Abstract

Explicit and compact expressions describing the reflection and the transmission of a Gaussian beam by anisotropic parallel plates are given. Multiple reflections inside the plate are taken into account as well as arbitrary optical axis orientation and angle of incidence.

© 2003 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. J. Poirson, T. Lanternier, J.-C. Cotteverte, A. L. Floch, F. Bretenaker, “Jones matrices of a quarter-wave plate for Gaussian beams,” Appl. Opt. 34, 6806–6818 (1995).
    [CrossRef] [PubMed]
  3. E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
    [CrossRef]
  4. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. I. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
    [CrossRef]
  5. P. Yeh, “Electromagnetic propagation in birefringent media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  6. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Cross-polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
    [CrossRef] [PubMed]
  7. H. Bacry, M. Cadihac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
    [CrossRef]
  8. Y. Fainman, J. Shamir, “Polarization of nonplanar wave fronts,” Appl. Opt. 23, 3188–3195 (1984).
    [CrossRef] [PubMed]
  9. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell beams: transformation by general linear optical systems,” J. Opt. Soc. Am. A 2, 1291–1296 (1985).
    [CrossRef]
  10. S. Huard, Polarisation de la Lumière (Masson, Paris, 1993).
  11. M. Bass, E. W. Vanstryland, D. R. Williams, W. L. Wolfe, eds., Handbook of Optics, Vol. II (McGraw-Hill, New York, 1995).
  12. X. Zhu, “Explicit Jones transformation matrix for a tilted birefringent plate with its optic axis parallel to the plate interface,” Appl. Opt. 33, 3502–3506 (1994).
    [CrossRef] [PubMed]
  13. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
    [CrossRef]
  14. C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
    [CrossRef]
  15. K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik (Stuttgart) 70, 6–13 (1985).
  16. Maple® V software program (Waterloo Maple Inc., 57 Erb Street West, Waterloo, Ontario, Canada N2L 6C2).

1995 (1)

1994 (1)

1993 (1)

1987 (1)

1985 (2)

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell beams: transformation by general linear optical systems,” J. Opt. Soc. Am. A 2, 1291–1296 (1985).
[CrossRef]

K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik (Stuttgart) 70, 6–13 (1985).

1984 (1)

1983 (2)

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. I. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

1982 (1)

1981 (1)

H. Bacry, M. Cadihac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

1979 (1)

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bacry, H.

H. Bacry, M. Cadihac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bretenaker, F.

Cadihac, M.

H. Bacry, M. Cadihac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Cotteverte, J.-C.

Fainman, Y.

Floch, A. L.

Gu, C.

Huard, S.

S. Huard, Polarisation de la Lumière (Masson, Paris, 1993).

Lanternier, T.

Melle, H.

K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik (Stuttgart) 70, 6–13 (1985).

Moser, J.

K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik (Stuttgart) 70, 6–13 (1985).

Mukunda, N.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Cross-polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef] [PubMed]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell beams: transformation by general linear optical systems,” J. Opt. Soc. Am. A 2, 1291–1296 (1985).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. I. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Poirson, J.

Shamir, J.

Simon, R.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Cross-polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef] [PubMed]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell beams: transformation by general linear optical systems,” J. Opt. Soc. Am. A 2, 1291–1296 (1985).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. I. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Cross-polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef] [PubMed]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell beams: transformation by general linear optical systems,” J. Opt. Soc. Am. A 2, 1291–1296 (1985).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. I. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Yeh, P.

Zander, K.

K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik (Stuttgart) 70, 6–13 (1985).

Zhu, X.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Optik (Stuttgart) (1)

K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik (Stuttgart) 70, 6–13 (1985).

Phys. Rev. A (3)

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. I. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

H. Bacry, M. Cadihac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Other (4)

S. Huard, Polarisation de la Lumière (Masson, Paris, 1993).

M. Bass, E. W. Vanstryland, D. R. Williams, W. L. Wolfe, eds., Handbook of Optics, Vol. II (McGraw-Hill, New York, 1995).

