Effects of imperfect angular adjustment on plasmonic force
Leonid Frumin Alexander Tusnin Oleg Belai David Shapiro
physics.optics

	
	

Institute of Automation and Electrometry,  Russian Academy of Sciences, Siberian Branch, 1 Koptjug Avenue, Novosibirsk 630090, Russia
Novosibirsk State University, 2 Pirogov Street, Novosibirsk 630090, Russia

Institute of Automation and Electrometry, Russian Academy of Sciences, Siberian Branch, 1 Koptjug Avenue, Novosibirsk 630090, Russia
Novosibirsk State University, 2 Pirogov Street, Novosibirsk 630090, Russia

Institute of Automation and Electrometry,  Russian Academy of Sciences, Siberian Branch,  1 Koptjug Avenue, Novosibirsk 630090, Russia

Institute of Automation and Electrometry,  Russian Academy of Sciences, Siberian Branch, 1 Koptjug Avenue, Novosibirsk 630090, Russia
Novosibirsk State University, 2 Pirogov Street, Novosibirsk 630090, Russia



The attractive plasmon force between two metallic walls when the electromagnetic wave propagates through a narrow slit has been studied earlier for parallel plates and normal incidence. In present paper the effects of imperfect adjustment of plates and laser beam are analyzed. The change of force for non-parallel plates is shown to be of the first order in inclination angle when the wedge is along wave propagation and of the second order for transverse case. The small incidence angle leads to decrease in force due to the antisymmetric waveguide mode appearing in the slit.
	
03.50.De,42.50.Wk,85.85.+j







Effects of imperfect angular adjustment on plasmonic force
    David Shapiro
    December 30, 2023
==========================================================



§ INTRODUCTION


Since the 1980s, Micro-Electro-Mechanical Systems (MEMS) have rapidly developed and become an essential part of the technology for micro-devises <cit.>. This industry produces controllers, accelerometers, gyroscopes having various applications, from common cell phones <cit.> to biosensors <cit.>. Today a new type
of fast micro-controllers appears as a merge of MEMS and micro-optics: Micro-Opto-Electro-Mechanical Systems (MOEMS).
Being based on the interaction between electromagnetic field and solid body MOEMS provides new possibilities in the nano- and micro-particles manipulation.

In the case of metallic particle the surface plasmon polaritons are excited at the interface metal-dielectric, where the real part of dielectric function change its sign, and cause the plasmon force <cit.>.
The forces, with collective plasmon resonances excited by the laser field, are  studied experimentally for the dielectric sphere near a conducting plate <cit.> and theoretically between two close metallic nanospheres <cit.>. They are challenging for optical trapping and laser tweezers <cit.>. In particular case of slit between plane-parallel metallic plates the attractive plasmon force has been predicted <cit.>
with properties determined by the geometry, conductivity, and the light polarization. Magnitude of this force for gold walls and normal incidence is of the order of nanonewtons, hence it becomes an important experimental issue. To study this effect closely an experiment is carried out with the Nano Force Facility <cit.>. At the same time previous theoretical studies did not consider the possible experimental uncertainties such as  misalignment of the plates or laser beam.

The goal of present paper is to calculate corrections to the force at small deviation from the plane-parallel geometry. In the first order the perturbation is a sum of corrections over different uncertainties, and then the additions can be estimated separately. In Sec. <ref> the wedge  parallel (A) and perpendicular (B) to the propagation is studied. The oblique incidence is treated in Sec. <ref>. The field (A) and Maxwell's tension (B) are calculated within the lower modes approximation.
Sec. <ref> summarizes the results.



