Abstract

From a matrix formulation of the boundary conditions we obtain the fundamental invariant for an interface and a remarkably simple factorization of the interface matrix, which enables us to express the Fresnel coefficients in a new and compact form. This factorization allows us to recast the action of an interface between transparent media as a hyperbolic rotation. By exploiting the local isomorphism between SL(2, C) and the (3+1)-dimensional restricted Lorentz group SO(3, 1), we construct the equivalent Lorentz transformation that describes any interface.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
    [CrossRef]
  2. J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
    [CrossRef]
  3. J. M. Vigoureux, “The reflection of light by planar stratified media: the grupoid of amplitudes and a phase ‘Thomas precession,’ ” J. Phys. A 26, 385–393 (1993).
    [CrossRef]
  4. J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
    [CrossRef]
  5. J. J. Monzón, L. L. Sánchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
    [CrossRef]
  6. J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
    [CrossRef]
  7. J. J. Monzón, L. L. Sánchez-Soto, “Origin of the Thomas rotation that arises in lossless multilayers,” J. Opt. Soc. Am. A 16, 2786–2792 (1999).
    [CrossRef]
  8. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.
  9. J. J. Monzón, L. L. Sánchez-Soto, “On the concept of absorption for a Fabry–Perot interferometer,” Am. J. Phys. 64, 156–163 (1996).
    [CrossRef]
  10. R. M. A. Azzam, “Transformation of Fresnel’s interface reflection and transmission coefficients between normal and oblique incidence,” J. Opt. Soc. Am. 69, 590–596 (1979);“Direct relation between Fresnel’s interface reflection coefficients for the parallel and perpendicular polarizations,” J. Opt. Soc. Am. 69, 1007–1016 (1979) and references therein.
    [CrossRef]
  11. J. J. Monzón, L. L. Sánchez-Soto, “Algebraic structure of Fresnel reflection and transmission coefficients at an interface,” Optik (Stuttgart) 110, 309–312 (1999).
  12. J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Chap. 15.
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980), Sec. 1.5.
  14. A. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).
  15. P. Hariharan, Optical Interferometry (Academic, Sydney, 1985), Appendix A4.
  16. J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987), Chap. 2.
  17. M. A. Khashan, “A Fresnel formula for dielectric multilayer mirrors,” Optik (Stuttgart) 54, 363–371 (1979).
  18. S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
    [CrossRef]
  19. I. V. Lindell, A. H. Sihvola, “The quotient function and its applications,” Am. J. Phys. 66, 197–202 (1998).
    [CrossRef]
  20. A. Mischenko, A. Fomenko, A Course of Differential Geometry and Topology (Mir, Moscow, 1988), Sec. 1.4.
  21. A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN-Polish Scientific, Warszaw, 1977), Chap. 17.
  22. A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Dover, New York, 1980), Chap. 1.
  23. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).
  24. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980), Sec. 10.8.

1999 (4)

J. J. Monzón, L. L. Sánchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Origin of the Thomas rotation that arises in lossless multilayers,” J. Opt. Soc. Am. A 16, 2786–2792 (1999).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Algebraic structure of Fresnel reflection and transmission coefficients at an interface,” Optik (Stuttgart) 110, 309–312 (1999).

1998 (1)

I. V. Lindell, A. H. Sihvola, “The quotient function and its applications,” Am. J. Phys. 66, 197–202 (1998).
[CrossRef]

1996 (1)

J. J. Monzón, L. L. Sánchez-Soto, “On the concept of absorption for a Fabry–Perot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

1993 (2)

J. M. Vigoureux, “The reflection of light by planar stratified media: the grupoid of amplitudes and a phase ‘Thomas precession,’ ” J. Phys. A 26, 385–393 (1993).
[CrossRef]

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

1992 (1)

1991 (2)

J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
[CrossRef]

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

1979 (2)

Azzam, R. M. A.

Barut, A. O.

A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN-Polish Scientific, Warszaw, 1977), Chap. 17.

A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Dover, New York, 1980), Chap. 1.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980), Sec. 1.5.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980), Sec. 10.8.

Coldren, L. A.

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

Corzine, S. W.

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

Fomenko, A.

A. Mischenko, A. Fomenko, A Course of Differential Geometry and Topology (Mir, Moscow, 1988), Sec. 1.4.

Grossel, Ph.

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

Hariharan, P.

P. Hariharan, Optical Interferometry (Academic, Sydney, 1985), Appendix A4.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

Khashan, M. A.

M. A. Khashan, “A Fresnel formula for dielectric multilayer mirrors,” Optik (Stuttgart) 54, 363–371 (1979).

Lekner, J.

J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987), Chap. 2.

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, “The quotient function and its applications,” Am. J. Phys. 66, 197–202 (1998).
[CrossRef]

Mischenko, A.