Maple® V software program (Waterloo Maple Inc., 57 Erb Street West, Waterloo, Ontario, Canada N2L 6C2).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

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Figures (7)

Fig. 1
Fig. 1

Schematic view of plane-wave propagation in the anisotropic slab. For the sake of clarity, some of the inner reflected rays are represented by small arrows. The plane of incidence coincides with the plane yz. Symbols a± and b± correspond to the four possible propagation directions for the incident angle θ1. The four elementary transverse walk-offs Δa± and Δb± are indicated. The different vector bases used throughout this paper are also shown: {, ŷ, } for the incident and transmitted beams, {, ŷ, } for the reflected beam, {xˆI, yˆI, zˆI} to perform calculations of the birefringence effects. Associated with the first two bases are the axes of propagation: z axis for the incident beam, z axis for the transmitted beam, and z axis for the reflected beam.

Fig. 2
Fig. 2

Intensity of a Gaussian beam measured after a first-order quarter-wave plate and a perfect linear polarizer (a) for an optical axis in the plane of interface, (b) for an optical axis inclined by π/4 with respect to the plane of interface. The calculations are performed with the scalar Fourier approximation and are shown as function of the angle of incidence θ1 and the azimuth angle of the optical axis ϕc.

Fig. 3
Fig. 3

Same as Fig. 2 for a tenth-order quarter-wave plate.

Fig. 4
Fig. 4

Intensity of a Gaussian beam measured after a tenth-order quarter-wave plate (QWP) and a perfect linear polarizer as function of the angle of incidence θ1. Orientations of the QWP optical axis are fixed to (a) θc=π/2 and ϕc=0, (b) θc=π/4 and ϕc=π/4. Solid curves and dashed curves represent the calculations performed with the scalar Fourier approximations for w0=100 µm and w0=200 µm, respectively. Dotted curves show the calculations performed with the plane-wave approximation.

Fig. 5
Fig. 5

Relative difference between the quantity of Fig. 2 using the plane-wave approximation and the scalar Fourier approximation.

Fig. 6
Fig. 6

Same as Fig. 5 but for the quantity of Fig. 3.

Fig. 7
Fig. 7

(a) Inverse of the intensity measured after a tenth-order quarter-wave plate and a perfect circular left polarizer. Calculations are performed using the scalar Fourier approximation and are shown as a function of the incident angle θ1 and the azimuth angle of the optical axis ϕc. (b) Relative difference between the quantity shown in the top plot calculated with the scalar Fourier approximation and the plane-wave approximation. For these figures, the optical axis is taken in the plane of interface (θc=π/2).

Equations (61)

