Modeling of a focusing grating coupler using vectorscattering theory
Joel Seligson
The diffraction efficiency of a focusing grating coupler has been modeled using a vector scattering treatment.The diffracted field components have been solved for the case of a TE-polarized guided wave. Theamplitudes have been compared with those at the exit pupil of an aplanatic lens. The field distribution at thefocal plane has been computed for field angles of 0 and 150.
1. Introduction
Focusing grating couplers are used in guided-waveoptics to couple light out of a waveguide into a focusedspot above the guide. The focusing grating couplertakes the form of a chirped diffraction grating withcurved grooves where the functional form of thegrooves is determined by the phase matching condi-tion between the guided wave and radiated focusedwave.' Heitmann and Ortiz2 produced a focusinggrating coupler using holographic exposure of photo-resist. The exposure was done at a wavelength of 458nm, and the geometry of the two interfering wave-fronts, a plane wave and a spherical wave, was chosento minimize the aberrations in the outcoupled wave-front at the wavelength of 633 nm. The focal spotdiameter for the grating with a numerical aperture of0.32 was 2.2 jim. Electron-beam writing has beenemployed to produce focusing grating couplers used indemonstration devices for applications in optical datastorage 3 and printing. 4 A typical focusing grating cou-pler3 had a focal length of 2.0 mm, aperture of 1.0 X 1.0mm, and a focal spot width of 3.5 gim at the wavelengthof 790 nm. The focusing grating coupler was modeledassuming a uniform diffraction efficiency across thegrating and taking into account the exponential decayof the power in the guided wave due to the outcoupling.The corresponding focal spot width in the diffraction
When this work was done the author was with Eastman KodakCompany, Research Laboratories, Diversified Technologies Group,Rochester, New York 14650; he is now with SRD, Ltd., MoshavShorashim, D. N. Misgav 20164, Israel.
limit was calculated to be 1.4 Am, considerably smallerthan the observed one. It is to be noted that thenumerical aperture of this focusing grating coupler wasonly 0.24, whereas high-density optical data recordingcurrently requires a numerical aperture of 0.5-0.6.
Katzir et al. 5 have modeled a chirped diffractiongrating with linear grooves. The amplitude of theradiated field due to the outcoupling of the guided TEmode was calculated using coupled-mode theory.6Similarly, Lin et al. 7 modeled the in-plane diffractioncaused by a chirped curved Bragg-grating using cou-pled-mode theory. In the first case, the guided wavevector and the projection of the diffracted wave vectoronto the plane of the waveguide were collinear, andconsequently there was no TE-TM mode conversion.In the second case, mode conversion was neglected dueto the small diffraction angles. In a focusing gratingcoupler with a high numerical aperture, however, thedeparture from collinear diffraction is significant, andcoupling between TE and TM modes has to be includ-ed.8 It has been shown9 that coupled-mode theory isnot the appropriate method when TE-TM coupling ispresent. As the computational method, we chose aperturbation expansion method, first introduced toguided-wave optics in the treatment of scattering fromsurface roughness.'0 This method will be applied tothe specific problem of obtaining the diffracted field atthe exit pupil of high N.A. focusing grating couplers.Fourier transform is subsequently employed to com-pute the field at the focal plane of the grating coupler.
11. Theory
A. Arbitrary Surface CorrugationLet us consider a focusing grating coupler depicted
in Fig. 1. The grating coupler lies in the y-z plane withthe origin 0 in the center of the grating. The guidedwave propagates in the y-z plane and is coupled by thefocusing grating coupler out of the waveguide and fo-
cused into the focal spot F. The focal spot is located inthe x-z plane, a distance f from the origin. The lineOF, chief ray in the geometrical optics terminology,forms a field angle 6 with the positive x axis.
Following the notation of Ref. 10, we consider thewaveguide shown in Fig. 2. The film-substrateboundary is located at the x = 0 plane, and the film-cover interface is given by
x = h + nj(y,z). (1)
Here h is the average thickness of the waveguide, i7 isrelated to the amplitude of the corrugation, and givesthe functional dependency of the surface corrugationon the y and z coordinates. The electrical and mag-netic fields are expressed in terms of the vector poten-tial A, which is expanded in the three regions (cover =I, film = II, and substrate = III) as a series in powers of
dent of the y coordinate. The incident wave can beexpanded in region I around x = h to yield
aA A A I1 ax xa2A, 7 + * -- -
x aX =h(6)
A similar expansion is obtained in region II. By sub-stituting Eqs. (2)-(6) into the boundary conditions andequating terms of equal power of , one obtains thefollowing equations for the first-order scattered fields:
!x (Al -AsD)x=h -0, (7)
x (v x A- v x AII)x=h y,)(Y9 (aV - d )(8)
x x (AII- - AI`)x= = 0,
X x (v x AII - V X AI'I)x=o = .