§ NON-PARALLEL PLATES



The light-induced force had been studied theoretically for normal incidence in plane-parallel geometry <cit.>. The geometrical scheme is shown in Fig. <ref>.  The plane wave has p-polarization with magnetic field vector along axis y, since only this state excites surface plasmons. The parallel metal plates at the distance 2ℓ are considered as infinite in x,y,z directions. The wavelength of radiation λ is assumed much greater than the half-width ℓ. In the opposite case waveguide modes are excited, and then decrease the amplitude h_0 of field at the slit entrance <cit.>. Plasmons are generated at the surface of metal with dielectric permittivity _M=_1+i_2, where _1<0,|_1|≫_2∼1.

Remind the main relations for a perfect plane-parallel slit. Write electric and magnetic fields of monochromatic wave with frequency ω as

    E_i=e_i e^-iω t+c.c.,   H_i=h_i e^-iω t+c.c.,

where i=x,y,z is the Cartesian index, c.c. means complex conjugated terms. The Maxwell tension tensor is

    σ_xx=|e_x|^2-|h_y|^2-|e_z|^2/4π≃h_0^2e^-2β_2z/4π√(|_1|)k_0l,

where k_0=ω/c is the wavenumber of incident field in free space, c is the speed of light, h_0 is the amplitude.
The propagation constant in the slit along z is β=β_1+iβ_2, where

    β_1≃ k_0+1/2l√(|_1|),  β_2=_2/4l|_1|^3/2.

If the plates are infinite along z the unperturbed force per unit length in y-direction of is

    f_0≃σ_xx/2β_2=
    h_0^2λ|_1|/4π^2_2.

The value h_0 at the entrance of slit (x,z)=(0,0) is determined by the interference of incident and reflected waves. In the limiting case  of very narrow slit ℓ/λ≪1 their amplitudes are equal and h_0=2H_0, where H_0 is the amplitude of plane incident wave H_y(r⃗)=H_0e^ik⃗·r⃗. For wider slit h_0 has oscillating ℓ-dependence die to excitation of higher waveguide modes <cit.>.



 §.§ Longitudinal wedge






Let the walls are inclined symmetrically, as shown in Fig. <ref> (a), where α≪1 is the wedge half-angle. The walls  can be described by a pair of linear equations
x=ℓ_0+α z, x=-ℓ_0-α z. The Helmholtz equation for magnetic field in (x,z)-plane is

    (∂_x^2+∂_z^2+k^2)
    h=0,

where for p-wave h=h_y, k^2= k_0^2, =1 in the slit and  _M in the metal. We introduce the polar coordinates with the center at intersection point of the walls extensions, Fig. <ref>. The Helmholtz equation (<ref>) in free space φ^2<α^2  reduces to

    [1/r∂/∂ r(r∂/∂ r)+
    1/r^2∂^2/∂φ^2+k_0^2]H=0,

where k_0=ω/c.
We look for the solution of the form
h(r,φ)=R(r)Φ(φ) and get from (<ref>):

    (rR')'/rR+1/r^2Φ”/Φ+k_0^2=0.


The separation of variables occurs; the angular equation has form Φ”+μΦ=0, where μ is the separation parameter. At the boundaries of perfect conductor (lnΦ)'(φ=±α)=0.  The even and odd modes with μ=-π^2m^2/α^2 are:

    Φ_m(φ)=cosπ mφ/2α,   m=0,2,…,
    sinπ mφ/2α,   m=1,3,…

The general solution is a Fourier series, the decomposion over angular eigen functions
Φ_m(φ). Zero mode m=0 is a constant Φ_0=1. The norm of solution is determined by the magnetic field h_0 at the entrance, like in unperturbed Eq. (<ref>). For small α we can approximately assume that the entrance is bounded from below by a horizontal line, and not by an arc.

For real metal the solution is more complicated, since the boundary conditions are different, namely, continuity of weighted normal derivative. Denoting separation parameter μ=-p^2 in the slit and μ=-p_M^2 in the metal we take into account the zero mode only for the sub-wavelength slit. Angular function for the zero mode at φ>0 is

    Φ_0(φ)=cosh(pφ/α),   φ<α,
    exp(-p_Mφ-α/α),   α<φ.

