Abstract

Spectral analysis of device (instrument) operators, as an alternative approach to dynamic polarization phenomena, is presented by means of the example of a classical time-varying optical device: the rotating bi refringent plate. The mutual coherence matrices of right and left circularly polarized light are emphasized in the spectral structure of the rotating-plate matrix, and their physical significance is analyzed.

© 2001 Optical Society of America

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References

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  1. J. Garcia-Mateos, F. Canal, M. Haelterman, “Passive fiber ring flip-flop memory based on polarization dynamics,” Opt. Commun. 137, 427–436 (1997).
    [CrossRef]
  2. A. D. May, I. Stephan, “Polarization dynamics in quasi-isotropic lasers,” in Coherence and Quantum Optics, J. H. Eberly, ed. (Plenum, New York, 1990).
  3. A. P. Voitovich, A. M. Kulminskii, V. N. Severikov, “Nonlinear dynamics of laser system at intracavity modulation of polarization,” Opt. Commun. 126, 152–166 (1996).
    [CrossRef]
  4. L. P. Svirina, V. G. Gudelev, Yu. P. Zhurik, “Spontaneous pulsations in gas class-A lasers with weakly anisotropic cavities,” Phys. Rev. A 56, 5053–5064 (1997).
    [CrossRef]
  5. D. E. Aspnes, P. S. Hauge, “Rotating compensator/analyzer fixed-analyzer ellipsometer: analysis and comparison to other automatic ellipsometers,” J. Opt. Soc. Am. 66, 949–954 (1976).
    [CrossRef]
  6. R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
    [CrossRef]
  7. G. E. Jellison, “Two-channel polarization modulation ellipsometers,” Appl. Opt. 29, 959–974 (1990).
    [CrossRef] [PubMed]
  8. T. Tudor, “The evolution of light polarization state along a KDP electro-optical modulator,” Optik 109, 27–34 (1998).
  9. T. Tudor, I. Vinkler, “Time-varying coherency matrices and spectral coherency matrices,” Pure Appl. Opt. 7, 1451–1457 (1998).
    [CrossRef]
  10. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), p. 363.
  11. C. Brosseau, Fundamentals of Polarized Light. A Statistical Optics Approach (Wiley, New York, 1998), p. 93.
  12. A. S. Marathay, “Operator formalism in the theory of partial polarization,” J. Opt. Soc. Am. 55, 969–980 (1965).
  13. J. Ben Uri, “Polarization and interference in optics: Part II,” Optik 47, 405–420 (1977).
  14. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).
  15. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
  16. J. W. Simmons, M. J. Guttmann, States, Waves and Photons: A Modern Introduction to Light (Addison-Wesley, Reading, Mass., 1970).
  17. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1996).
  18. I. P. Kaminow, E. H. Turner, “Electro-optic light modulators,” Appl. Opt. 5, 1612–1628 (1966).
    [CrossRef] [PubMed]
  19. P. Connes, D. H. Tuan, J. Pinard, “Décomposition de raises spectrales par modulation en haute fréquence,” J. Phys. Radium 23, 173–183 (1962).
    [CrossRef]
  20. A. Linke, ed., Special issue on coherent communication, J. Lightwave Technol. LT-8(3) (1990).
  21. F. Durst, A. Melling, A. Whitelaw, Principles and Practice of Laser-Doppler Velocimetry (Academic, London, 1981).
  22. J. P. Campbell, “Rotating waveplate optical frequency shifting in lithium niobate,” IEEE J. Quantum Electron. QE-7, 450–457 (1971).
    [CrossRef]
  23. I. W. Stallard et al., “Electro-optic frequency translators and their applications in coherent optical fiber systems,” Br. Telecom. J. 4, 16–22 (1986).
  24. C. F. Buhrer, D. Baird, E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
    [CrossRef]
  25. C. J. Peters, “Optical frequency translator using two phase modulators in tandem,” Appl. Opt. 4, 857–861 (1965).
    [CrossRef]
  26. L. E. Drain, B. C. Moss, “The frequency shifting of laser light by electro-optic techniques,” Opto-Electron. 4, 429–439 (1972).
    [CrossRef]
  27. M. Izutsu, S. Shikama, T. Sueta, “Integrated optical SSB modulator/frequency shifter,” IEEE J. Quantum Electron. QE-17, 2225–2229 (1981).
    [CrossRef]
  28. P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, Oxford, UK, 1947), Preface, p. VIII.