(9)
(10)
The difference between the second derivatives in Eq.(8) can be rewritten using the fact that the initial fieldAi satisfies the two wave equations:
AI = A + nAI
n=1
x > h, (2)
0nAII < x < h, (3)n=1
All, = Al,' + E NnAI, x < 0, (4)n=1
where the subscript i refers to the unperturbed field.The subscript sn refers to the nth-order scatteredfield. The equations to be satisfied by A, AI,, and A,,,
are the wave equations in the respective regions, thetransversality condition v * A = 0, and the boundaryconditions stating that h X A and h X (v X A) arecontinuous across all interfaces. Here h denotes thelocal surface normal, which for the lower surface is h =
x, and for the upper surfaceh= [1+ (a7 a)2 + (71 ] (x 9 7 y
L \ ay ) \ ezJ k X ay ??az)
x, 9, and i denote the unit vectors along the threecoordinate axes. We assume an incident guided TE-polarized wave with Ai = Aiy, where Ai is indepen-
d 2 = (2 - n2k)A
d Al = (p2 - 2ax2 n k )A ,
(11)
(12)
where f is the propagation constant of the guidedwave, n, and nf are the refractive indices of the coverand film, respectively, and ko is the free-space wavevector. At the boundary x = h the two amplitudes A'
and A'I of the guided TE wave have to be equal tosatisfy the boundary conditions. This common valuecan be written as A exp(i3z). The result for the differ-ence of the derivatives in Eq. (8) can now be written as( Al' a2')A
ax 2 a_ 2)'x=h = g ~ e p i 3(13)
whereAe = n2 n2 (14)
The solutions for the first-order scattered fields arewritten as plane-wave expansions:
where q = qyy + qzz is the projection of the wave vectoronto the y-z plane, and R = y5 + z2 is the projection ofthe position vector onto the y-z plane. The transversepropagation constants are given by
q = (e k2 _ q2 )1/2, (18)
qf = (k2 - q2)1/2, (19)
q = (ek 2 - q2 )1/2, (20)
where sc, ef, and are the dielectric constants of thecover, film, and substrate, respectively. The plane-wave expansions (15)-(17) are substituted into Eqs.(7)-(10).
Additional equations are generated by applying thetransversality conditions v A = 0 to the plane-waveexpansions. This generates the following twelve equa-tions for the components of the plane-wave amplitudesQ, R S and T:
qQ + Q + qzQz = 0, (21)
q fR + qYR + qR = 0, (22)
-qfS + qS + qSz = 0, (23)
-q.T + qYT + qzT = 0, (24)
QY exp(iqch) - R exp(iqfh) - Sy exp(-iqfh) = 0, (25)
Q, exp(iqh) - R exp(iqfh) - S exp(-iqfh) = 0, (26)
qR - q1R + qzS. + qfSz - qzT - qTz = 0. (32)The equations are homogeneous, except for Eq. (27),
which contains the driving term consisting of the inter-action between the incident field and surface corruga-tion. It would be useful to be able to blaze the gratingso that-the amplitude of the diffracted wave in thecover, Q is enhanced at the expense of the wave dif-fracted into the substrate, T. It is to be noted that inthe TE case under consideration the shape of the sur-face corrugation (y,z) enters the driving terms onlythrough its Fourier components rather than through adirect shape factor, such as a slope. The higher-orderFourier components would cause diffraction to higher-order foci. It is not currently clear, however, if thisalso affects the power balance between the two maindiffracted orders in the cover and substrate.