At φ<0 the solution can be continued as an even function.
Then the radial equations in free space and metal are

    R”+1/rR'+(k_0^2+p^2/α^2r^2)R=0,  
    R”+1/rR'+(k_M^2+p^2_M/α^2r^2)R=0,

respectively. In order to provide the solution as a cylindrical wave at infinity one must chose the Hankel function of the second kind:

    R(r)=
    H^(1)_ip/α(k_0r),   φ<α,
    
    H^(1)_ip_M/α(k_Mr),   α<φ.

We find the first relation between p and p_M  equating the weighted logarithmic derivatives of angular function (<ref>):

    ptanh p=-p_M/_M.


The second relation follows from radial solution (<ref>). Since the entrance at α≪1 is far from the polar coordinate origin r_0=ℓ_0/α, where ℓ_0 is the slit half-width, we have to use asymptotic formula for Hankel function (<ref>). 
There are different asymptotic expansions of cylindrical functions H^(1)_ν(ξ). The choice of asymptotics is dictated by the relative rate of increasing in argument ξ and parameter (the order) ν. We need the Debye expansion <cit.> while both the argument and parameter tend to infinity at fixed ratio ν/ξ: 

    H_ν^(1)(ν/cosψ)≈√(2/πνtanψ)exp[iνtanψ-iνψ-iπ/4],  ν→∞.


Changing the variable r=ℓ_0/α+z we write the argument of Hankel function in asymptotic formula  (<ref>) as

    k_0(ℓ_0/α+z)
    =ν/cosψ.

We expand ψ to the Taylor series over the powers of z 
limited ourselves by the main terms 
provided z≪ r_0 or

    α z≪ℓ_0.

Factor at the exponent (<ref>) in main order does not depend on z and is included in an arbitrary wave amplitude A. The zeroth order in the exponent is also included in the amplitude. In the first approximation, the wave is described by the expression:

    H_ν^(1)(ν/cosψ)≈
    Ae^iην z,  η=.d(tanψ-ψ)/dz|_z=0.

Differentiating expression (<ref>), we get

    H_ν^(1)(ν/cosψ)≈
    Aexp(± iβ z ),  β = 1/ℓ√(a^2+p^2).

Here a=k_0ℓ<1 is the dimensionless small parameter.
Analogously, for metal we have β=√(ε a^2+p_M^2)/ℓ, where ℓ(z)=ℓ_0+α z is the slit half-width at height z, hence 

    p^2+a^2=p_M^2+_Ma^2.

This is the second relation that means the equal propagation constants along z in metal and free space 
or equal phase velocities. Substituting p_M from (<ref>) to (<ref>) we obtain the dispersion relation

    ptanh p=-√(p^2+a^2(1-_M))/_M.

The similar relation has been derived for coupled surface plasmons at the parallel boundaries of metal plate surrounded by free space <cit.>.

The dispersion relation turns to be the same as for plane-parallel plates. The only difference consists in z-dependence of the half
width ℓ(z). Denote L_z the height of the plates in z-direction. The variation of width up to L_z should be small (<ref>), i.e.

    α L_z≪ℓ_0.

For α≠0 the Maxwell's tension  (<ref>) can be integrated  over z:

    f=2f_0∫_0^L_zexp(
    		--2β_2z/1+α z/ℓ_0)/1+α z/ℓ_0β_2 dz
    =
    	2f_0β_2e^-2β_2ℓ_0/α/α[
    	(-2β_2ℓ_0/α)-
    	(-2β_2ℓ_0/α+α^2L_z/ℓ_0)
    	],

where f_0 is the unperturbed force (<ref>),  and is the integral exponent <cit.>:

    (ζ)=∫_-∞^ζe^t/tdt.


For small angle α≪1 the arguments of integral exponents become large parameters. The asymptotic of integral exponent gives a series for force (<ref>): 

    f=f_0(1-e^-2β_2L_z){1+α[(2β_2L_z)^-1+1+(2β_2L_z)]
    }+O(α^2).