A. Mischenko, A. Fomenko, A Course of Differential Geometry and Topology (Mir, Moscow, 1988), Sec. 1.4.

Monzón, J. J.

J. J. Monzón, L. L. Sánchez-Soto, “Origin of the Thomas rotation that arises in lossless multilayers,” J. Opt. Soc. Am. A 16, 2786–2792 (1999).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Algebraic structure of Fresnel reflection and transmission coefficients at an interface,” Optik (Stuttgart) 110, 309–312 (1999).

J. J. Monzón, L. L. Sánchez-Soto, “On the concept of absorption for a Fabry–Perot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

Perelomov, A.

A. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).

Ra¸czka, R.

A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN-Polish Scientific, Warszaw, 1977), Chap. 17.

Sánchez-Soto, L. L.

J. J. Monzón, L. L. Sánchez-Soto, “Origin of the Thomas rotation that arises in lossless multilayers,” J. Opt. Soc. Am. A 16, 2786–2792 (1999).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Algebraic structure of Fresnel reflection and transmission coefficients at an interface,” Optik (Stuttgart) 110, 309–312 (1999).

J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “On the concept of absorption for a Fabry–Perot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, “The quotient function and its applications,” Am. J. Phys. 66, 197–202 (1998).
[CrossRef]

Stone, J. M.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Chap. 15.

Vigoureux, J. M.

J. M. Vigoureux, “The reflection of light by planar stratified media: the grupoid of amplitudes and a phase ‘Thomas precession,’ ” J. Phys. A 26, 385–393 (1993).
[CrossRef]

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
[CrossRef]

J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980), Sec. 1.5.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980), Sec. 10.8.

Yan, R. H.

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

Am. J. Phys. (3)

J. J. Monzón, L. L. Sánchez-Soto, “On the concept of absorption for a Fabry–Perot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

I. V. Lindell, A. H. Sihvola, “The quotient function and its applications,” Am. J. Phys. 66, 197–202 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Phys. A (1)

J. M. Vigoureux, “The reflection of light by planar stratified media: the grupoid of amplitudes and a phase ‘Thomas precession,’ ” J. Phys. A 26, 385–393 (1993).
[CrossRef]

Opt. Commun. (1)

J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[CrossRef]

Optik (Stuttgart) (2)

M. A. Khashan, “A Fresnel formula for dielectric multilayer mirrors,” Optik (Stuttgart) 54, 363–371 (1979).

J. J. Monzón, L. L. Sánchez-Soto, “Algebraic structure of Fresnel reflection and transmission coefficients at an interface,” Optik (Stuttgart) 110, 309–312 (1999).

Other (11)

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Chap. 15.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980), Sec. 1.5.

A. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).

P. Hariharan, Optical Interferometry (Academic, Sydney, 1985), Appendix A4.

J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987), Chap. 2.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

A. Mischenko, A. Fomenko, A Course of Differential Geometry and Topology (Mir, Moscow, 1988), Sec. 1.4.

A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN-Polish Scientific, Warszaw, 1977), Chap. 17.

A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Dover, New York, 1980), Chap. 1.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980), Sec. 10.8.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Wave vectors of the incident, reflected, and transmitted fields at interface 01.

Fig. 2
Fig. 2

Plot of the factor ξ01 and r01 as functions of the angle of incidence θ0 (in degrees) for both p and s polarizations for an air–glass interface (N0=1, N1=1.5). The marked points correspond to the Brewster angle.

Equations (76)

Equations on this page are rendered with MathJax. Learn more.