The solutions for Eqs. (21)-(32) are written-in termsof 3 X 3 complex matrices p, &, and ~, defined by
(33)
(34)
(35)
where R = (RxRyRs), S = (Sx,SySz), and Q =(QxQyQz). Simple but tedious algebra leads to thefollowing expressions for p and &:
a2y = y exp(iqh) - (qT-y - qtiy-) exp(iqfh)
- (qz2 y~ + qfo12 ,) exp(-iqfh),
a 2 z = - (- + qc) exp(iqch) - (qTXz - qfTzz) exp(iqfh)
- (qzyxz + qfazz) exp(-iqfh).
E _ EIf
P ( s t
qz (1 q2 )qf \ qsqf/
2qyqz
q2/ q2
sqsf
0 F Y F-qf qf
0 pzz exp(iqfh) + exp(-iqfh) -pyz exp(iqfh)
0 -pzy exp(iqfh) pyy exp(iqfh) + exp(-iqj
where
F = f +s exp(iqfh) + exp(-iqfh),qf+ q,
D = [pyy exp(iqfh) + exp(-iqfh)1E[p, exp(iqfh) + exp(-iqfh)]
- Pyzpzy exp(i2qfh).
The Qy and Q, components are solved in terms ofdriving term as
alya 2z - a 2yalz
x JJ r(yz) exp(-iqyy) exp[-i(qz - O)z]dydz,
a2y ifoalya2z -a2yalz
x f E t(y z) exp(-iqy) exp[-i(qz - )zdydz,
where the a terms are given by/2
aly =-I + qc) exp(iqch) - (qyTi-y - qfi-r) exp(iqfh)
-(qy xy + qfayy) exp(-iqfh),
a1z = -- exp(iqh) - (qy,.z - qtTy,) exp(iqfh)
- (qyfafz + qfo-yz) exp(-iqfh),
(36)
(37)exp(iqch)
J D
fh/
In the case of a focusing grating coupler, one is primari-ly interested in the fields in the cover. (Using a re-
(38) flecting layer in the substrate, one includes the sub-strate fields as well,3 but this case is not considered inthe current work.) Once Qy and Q, are calculated fromEqs. (40)-(41), Qx is given by Eq. (21).
(39) The total field in the cover is obtained by substitut-the ing Qx, Qy, and Qz into Eq. (15) and substituting the
resulting Al into Eq. (2). For the purpose of examin-ing the performance of a focusing grating coupler, onecan ignore the evanescent contribution of the incidentfield A!, and one obtains for the scattered field to thefirst order in q the result
R(y,z) is the distance from point (0,y,z) to the focalpoint. The diffraction efficiencies for the three com-ponents of the diffracted field are now defined as
/\y~~q2= a (51)-ya2z-a2yalZ
A2(q,,,q2) = _- a , (52)alya2.- -a2yaliZ
A 2 (qyq,) =1- (qYAY + qzAz). (53)
The focusing grating coupler is constructed so thatat each point of the grating the diffracted wave vectorpoints toward the focal point. This determines thevalues of qy and q at each point (y,z) of the gratingthrough the equations
qY= -, R i'(54)
q= (f sinO - z). (55)
Ill. Results
A. Pupil
As a numerical example, we chose a waveguide withthe following parameters: The refractive index of thesubstrate n is 1.47, the refractive index of the film n is1.56, and the refractive index of the cover n is 1.00.The thickness of the film h is 0.9 gm. The wavelengthof the guided wave is assumed to be 633 nm. The focallength of the grating f is 1000 gim, and the aperture ofthe grating is 1500 gn X 1500 gm, giving a numericalaperture of 0.6. The diffraction efficiencies Ax, Ay,and Az, have been calculated for the field angles = 0and 15°, and the results are presented in Figs. 3-6.Figure 3 shows the amplitude and phase (solid line) ofAy for field angle = 0 along the z axis (y = 0 gm).The amplitude declines somewhat toward the center ofthe grating. A least-squares second-order fit (dottedline) to the phase shows it to be largely parabolic andthus removable by refocusing. The remaining devi-ation is of fourth order in z. Due to symmetry, both Axand A vanish along the z axis. The amplitudes andphases of Ax, Ay and Az at y = 750 Aim are presented inFig. 4. The behavior of Ay is similar to that at y = 0gim, and the appearance of Ax can be attributed to thetilting of the diffracted wave vector toward the x-zplane. A z component A is produced by the obliqueincidence of the guided wave on the grating. Its mag-nitude is zero along both y and z axes and increasestoward the corners of the grating. The phases of allthree diffraction efficiencies behave identically, ex-cept for the halfwave jump for Az at origin due to thechange of sign. The results for field angle = 150 areshown in Figs. 5 and 6. They are similar to the resultsfor 0 = 0, only shifted. The nonparabolic behavior ofthe phase is also more pronounced farther away fromthe origin. It should be possible to compensate for thisphase aberration by a suitable correction in the groovepattern.