A factor in front of curly bracket is a force between finite
parallel walls, the next term is linear correction for inclination:

    δ f=-α f_0/2ℓ_0β_2L_z^2.

The correction is negative, that means the decreasing force in broadened slit. Fig. <ref> shows the dependence on  L_z. With decreasing angle α the curves tend to unperturbed dependence. In the domain of short slit the curves lie below the solid line, that corresponds to the negative correction (<ref>). For tall slit, where the curves intersect solid line, our Taylor series obtained for small z is not valid, since condition (<ref>) being violated. 





 §.§ Transverse wedge


We consider V-slit with wedge perpendicular to wave vector, Fig. <ref> (b). Let us introduce the new variables at y>0

    x'=ℓ x/ℓ+α y,   y'=y

to make the plates parallel in new coordinates.
The first-derivative operators transform to

    ∂_x=ℓ/ℓ+α y∂_x',  ∂_y=-αℓ x/(ℓ+α y)^2∂_x'+∂_y'.

Hereafter the half-width ℓ≡ℓ_0 is independent of z.
Then the Laplace operator in old coordinates
acquires crossed terms in non-orthogonal varables:

    △=ℓ^2/(ℓ+α y)^2[
    1+α^2 x^2/(ℓ+α y)^2]∂_x'^2
    +∂_y'^2-2αℓ x/(ℓ+α y)^2∂_x'∂_y'+2α^2ℓ x/(ℓ+α y)^3∂_x'+∂_z^2.


For α≪1 the Migdal's perturbation theory <cit.> can be applied for geometry slightly different 
from exactly solvable (see also <cit.>).
We seek the solution to (<ref>) as a series:

    h(x',y',z)=g_0+α g_1+α^2g_2+…

Substituting (<ref>) into (<ref>) and equating terms with the same power of α we get the chain.  Zero order yields the unperturbed equation (△+k^2)g_0=0 with the known solution

    g_0=h_0e^iβ zcoshϰ x',   0<x'<l,
    coshϰ le^-ϰ_M(x'-l),   l⩽ x'.

The first order is (△+k^2)g_1∝∂_x'∂_y'g_0=0, since the zero-order solution is independent of y', then g_1=0.  Correction g_2 has the second order in α and may be neglected. We do not present the second-order term here since it is negligibly small at α≪1.



§ OFF-NORMAL INCIDENCE


As it was shown in the previous studies <cit.>, excitation of the zero mode in the narrow slit (k_0 l < 1) leads to attraction between the plates. Since in this case electro-magnetic pressure is proportional to |β^(0)|^2 - |k_0|^2 (where β^(0) is a propagation constant of this mode), the fact that |β^(0)|>k_0 plays a crucial role. But for the others modes |β^(i)|<k_0, and then they are evanescent. Deviation of angle γ between k⃗_0 and surface normal increases the contribution of odd modes, and then  decreases the attraction. We aim to calculate the effect of off-normal incidence below taking into account the first antisymmetric mode.



 §.§ Field in the slit

Here we consider an incident p-wave, Fig. <ref>. The incident plane wave has only y-component of the magnetic field, and the wave vector k⃗_⃗0⃗ = (k_x, 0, k_z). The total field is H(x,z,t) = H(x,z)e^iω t+c.c. Solution of the problem is based on  the two-dimensional Helmholtz equation (<ref>) with the boundary conditions: continuity of tangential components of electric and magnetic fields at the boundaries.

We imply Fourier-transformation to the field in order to get algebraic equations instead of differential. Thus, the field in the slit transforms into Fourier series

    H^> = ∑_ν h_ν b_ν(x) e^iβ^(ν) z,

where b_ν(x) = cosq_νx (sinq_νx) for the even (odd) modes, β^(ν)2 = k_0^2-q_ν^2.
The field in free space transforms into Fourier integral:

    H^< = (e^i k_0z z+R e^-i k_0z z) e^i k_0x x + ∫ a_k e^i k x - i κ z dk,


    R=cosγ-√(-sin^2γ)/cosγ+√(-sin^2γ)

is the Fresnel reflection coefficient for ℓ→0, κ^2 = k_0^2-k^2.

