Talgar District, the administrative center is Talgar;
 Uygur District, the administrative center is Chundzha;
 Zhambyl District, the administrative center is Uzynagash
 Taldyqorğan city
 Konaev city
 Tekeli city

Climate
The natural conditions of the Almaty region include 5 climatic zones - from deserts to eternal snows. The climate is sharply continental, the average January temperature in the lowlands is -15 C, in the foothills - 6-8 C; July - +16 C and + 24 + 25 C, respectively. The annual rainfall on the plains is up to 300 mm, in the foothills and mountains - from 500-700 to 1000 mm per year.

Ethnic composition
The share of Kazakhs, Russians, Uigurs by region census of 2009 year.
Subsequently, the ethnic composition underwent strong changes with the advent and collapse of the Soviet Union. At present, it is still multi-ethnic, however, Kazakhs absolutely prevail (72.07%), the share of Russians has greatly reduced to 13.33% of the region's inhabitants. The number of Uigurs is also significant (7.73%). The number of other Asian peoples - Turks, Kurds, Uzbeks, Kyrgyz, as well as Koreans does not exceed 4% in total. The numerous communities of Germans, Greeks, Poles, Ukrainians, and other European peoples, who were once living in Kazakhstan during the Soviet era, significantly decreased due to mass emigration to other countries after 1991.

Demographics

As of 2020, the Almaty Region has a population of 2,055,724.

Ethnic groups (2020):
Kazakh: 72.32%
Russian: 13.12%
Uyghur: 7.74%
Turkish: 1.84%
Azerbaijani: 0.86%
Others: 4.12%

Note: the population and ethnicity figures exclude the City of Almaty, which is administratively separate from the Almaty Region.

Historical Ethnic Composition
In the 1989 census (excepting Alma-Ata):
Kazakhs: 54.2%.
Russians: 28.6%.
Uigurs: 7.8%.
Others: 9.4%.

In the 1999 census (excepting Almaty):
Kazakhs: 66.4%.
Russians: 23.8%.
Uigurs: 7.0%.
Others: 4.6%.

In the 2009 census (excepting Almaty):
Kazakhs: 68.2%.
Russians: 22.8%.
Uigurs: 6.2%.
Other: 3.8%.

Sights
In 2021, the mountain cluster of the Almaty Region was selected as one of the top 10 tourism destinations in Kazakhstan. Other popular sights include:
Lake Alakol - healing lake
Burkhan Bulak - highest waterfall
Altynemel - national park
Besqaynar - Ski resorts located 1500 metres above sea level
Kolsay
Khan Taniry
Kaindy
Balkhash
Tamgaly - a World Heritage Site (petroglyphs)
Charyn Canyon - Charyn National Park
Talgar - a region where the Issyk kurgan is located
Medeo - one of the most famous ice rinks in the world
Dostyk - a point of entry from China to Kazakhstan
Kapchagay Dam on the Ili River
Also in 2021, the tourism department of Kazakhstan announced that 15 huts for tourism shelter equipped with Wi-Fi, charging stations, and first-aid kits are planned to be constructed in the Almaty mountain area in the near future. The huts will be located in the middle and left Talgar, Kimasar, Tuyuk-Su, Upper Gorelnik, Big Almaty, Shukur and Prokhodnoye gorges.

Culture

- Taldyqorğan Drama Theatre named after B.Rimova
Drama Theatre named after B.Rimova was opened on November 4, 1975. A group of young graduates of the Almaty State Conservatory named after Kurmangazy, who graduated from class of People's Artist of Kazakhstan Sholpan Zhandarbekova, opened the curtain of the theatre with the performance of G. Musrepov "Kozy Kurpesh-Bayan Sulu".
On August 31, 2000, the theatre was named after People's Artist of Kazakhstan Biken Rimova, laureate of State Prize.
On October 30, 2018, they moved to a new building.

-«Museum of Applied Arts named after Master Darkembay»  PUSE
The total area is 3.0 hectares. Internal area of the museum - 375 square meters. The number of exhibits - 2437. The museum consists of 6 exhibition halls.
The number of people for excursion is 25 per a group.
Address: Almaty region, Enbekshikazakh district, Akshi village, Darkembai  avenue, house 1.

- Museum «Batyr babalar»
(Heroic ancestors)
The museum was built in 2016. It was opened in 2017within the framework of the program of the First President of the Republic of Kazakhstan Nursultan Nazarbayev «Course towards the future: modernization of Kazakhstan's identity».