1. Comparison with a LensIt is interesting to compare the field components at
the exit pupil of the grating with those at the exit pupilof a lens of equal numerical aperture and zero fieldangle. An aplanatic lens has been examined by Rich-ards and Wolf.'" The three-field components are giv-en in our notation by
E = co s 2
y + z2
Ez =cos yz(Cos - 1)y2 + z 2
(56)
(57)
(58)
where the constant amplitude factors have been omit-ted. The angle is the angle between the optical axisand a ray from point (y,z) to the focal point. The lensis illuminated with a plane wave of constant amplitudepolarized in the y direction. In the numerical calcula-tions the focal length of the lens is 1000 m, and itsaperture is 1500 m X 1500 gm. Again, on the z axis,only Ey is nonzero, and it is plotted in Fig. 7. Com-pared to the corresponding results for the grating, Fig.3, it is observed that for the lens the amplitude de-creases away from the center, whereas for the gratingthe opposite takes place. Figure 8 shows the resultsfor the edge of the pupil, y = 750 gim. The amplitudesfor the three polarizations behave similarly to those forthe grating shown in Fig. 4, except again for the stron-ger tendency to decrease away from the center.
B. Focus
In a focusing grating coupler of constant corrugationheight , the guided wave decays exponentially due todiffraction if one ignores the slight changes in diffrac-tion efficiency across the grating as shown above.This leads to a diminished effective numerical aper-ture and consequently to a larger focal spot. By ad-justing X along the grating, the diffracted power can begiven a more uniform distributions We assume herethat this has been done, so that the diffraction efficien-cies A, Ay, and A also describe the actual diffractedfield at the exit pupil of the grating. For the purposeof the numerical calculations the field componentshave been normalized by setting AEk Aw = 1. Thefactor comes from E = - (dA/dt), assuming a timebehavior of exp(-iwt) for the fields. We also assumethat aq remains sufficiently small for the perturbationtreatment still to be valid. (This may not be the casewhen one strives for a very high total diffraction effi-ciency of the grating, since a high local diffractionefficiency at the depleted tail end of the guided wave isthen required.) The field at the focal plane is obtainedthrough a Fourier transform of the field at the exitpupil. The focal plane is perpendicular to the chiefray OF. The origin of the Cartesian coordinate systemin the focal region is atF. The x axis lies along OF, andthe y axis is parallel to the y axis in the pupil. Beforeperforming the transform, the defocus is corrected byfinding a least-square fit of the form
(m)Fig. 3. Amplitude and phase of diffraction efficiency A, for 0 = 0°,
y = 0 Am. Solid line: computed amplitude and phase. Dotted line:least-squares fit to the phase.
(1n1)
Fig. 5. Amplitude and phase of diffraction efficiency A, for 0 = 15°,y = 0 Am.
a)-o
E
0 05
0.04
0.03
0.02
0.01
Q.vU-500 0 500
a)CZ
a,a)
n
0.6
0.4
0.2
0.0
-0.2-500 0 500
z (m)
Fig. 4. Amplitude and phase of diffraction efficiencies A., Ay, andA, for 0 = 00, y = 750 Am.
0(y Z) = (yl + Z2) + fly + ,YZ + ^ (59)
for the phase of Ay and subtracting this phase termfrom the phase of Ax Ay, and Az. The resulting ycomponent of the electric field Ey in the focal planealong the y and z axes is shown in Fig. 9 for field angle 0= 00. E is nearly identical to that of a diffraction-limited result. The ratio of the maximum of the firstsidelobe on the z axis to the center value is 0.244,whereas for a diffraction-limited system it is 0.212.The calculation also yields a small x component due tothe high numerical aperture of the grating coupler.The z component Ez is zero along the y and z axes due
0 05
0 0411
_01
-Q-F-
0.03
002
0.01
0.00
EU)a)
a)U)CZ-C0-
08
0.6
0.4
0.2
0.0
-0.2- 500 0 500
(m)
Fig. 6. Amplitude and phase of diffraction efficiencies A, Ay, andA, for O = 15', y = 750 tkm.
to the fact that in the pupil E, has the following sym-metry properties:
E 2(-y,z) =-E,(yz),
Ez(y,-z) = -E,(y,z).