The boundary conditions in the slit reveals dispersion equations for even modes:

    tan(q_νl) = √((1-ε)a^2 - (q_νℓ)^2)/ε q_νℓ,

for odd modes the trigonometric function has to be replaced: tan→ -.

As it was shown for perfect conductor, in the case of sub-wavelength slit and normal incidence it is sufficient to consider only the first mode, and the others are negligible <cit.>. We imply the same assumption to the odd modes to simplify the calculations. We consider only the first even (h_0) and odd (h_1) modes:

    H^> = h_0cosh(q_0 x) e^iβ^(0) z + h_1sin(q_1 x)e^iβ^(1) z,

where β^(0),β^(1) are the propagation constants of zero and first modes, respectively.
To satisfy the conditions at z=0, we follow the procedure  <cit.> E_x^<(x) = E_x^>(x) expressing a_k in terms of h_0 and h_1. Then the continuity condition H^<(x)=H^>(x), |x|<l gives coefficients


    h_0 = (1+R)f_q_0,k_0x + ℓ k_0z(1-R)/π∫((k_0x-k)ℓ) f_q_0,k/κ dk /f_q_0,q_0 + β^(0)ℓ/(2π){∫f_q_0,k f_q_0,k/κ dk + cosh(q_0 ℓ)/ε∫G^ev(q_0M,k) f_q_0,k/κ dk  },
    
    h_1 = (1+R)f_q_1,-k_0x + ℓ k_0z(1-R)/π∫((k_0x-k)ℓ) f_q_1,-k/κ dk /i f_q_1,q_1 + β^(1)ℓ/(2π) {∫f_q_1,k f_q_1,-k/κ dk + sin(q_1 ℓ)/ε∫G^odd(q_1M,k) f_q_1,-k/κ dk  }.




    f_q_0,k = ((i q_0+k)ℓ) + ((iq_0 - k)ℓ),
    
    f_q_1,k = i(((q_1 + k)ℓ) - ((q_1 - k)ℓ)),
    
    G^ev(x,y)= 2(xcosyℓ - 2ysinyℓ)/ℓ(x^2+y^2),
    
    G^odd(x,y)=-2i(xsinyℓ + 2ycosyℓ)/ℓ(x^2+y^2)

and q_iM =√(β^(i)2-_M k_0^2), i=0,1 is the transverse wave vector in metal.

If γ≪ 1 it is possible to estimate the amplitudes. For perfect metal q_0 ℓ→ 0, q_1 ℓ→π/2 then

    h_0 ≈ 2-(k_0ℓγ)^2/3, h_1 ≈  2 i  k_0ℓγ.

Hence, the amplitude h_1∼γ while the correction to h_0 is of the order of γ^2.



 §.§ Maxwell tension tensor




Maxwell's tensor (<ref>) determines attractive force between the walls. Non-zero γ leads to excitation of the odd modes in the slit. But according to orthogonality of the even and odd modes, the set of equations on the Fourier coefficients splits into the two independent sets. We calculated the amplitudes of lower even and odd modes h_0,h_1. The presence of second mode adds the interference term h_0^*h_1 of the first order in γ.

We find the dependence of σ_xx on k_0ℓ for different γ numerically. Tension σ_xx is shown in Fig. <ref>  for gold as a function of parameter k_0ℓ. The first asymmetric mode decreases the force, especially for wider slit. Fig. <ref> demonstrates that the attraction can be changed by repulsion at higher width. When the curve intersects axis x an equilibrium  distance appears being large for small angles and decreasing with γ.