N0sin θ0=N1sin θ1.
E1(+)=E0(t)+E1(r)=t01E0(+)+r10E1(-),
E0(-)=E0(r)+E1(t)=r01E0(+)+t10E1(-),
cos θ0E0(+)N0E0(+)=cos θ0cos θ1-N0N1E0(r)E0(t).
cos θ1E1(-)N1E1(-)=cos θ1cos θ0-N1N0E1(r)E1(t).
cos θ0-cos θ0N0N0E0(+)E0(-)=cos θ1-cos θ1N1N1E1(+)E1(-).
cos θ0[E0(+)-E0(-)]=cos θ1[E1(+)-E1(-)],
N0[E0(+)+E0(-)]=N1[E1(+)+E1(-)],
N0cos θ0{[E0(+)]2-[E0(-)]2}
=N1cos θ1{[E1(+)]2-[E1(-)]2},
N0cos θ0-N0cos θ011E0(+)E0(-)
=N1cos θ1-N1cos θ111E1(+)E1(-),
N0cos θ0[E0(+)-E0(-)]=N1cos θ1[E1(+)-E1(-)],
E0(+)+E0(-)=E1(+)+E1(-).
cos θi-cos θiNiNi=cos θi00Ni 1-111Ti(p)2R(π/4),
T0(p)R(π/4)E0(+)E0(-)=T1(p)R(π/4)E1(+)E1(-).
T0(s)R(π/4)E0(+)E0(-)=T1(s)R(π/4)E1(+)E1(-),
Nicos θi-Nicos θi11=Nicos θi001 1-111Ti(s) 2R(π/4).
Ti(s)=Ni001/NiTi(p).
E0(+)E0(-)=I01E1(+)E1(-),
I01=1t011-r10r01t01t10-r01r10.
I01(p)=R-1(π/4)T0-1(p)T1(p)R(π/4)=R-1(π/4)×cos θ1/cos θ000N1/N0R(π/4),
I01(s)=R-1(π/4)T0-1(s)T1(s)R(π/4)=R-1(π/4)×(N1cos θ1)/(N0cos θ0)001R(π/4),
I01=abba,
r10=-r01,
1+r01r10=t01t10,
I01=1t011r01r011.
det I01(p)=det I01(s)=N1cos θ1N0cos θ01.
e0(±)=N0cos θ0E0(±),
e1(±)=N1cos θ1E1(±).
e0(+)e0(-)=i01e1(+)e1(-),
i01=R-1(π/4)1/ξ0100ξ01R(π/4)=12ξ01+1/ξ01ξ01-1/ξ01ξ01-1/ξ01ξ01+1/ξ01,
ξ01(p)=N1cos θ0N0cos θ11/2,
ξ01(s)=N0cos θ0N1cos θ11/2.
ξ01(p)ξ01(s)=cos θ0cos θ1,
ξ01(s)ξ01(p)=N0N1.
i01=1t^011r^01r^011,
t^01=2ξ01+1/ξ01,
r^01=ξ01-1/ξ01ξ01+1/ξ01,
1-r^012=t^012.
t01=N1cos θ1N0cos θ01/2t^01,
r01=r^01.
t^01=t^10,
r^01=-r^10.
ξ01=exp(ζ/2),
e0(+)e0(-)=cosh(ζ/2)sinh(ζ/2)sinh(ζ/2)cosh(ζ/2)e1(+)e1(-),
t^01=1cosh(ζ/2),
r^01=tanh(ζ/2).
e˜(+)e˜(-)=R(π/4)e(+)e(-)
e˜0(+)e˜0(-)=1/ξ0100ξ01e˜1(+)e˜1(-).
e˜0(+)e˜0(-)=e˜1(+)e˜1(-),
[e0(+)]2-[e0(-)]2=[e1(+)]2-[e1(-)]2,
xμ=Λνμxν,
(x0)2-(x1)2-(x2)2-(x3)2
X=xμσμ=x0+x3x1-ix2x1+ix2x0-x3,
xμ=12tr(Xσμ).
X=WXW,
Λνμ(W)=12tr(σμWσνW).
Λ=LR.
W=HU,
ξ01=|ξ01|exp(iϕ)=exp(ζ/2)exp(iϕ).
i01=cosh(ζ/2)sinh(ζ/2)sinh(ζ/2)cosh(ζ/2)×cos ϕi sin ϕi sin ϕcos ϕ=HU.
Λ=LR=γγβ00γβγ0000100001×1000010000cos 2ϕsin 2ϕ00-sin 2ϕcos 2ϕ,
γ=12(|ξ01|2+1/|ξ01|2)=coshζ,
β=|ξ01|2-1/|ξ01|2|ξ01|2+1/|ξ01|2=tanh ζ,
e=e(+)e(-),
E=ee=|e(+)|2e(+)e(-)*e(+)*e(-)|e(-)|2,
E0=i01E1i01,
x0x1x2x3[|e(+)|2+|e(-)|2]/2Re[e(+)*e(-)]Im[e(+)*e(-)][|e(+)|2-|e(-)|2]/2,
(x0)2-(x1)2-(x2)2-(x3)2
=(x0)2-(x1)2-(x2)2-(x3)2=0,
(x0)2-(x1)2=(x0)2-(x1)2,
(x2)2+(x3)2=(x2)2+(x3)2,
(x0)2-(x1)2=(x2)2+(x3)2
=(x0)2-(x1)2=(x2)2+(x3)2.
{[|e0(+)|2+|e0(-)|2]/2}2-{Re[e0(+)*e0(-)]}2={[|e1(+)|2+|e1(-)|]/2}2-{Re[e1(+)*e1(-)]}2={[|e0(+)|2-|e0(-)|]/2}2+{Im[e0(+)*e0(-)]}2={[|e1(+)|2-|e1(-)|2]/2}2+{Im[e1(+)*e1(-)]}2.

Metrics