(60)
(61)
Outside the coordinate axes, however, Ez is not zero,as can be seen in Fig. 10, where Ey and Ez have beenplotted along the line y 1.27 ,gm. Still the maximumamplitude of Ez is less than the first sidelobe of Ey.Changing the field angle 0 from 0 to 15° breaks thesymmetry, as can be observed from Figs. 11 and 12. InFig. 11, the asymmetry of Ey is evident, especially in
Fig. 7. Amplitude of the field component Ey at the exit pupil of alens, y = 0 m.
0-5 -4 -3 -2 -I 0 1 2 3 4 5
(m)
Fig. 10. Field components E, and E, at the focal plane of thegrating along y = 1.27 m, field angle 0 = 0°.
-500 -250 0
(m)
x10-33
2
a)
-C:
250 500 750
Fig. 8. Amplitude of the field components E., Ey, and E, at the exitpupil of a lens, y = 750,um.
-4 -2 0y (p m)
-4 -2 0
Z (m)
2 4
2 4
Fig. 9. Field component E, at the focal plane of the grating alongthe y and z axes, field angle 0 00.
0
3
2
-4 -2 0 2 4Z (m)
Fig. 11. Field components Ey and E, at the focal plane of thegrating along y and z axes, field angle = 15°.
the first sidelobe in the positive z direction, where theratio of its maximum to the center value is now 0.301.Along the y axis the breaking of the symmetry yields anonzero, but still very small, E In Fig. 12, the y and zcomponents are again plotted along the line y = 1.27Am. The asymmetry is evident for both Ey and E,although E is still smaller than the first sidelobe of Eyat positive z.
IV. Conclusions
The diffracted electrical field components of a fo-cusing grating coupler have been calculated for anincident collimated TE-polarized guided wave usingscattering theory. The behavior of the amplitude ofthe field is similar but not identical to that of the fieldat the exit pupil of an aplanatic lens. The phase front
Fig. 12. Field components Ey and Ez at the focal plane of thegrating along y = 1.27 Am, field angle 0 = 15°.
of the diffracted field exhibits a small deviation from aspherical one. The amplitudes of the field compo-nents at the focal plane have been computed using theFourier transform. At a zero field angle the majorpolarization component Ey forms a nearly diffraction-limited focal spot, and the amplitude of the orthogonalpolarization E, is less than the first sidelobe of the Eyfield. At a field angle of 15° the effects of the above-mentioned phase aberration become noticeable inboth polarization components. An analysis of the op-tical disk write and read processes will have to deter-mine whether these effects are tolerable and whetherthey need to be corrected by a suitable design of thegrating.
The author did this work as an Eastman Kodak Co.sponsored Industrial Resident at the New York StateCenter for Advanced Optical Technology at the Insti-tute of Optics, University of Rochester, Rochester,NY. Useful discussions with Joseph F. Revelli andDennis L. Venable are gratefully acknowledged.
Appendix: Computing the Two Double Integrals
The first double integral is
I, (q,,,q) = JJ t(yZ)
X exp(-iqyy) exp[-i(q, - O)z]dydz. (Al)
An easy way to obtain the functional form of (y,z) is toconsider the grating to have been formed holographi-cally through interference of a spherical wave emanat-ing from the focal point F and a guided wave propagat-ing in the negative z direction.' The first is describedby exp[ikoR(y,z)], where R(y,z) is the distance from thepoint (y,z) to the focal point. The second one is of theform exp(-if3z). The amplitudes have been omitted,since we are interested only in the phase relationships.The resulting form for the surface corrugation is
In substituting A(y,z) into Eq. (Al), the complex conju-gate produces only high-frequency terms, which do notcontribute to the integral. The result is
where
F(y,z) = -R(y,z)- y - Z.k, k,
(A3)
(A4)
Both qy and qz are proportional to ko, and thus F(y,z) isindependent of ko. Consequently, the integral Il(y,z)can be evaluated using the method of stationaryphase.'3 Since any focusing grating coupler with anonzero field angle can be thought of as a part of alarger coupler with zero field angle, we can, withoutloss of generality, assume the coordinates of the focalpoint to be (XF,0,0). The critical point (yo,zo) satisfiesthe equations
From Eqs. (40), (41), and (21) one can write the planewave amplitude Q(q) as
Q(qyq,) = U(qyqz)I1(q,,qz), (A8)
where U(qy,qz) is obtained by comparing Eq. (A8) withthe results for Qx, Qy, and Q, in Sec. II.A.