- Ilyas Zhansugurov literary museum
It was opened in 1984 on the occasion of the 90th anniversary of the poet I. Zhansugurov. The museum is dedicated to the life and work of the poet.
The building of museum was constructed in 1905-1907. From 1918 to 1976 it was used as various institutions of the Soviet government: district branch of the regional union «Kosshy», the editorial office of the regional newspaper «October Flag», a laboratory for insemination of livestock, in 1976-1980 the regional museum of local lore. The total area of the museum is 271 square meters. The total number of exhibits is 6407. The building is under state protection as an example of architecture of the 19th century.

- Museum of victims of political repression
The building of  museum was constructed in 2018 in accordance with the decision of the akim of Almaty region A. Batalov in the framework of the program «Course towards the future: modernization of Kazakhstan's identity» of the First President of the Republic of Kazakhstan N. Nazarbayev.
The total area of the building is 535 square meters. Museum departments:
1. The history of repression in Kazakhstan in 1916-1986;
2. In memory of victims of political repression;
3. Historical and educational society «Adilet»  and its activities.
Museum funds are being replenished with materials relating to citizens who were subjected to political repression from the 1920s to the 1950s.

- Museum «Respect for Mother»
The museum was built in 2014, opened in 2015. The building area is 4987 square meters. Departments of the museum: on 2 floors there are 2 thematic halls:
1. craft and jewelry
2. Mother heroines.
Design capacity: 1 group of 30 people.

- «Abylkhan Kasteev Fine Art Gallery»
The museum was created in 2004 for the 100th anniversary of the national artist Abylkhan Kasteev.
Total area - 427.0 square meters, Internal area of the gallery - 1200 square meters. The number of materials of museum significance is 166. The gallery consists of 9 rooms.
The number of people on excursion is 10 per group.
Address: Almaty region, Panfilovsky district, Zharkent, Yuldashev street, house number 9

- «Gallery of Fine Art» PUSE
It was founded in 2003. The number of materials of museum significance is 404. The total area is 1011.9 square meters, of which 10 exhibition halls - 674.0 square meters.
The number of people on excursion is 45 - 50 people per a group.
The gallery works in five areas:
• Research and assembly work.
• Scientific and fund work.
• Scientific and expositional work.
• Cultural and educational work.
Work with the media
Address: Almaty region, Taldyqorğan, st. Kabanbay batyr 27 A

- PUI «Historical and Cultural Center «Atameken» of the First President of the Republic of Kazakhstan-Elbasy»
Put into operation on December 1, 2017.
All the exhibits collected in the golden fund of the center tell about the life of President Nursultan Nazarbayev from his childhood to the present day, when he became the Leader of independent Kazakhstan.
The total area is 3.5000 ha. The internal volume of the museum is 1249.5 square meters. The number of materials of museum significance is 649.
The center consists of 4 exhibition halls, there is a conference room for 16 seats.
Address: Almaty region, Karasai district, village Shamalgan, Zhibek Zholy street 158 / A.

http://kultura.zhetisu.gov.kz/

References

External links

Official regional website

Regions of Kazakhstan

Almaty Region


The Royal Standard of Scotland, (), also known as the Banner of the King of Scots, or more commonly the Lion Rampant of Scotland, is the Scottish Royal Banner of Arms. Used historically by the King of Scots, the Royal Standard of Scotland differs from Scotland's national flag, the Saltire, in that its correct use is restricted by an Act of the Parliament of Scotland to only a few Great Officers of State who officially represent the Sovereign in Scotland. It is also used in an official capacity at royal residences in Scotland when the Sovereign is not present.

The earliest recorded use of the Lion rampant as a royal emblem in Scotland was by Alexander II in 1222; with the additional embellishment of a double border set with lilies occurring during the reign of Alexander III (1249–1286). This emblem occupied the shield of the  royal coat of arms of the ancient Kingdom of Scotland which, together with a royal banner displaying the same, was used by the King of Scots until the Union of the Crowns in 1603, when James VI acceded to the thrones of the Kingdom of England and Kingdom of Ireland. Since 1603, the Lion rampant of Scotland has been incorporated into both the royal arms and royal banners of successive Scottish then British monarchs in order to symbolise Scotland; as can be seen today in the Royal Standard of the United Kingdom. Although now officially restricted to use by representatives of the Sovereign and at royal residences, the Royal Standard of Scotland continues to be one of Scotland's most recognisable symbols.

National Flag of Scotland

The Flag of Scotland, also known as the Saint Andrew's Cross or more commonly The Saltire, is the national flag of Scotland. The Saltire is the correct flag for everyone to fly in order to show both their loyalty and Scottish nationality. It is also, where possible, flown from Scottish Government buildings every day from 8am until sunset, with certain exceptions; for example United Kingdom National Days.

References

Flags of the United Kingdom
13th century in Scotland
1222 establishments
Establishments in Scotland
1220s establishments in Europe
Lions
National symbols of Scotland

Royal Banner of Scotland