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ψ(r, 0)=F[ψ¯(k)]=12π ψ¯(k)exp(-ik·r)d2k,
ψ(r, z)=exp-iz2k (Px2+Py2)ψ(r, 0),
Px=-i x,Py=-i y.
ψ(r, z)=F[ψ¯(k, z)],
ψ¯(k, z)=ψ¯(k)expiz2k k2.
ψ(r, 0)=2πw021/2 exp-r2w02ψ(r, z)=2πw021/2 -izRq(z) exp-ikr22q(z),ψ¯(k, z)=w0(2π)1/2 expiq(z)k22k,
kpw=k+kzzˆk+k1-|k|22k2zˆ
F(r, 0)=Ex(r, 0)Ey(r, 0)Ez(r, 0)Bx(r, 0)By(r, 0)Bz(r, 0)
F(r, 0)=12π ψ¯(k)exp(-ik·r)×expi kxk Gx+i kyk Gya1a20-a2a10d2k,
F(r, z)=12π ψ¯(k, z)×exp(-ik·r)M6×6a1a20-a2a10d2k,
M6×6=1+ky2-kx28k2-kykx4k2kx2kkykx4k2ky2-kx28k2-ky2k-kykx4k21-ky2-kx28k2ky2kky2-kx28k2-kykx4k2kx2k-kx2k-ky2k1ky2k-kx2k0-kykx4k2-ky2+kx28k2ky2k1+ky2-kx28k2-kykx4k2kx2k-ky2+kx28k2kykx4k2-kx2k-kykx4k21-ky2-kx28k2ky2k-ky2kkx2k0-kx2k-ky2k1.
Ex(r)Ey(r)Ez(r)=F[ψ¯(k, z)M3×3E0]=12π ψ¯(k, z)×exp(-ik·r)M3×3E0d2k,
M3×3=1+ky2-kx24k2-kykx2k20-kykx2k21-ky2-kx24k20-kxk-kyk0.
Et(r)=exp-iz2k (Px2+Py2)×FM¯t exp-izin2k (Px2+Py2)×ψ¯(k, 0)M3×3E0.
sˆpw=kˆpw×nˆ|kˆpw×nˆ|,pˆpw=kˆpw×sˆpw,
M¯t=ΩMtΩT,
Ω=xˆ·sˆpwxˆ·pˆpwxˆ·kˆpwyˆ·sˆpwyˆ·pˆpwyˆ·kˆpwzˆ·sˆpwzˆ·pˆpwzˆ·kˆpw,
Mt=0Mt0000,
Reflection1aatinterfaceI:R1a+,
Transmission1aatinterfaceI:P+-1T1a+,
Reflectiona1atinterfaceI:P+-1Ra1-P-,
Transmissiona1atinterfaceI:Ta1-P-,
Reflectiona1atinterfaceII:Ra1+,
Transmissiona1atinterfaceII:Ta1+,
P±=exp(idka±·nˆ)00exp(idkb±·nˆ),
Epw,t=(Ta1+P+-1T1a++Ta+P+-1Ra1-P-Ra1+P+-1T1a++)Epw,i.
Mt=Ta1+[1-P+-1Ra1-P-Ra1+]-1P-1T1a+,
Eout(r)=w02π Fexpi (zin+z)k2k×exp-w02k24JM¯tM3×3E0,
J=0J0000,
Iout=|Eout|2d2r=w02(2π)3    exp-w02(k2+k2)4×expi (zin+z)(k2+k2)k×[O3×3E0]·[O3×3*(k)E0*]×exp[i(k-k)·r]d2kd2kd2r,
Iout=w022π  exp-w02k22|O3×3E0|2d2k,
δ2(k-k)=(2π)2×exp[i(k-k)·r]d2r,
E1t(r)=2πω021/2 -izRq(z+zin) exp-ikx22q(z+zin)Ta1+exp-ik(y-Δa+)22q(z+zin)00exp-ik(y-Δb+)22q(z+zin)P+-1T1a+e0,
Δa±=cω N1d sin θ1 cos θ1|k¯a±·nˆ|,
Δb±=cω N1d sin θ1 cos θ1|k¯b±·nˆ|,
ψ(r, z)=F[ψ¯(k, z)]ψ(r-Δyˆ, z)=F[ψ¯(k, z)exp(iΔky)],
F-1[E1t(r)]=w022π1/2×expi (zin+z)k2kexp-w02k24×Ta1+exp(ikyΔa+)00exp(ikyΔb+)×P+-1T1a+e0.
W±=exp(iΔa±ky)00exp(iΔb±ky),
Et(r)=w022π1/2Fexpi (zin+z)k2k×exp-w02k24(Ta1+P˜+-1T1a++Ta+P˜+-1Ra1-P˜-Ra1+P˜+-1T1a++)e0=w022π1/2Fexpi (zin+z)ky2k×exp-w02ky24M˜te0,
P˜+-1=W+P+-1,
P˜-=W-P-,
M˜t=Ta1+(1-P˜+-1Ra1-P˜-Ra1+)-1P˜+-1T1a+.