§ CONCLUSION


Three possible experimental uncertainties are considered: non-parallel plates along or transverse to wave propagation and off-normal incidence. We calculate their contribution to the force acting between the plates. Since in the first order force perturbation is a sum of all corrections, we calculate them separately. We get the analytic expression of field for wedge along the wave propagation. Correction to the force is of the first order in angle. For the transverse wedge  the correction is shown to be of the second order. Then it is necessary first to consider the contribution of longitudinal wedge. For non-parallel plates the sub-wavelength assumption a=k_0ℓ<1 allows us to consider only the zero mode, but for the off-normal incidence we have to include the first odd mode. We calculate numerically the pressure acting on the walls.  Presence of the first odd mode decreases the pressure and could change attraction to repulsion.


We are grateful to V. Nesterov for excellent discussion of experimental parameters.
This work is supported by the Russian Foundation for Basic Research (# 16-52-12026)
and The Council for grants of President of Russian Federation (NSh-6898.2016.2).











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	[Gardner and Varadan(2001)]Gardner:2001
	author author J. W. Gardner and author V. K. Varadan, @noop  title Microsensors, Mems and
				Smart Devices (publisher John Wiley & Sons, address New York, year 2001)NoStop
	[Rebeiz and Muldavin(2001)]969936
	author author G. M. Rebeiz and author J. B. Muldavin, @noop  journal journal IEEE
				Microwave Magazine volume 2, pages 59
		(year 2001)NoStop
	[Rebeiz(2004)]rebeiz2004rf
	author author G. M. Rebeiz, @noop  title RF MEMS: theory, design,
				and technology (publisher John Wiley & Sons, address New York, year 2004)NoStop
	[Ko(2007)]ko2007trends
	author author W. H. Ko, @noop  journal journal Sensors
				and Actuators A: Physical volume 136, pages 62 (year 2007)NoStop
	[Grayson et al.(2004)Grayson, Shawgo, Johnson, Flynn, Li, Cima, and Langer]grayson2004biomems
	author author A. R. Grayson, author R. S. Shawgo,
		author A. M. Johnson, author N. T. Flynn, author
		Y. Li, author M. J. Cima,  and author R. Langer, @noop  journal journal Proceedings of the IEEE volume 92,		pages 6 (year 2004)NoStop
	[Arias-González and Nieto-Vesperinas(2003)]Arias-Gonzalez:03
	author author J. R. Arias-González and author M. Nieto-Vesperinas, @noop  journal
		journal J. Opt. Soc. Am. A volume
			20, pages 1201 (year 2003)NoStop
	[Volpe et al.(2006)Volpe,
		Quidant, Badenes, and Petrov]volpe2006surface
	author author G. Volpe, author R. Quidant,
		author G. Badenes,  and		author D. Petrov, @noop 
	journal journal Physical Review Letters		volume 96, pages 238101 (year 2006)NoStop
	[Chu and Mills(2007)]chu2007laser
	author author P. Chu and author D. Mills,	@noop  journal journal Physical Review
				Letters volume 99, pages 127401
		(year 2007)NoStop
	[Juan et al.(2011)Juan,
		Righini, and Quidant]juan2011plasmon
	author author M. L. Juan, author M. Righini, 		and author R. Quidant,	@noop  journal journal Nature Photonics		volume 5, pages 349 (year
		2011)NoStop
	[Jones et al.(2015)Jones,
		Maragò, and Volpe]jones2015optical
	author author P. H. Jones, author O. M. Maragò,  and author G. Volpe, @noop  title Optical tweezers:
				Principles and applications (publisher Cambridge University
		Press, year 2015)NoStop