Substituting Eq. (A8) into Eq. (46) leads to
AI(xyz) = ( ) JJf (qYqz)I1(qqz)
X exp(iqx) exp(iq - R)dqydqz. (A9)
We are interested in the fields in the exit pupil of thefocusing grating coupler and consequently compute AIat x = 0. This can be written as
AI(Oyz) = (1-2 g(qy,q2)
X exp[ik0f(qy,q2 )]dqydqz,
whereg(q,,,qz) = U(q,,,qz) -2irR~y z 0
,o-((q + qz)
f(q,,,q) = F(yo,zo) + y + Z.
(A10)
(All)
(A12)
The method of stationary phase can again be ap-plied, and it leads to the result
AI(Oyz) = inU(qy,qz) exp[-ik 0R(y,z)]- (A13)
Thus the field at the exit pupil of the focusing grat-ing coupler consists of a spherical wave converging tothe focal point, with its phase and amplitude modifiedby the function (qy,qz), and its amplitude propor-tional to the amplitude j of the surface corrugation.
References1. T. Suhara, H. Nishihara, and J. Koyama, "High-Performance
Focusing Grating Coupler Fabricated by Electron-Beam Wri-ting," in Technical Digest, Topical Meeting on Integrated andGuided- Wave Optics (Optical Society of America, Washington,DC, 1984), paper ThD4.
2. D. Heitmann and C. Ortiz, "Calculation and Experimental Veri-fication of Two-Dimensional Focusing Grating Couplers," IEEEJ. Quantum Electron. QE-17, 1257 (1981).
3. S. Ura, T. Suhara, H, Nishihara, and J. Koyama, "An Integrat-ed-Optic Disk Pickup Device," IEEE/OSA Lightwave Technol.LT-4, 913 (1986).
4. T. Suhara, N. Nozaki, and H. Nishihara, "An IntegratedAcousto-Optic Printer Head," in Proceedings, Fourth Europe-an Conference on Integrated Optics, C. D. W. Wilkinson and J.Lamb, Eds. (SETG, Ltd., Glasgow, 1987), pp. 119-122.
5. A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, "ChirpedGratings in Integrated Optics," IEEE J. Quantum Electron.QE-13, 296 (1977).
6. D. Marcuse, Theory of Dielectric Optical Waveguides (Academ-ic, New York, 1974), Chap. 3.
7. Z.-Q. Lin, S.-T. Zhou, W. S. C. Chang, S. Forouhar, and J.-M.
Delavaux, "A Generalized Two-Dimensional Coupled-ModeAnalysis of Curved and Chirped Periodic Structures in OpenDielectric Waveguides," IEEE Trans. Microwave Theory Tech.MTT-29, 881 (1981).
8. K. Wagatsuma, J. Sasaki, and S. Saito, "Mode Conversion andOptical Filtering of Obliquely Incident Waves in CorrugatedWaveguide Filters," IEEE J. Quantum Electron. QE-15, 632(1979).
9. J. Seligson, "The Orthogonality Relation for TE- and TM-Modes in Guided-Wave Optics," accepted for publication inIEEE/OSA J. Lightwave Technol.
10. D. G. Hall, "Scattering of Optical Guided Waves by WaveguideSurface Roughness: A Three-Dimensional Treatment," Opt.Lett. 6, 601 (1981).
11. B. Richards and E. Wolf, "Electromagnetic Diffraction in Opti-cal Systems. II. Structure of the Image in an Aplanatic Sys-tem," Proc. R. Soc. London Ser. A 253, 358 (1959).
12. A. V. Kukharev, M. Yu. Lipovskaya, A. A. Lipovskii, A. V.Pavlenko, and V. Yu. Petrunkin, "Prototype Integrated-OpticBeam Expander," Sov. Phys. Tech. Phys. 30, 964 (1985).
13. M. Born and E. Wolf, Principles of Optics (Pergamon, London,1975), Appendix III.
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