Sin=|a1|2+|a2|2|a1|2-|a2|2a1a2*+a1*a2i(a1a2*-a1*a2),Sout=(|Etx|2+|Ety|2)d2r(|Etx|2-|Ety|2)d2r(EtxEty*+Etx*Ety)d2ri(EtxEty*-Etx*Ety)d2r,
MS=(ρ112+ρ122)/2+(ρ212+ρ222)/2(ρ112+ρ212)/2-(ρ122+ρ222)/2ρ11,12 cos ϕ11,12+ρ21,22 cos ϕ21,22ρ11,12 sin ϕ11,12+ρ21,22 sin ϕ21,22(ρ112+ρ122)/2-(ρ212+ρ222)/2(ρ112+ρ222)/2-(ρ212+ρ122)/2ρ11,12 cos ϕ11,12-ρ21,22 cos ϕ21,22ρ11,12 sin ϕ11,12-ρ21,22 sin ϕ21,22ρ11,21 cos ϕ11,21+ρ12,22 cos ϕ12,22ρ11,21 cos ϕ11,21-ρ12,22 cos ϕ12,22ρ11,22 cos ϕ11,22+ρ12,21 cos ϕ12,21ρ11,22 sin ϕ11,22-ρ12,21 sin ϕ12,21-ρ11,21 sin ϕ11,21-ρ12,22 sin ϕ12,22-ρ11,21 sin ϕ11,21+ρ12,22 sin ϕ12,22-ρ11,22 sin ϕ11,22-ρ12,21 sin ϕ12,21ρ11,22 cos ϕ11,22-ρ12,21 cos ϕ12,21,
ρij2=w02π  exp-w02ky22|mij|2dky,ρij,kl=w02π  exp-w02ky22mijmkl*dky;
cos ϕij,kl=12ρij,kl  exp-w02ky2×[mijmkl*+mij*mkl]dky;sin ϕij,kl=12iρij,kl  exp-w02ky2×[mijmkl*-mij*mkl]dky,
JM˜t=m11m12m21m22.
δ(I)=I,Gauss-I,pwI,Gauss
IL=ρ112+ρ2122-ρ11,21 sin ϕ11,21,
δ(IL)=IL,Gauss-IL,pwIL,Gauss,
moo=t1oto1 exp[-i(kyΔo+δo)]1-ro2 exp[-2i(kyΔo+δo)],mee=t1ete1 exp[-i(kyΔe+δe)]1-re2 exp[-2i(kyΔe+δe)],moe=meo=0,
|ρoo|2=w02k22π |moo|2 exp-w02ky22dky,
|ρee|2=w02k22π |mee|2 exp-w02ky22dky,
ρoo;ee sin ϕoo;ee=12i [moomee*-moo*mee]×exp-w02ky22dky=w02k22π [R(mee)J(moo)-R(moo)J(mee)]exp-w02ky22dky,ρoo;ee cos ϕoo;ee=12 [moomee*+moo*mee]×exp-w02ky22dky=w02k22π [R(mee)R(moo)+J(moo)J(mee)]exp-w02ky22dky,
2a1002a2 exp-i arcsin a3a1a2,
|ρoo|22=tx22 1+rx21+2 cos(4φx)exp-2Δ2w02+2rx3cos(2φx)exp-Δ22w02+cos(6φx)exp-9Δ22w02+rx41+2 cos(4φx)exp-2Δ2w02+2 cos(8φx)exp-8Δ2w02+,
ρoo;ee2 sin ϕoo;ee=txty2 -sin(φx-φy)+exp-Δ22w02[rx sin(3φx-φy)-ry sin(3φy-φx)]+rxry-sin[3(φx-φy)]+exp-Δ22w02×[ry sin(5φy-3φx)-rx sin(5φx-3φy)]+exp-2Δ2w02ryrx sin(5φy-φx)-rxry sin(5φx-φy)+exp-9Δ22w02ry2rx sin(7φy-φx)-rx2ry sin(7φx-φy)+,
cˆ=sin θc cos ϕcxˆI+sin θc sin ϕcyˆI+cos θczˆI,
kpw=kxxˆI+(ky cos θ1+kz sin θ1)yˆI+(kz cos θ1-ky sin θ1)zˆI,k=k sin θ1yˆI+k cos θ1zˆI,
cos θ1pw=kzk cos θ1-kyk sin θ1,tan ϕpw=kz sin θ1+ky cos θ1kx.
sˆpw=Ns[(ky cos θ1+kz sin θ1)xˆ-kx cos θ1yˆ-kx sin θ1zˆ],pˆpw=Np[(kxkz cos θ1-kxky sin θ1)xˆ+(kykz cos θ1+kz2 sin θ1+kx2 sin θ1)yˆ+(-k2 cos θ1-kykz sin θ1)zˆ],kˆpw=Nk[kxxˆ+kyyˆ+kzzˆ],

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