	[Nesterov et al.(2011)Nesterov, Frumin, and Podivilov]0295-5075-94-6-64002
	author author V. Nesterov, author L. Frumin, 		and author E. Podivilov,	@noop  journal journal EPL (Europhysics
				Letters) volume 94, pages 64002
		(year 2011)NoStop
	[Nesterov and Frumin(2011)]0957-0233-22-9-094008
	author author V. Nesterov and author L. Frumin, @noop  journal journal
			Measurement Science and Technology volume 22,		pages 094008 (year 2011)NoStop
	[Nies et al.(2016)Nies,
		Buetefisch, Naparty, Wurm,
		Belai, Shapiro, and Nesterov]nies2016experimental
	author author D. Nies, author S. Buetefisch,
		author D. Naparty, author M. Wurm, author
		O. Belai, author D. Shapiro,  and author V. Nesterov, in @noop  booktitle SPIE Nanoscience+ Engineering (organization
	International Society for Optics and Photonics, year 2016)	p. pages 99222LNoStop
	[Nesterov et al.(2016)Nesterov, Belai, Nies, Buetefisch, Mueller, Ahbe, Naparty, Popadic, and Wolff]0026-1394-53-4-1031
	author author V. Nesterov, author O. Belai,
		author D. Nies, author
		S. Buetefisch, author
		M. Mueller, author T. Ahbe, author D. Naparty, author R. Popadic, 		and author H. Wolff,	@noop  journal journal Metrologia		volume 53, pages 1031 (year
		2016)NoStop
	[Shapiro et al.(2016)Shapiro, Nies, Belai, Wurm, and Nesterov]Shapiro16
	author author D. Shapiro, author D. Nies,
		author O. Belai, author M. Wurm,  and author
		V. Nesterov, @noop  journal journal Opt. Express volume
			24, pages 15972 (year 2016)NoStop
	[Watson(1952)]W
	author author G. N. Watson, @noop  title A treatise on the theory
				of Bessel functions (publisher The University press,	address Cambridge, year 1952)NoStop
	[Bateman and Erdelyi(1953)]BE2
	author author H. Bateman and author A. Erdelyi, @noop  title Higher transcendental
				functions, Vol. volume 2 (publisher
	McGraw-Hill, address New York, year
	1953)NoStop
	[Zayats et al.(2005)Zayats,
		Smolyaninov, and Maradudin]ZAYATS2005131
	author author A. V. Zayats, author I. I. Smolyaninov,  and author A. A. Maradudin, @noop  journal journal
			Physics Reports volume 408, pages 131
		 (year 2005)NoStop
	[Pitarke et al.(2007)Pitarke, Silkin, Chulkov, and		Echenique]0034-4885-70-1-R01
	author author J. M. Pitarke, author V. M. Silkin,
		author E. V. Chulkov,  and		author P. M. Echenique,	@noop  journal journal Reports on
				Progress in Physics volume 70, pages
		1 (year 2007)NoStop
	[Ovler et al.(2010)Ovler,
		Lozier, Boisvert, and Clark]Olver10
	author author F. W. J.			Ovler, author D. W. Lozier, author R. F. Boisvert,  and author C. W. Clark, @noop  title NIST Handbook of
				Mathematical Functions (publisher Cambridge University
		Press, address New York, year 2010)NoStop
	[Palik(1998)]Palik98
	editor E. D. Palik, ed.,	@noop  title Handbook of Optical Constants of
				Solids. Vol 1,2. (publisher Academic Press, address London, year 1998)NoStop
	[Migdal(1960)]jM59
	author author A. B. Migdal, @noop  journal journal JETP		volume 37, pages 176 (year
		1960)NoStop
	[Landau and Lifshitz(1977)]LL3E
	author author L. D. Landau and author E. M. Lifshitz, @noop  title Quantum Mechanics
				(Non-relativistic Theory), series Course of Theoretical
		Physics, Vol. volume 3 (publisher Elsevier,	address New York, year 1977)NoStop
	[Sturman et al.(2010)Sturman, Podivilov, and Gorkunov]sturman2010transmission
	author author B. Sturman, author E. Podivilov,
		 and author M. Gorkunov,	@noop  journal journal Physical Review
				B volume 82, pages 115419 (year 2010)NoStop