1998 (2)

T. Tudor, “The evolution of light polarization state along a KDP electro-optical modulator,” Optik 109, 27–34 (1998).

T. Tudor, I. Vinkler, “Time-varying coherency matrices and spectral coherency matrices,” Pure Appl. Opt. 7, 1451–1457 (1998).
[CrossRef]

1997 (2)

J. Garcia-Mateos, F. Canal, M. Haelterman, “Passive fiber ring flip-flop memory based on polarization dynamics,” Opt. Commun. 137, 427–436 (1997).
[CrossRef]

L. P. Svirina, V. G. Gudelev, Yu. P. Zhurik, “Spontaneous pulsations in gas class-A lasers with weakly anisotropic cavities,” Phys. Rev. A 56, 5053–5064 (1997).
[CrossRef]

1996 (1)

A. P. Voitovich, A. M. Kulminskii, V. N. Severikov, “Nonlinear dynamics of laser system at intracavity modulation of polarization,” Opt. Commun. 126, 152–166 (1996).
[CrossRef]

1990 (3)

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
[CrossRef]

A. Linke, ed., Special issue on coherent communication, J. Lightwave Technol. LT-8(3) (1990).

G. E. Jellison, “Two-channel polarization modulation ellipsometers,” Appl. Opt. 29, 959–974 (1990).
[CrossRef] [PubMed]

1986 (1)

I. W. Stallard et al., “Electro-optic frequency translators and their applications in coherent optical fiber systems,” Br. Telecom. J. 4, 16–22 (1986).

1981 (1)

M. Izutsu, S. Shikama, T. Sueta, “Integrated optical SSB modulator/frequency shifter,” IEEE J. Quantum Electron. QE-17, 2225–2229 (1981).
[CrossRef]

1977 (1)

J. Ben Uri, “Polarization and interference in optics: Part II,” Optik 47, 405–420 (1977).

1976 (1)

1972 (1)

L. E. Drain, B. C. Moss, “The frequency shifting of laser light by electro-optic techniques,” Opto-Electron. 4, 429–439 (1972).
[CrossRef]

1971 (1)

J. P. Campbell, “Rotating waveplate optical frequency shifting in lithium niobate,” IEEE J. Quantum Electron. QE-7, 450–457 (1971).
[CrossRef]

1966 (1)

1965 (2)

1962 (2)

C. F. Buhrer, D. Baird, E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

P. Connes, D. H. Tuan, J. Pinard, “Décomposition de raises spectrales par modulation en haute fréquence,” J. Phys. Radium 23, 173–183 (1962).
[CrossRef]

Aspnes, D. E.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1996).

Baird, D.

C. F. Buhrer, D. Baird, E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1996).

Ben Uri, J.

J. Ben Uri, “Polarization and interference in optics: Part II,” Optik 47, 405–420 (1977).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light. A Statistical Optics Approach (Wiley, New York, 1998), p. 93.

Buhrer, C. F.

C. F. Buhrer, D. Baird, E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Campbell, J. P.

J. P. Campbell, “Rotating waveplate optical frequency shifting in lithium niobate,” IEEE J. Quantum Electron. QE-7, 450–457 (1971).
[CrossRef]

Canal, F.

J. Garcia-Mateos, F. Canal, M. Haelterman, “Passive fiber ring flip-flop memory based on polarization dynamics,” Opt. Commun. 137, 427–436 (1997).
[CrossRef]

Collins, R. W.

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
[CrossRef]

Connes, P.

P. Connes, D. H. Tuan, J. Pinard, “Décomposition de raises spectrales par modulation en haute fréquence,” J. Phys. Radium 23, 173–183 (1962).
[CrossRef]

Conwell, E. M.

C. F. Buhrer, D. Baird, E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Dirac, P. A. M.

P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, Oxford, UK, 1947), Preface, p. VIII.

Drain, L. E.

L. E. Drain, B. C. Moss, “The frequency shifting of laser light by electro-optic techniques,” Opto-Electron. 4, 429–439 (1972).
[CrossRef]

Durst, F.

F. Durst, A. Melling, A. Whitelaw, Principles and Practice of Laser-Doppler Velocimetry (Academic, London, 1981).

Garcia-Mateos, J.

J. Garcia-Mateos, F. Canal, M. Haelterman, “Passive fiber ring flip-flop memory based on polarization dynamics,” Opt. Commun. 137, 427–436 (1997).
[CrossRef]

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Gudelev, V. G.

L. P. Svirina, V. G. Gudelev, Yu. P. Zhurik, “Spontaneous pulsations in gas class-A lasers with weakly anisotropic cavities,” Phys. Rev. A 56, 5053–5064 (1997).
[CrossRef]

Guttmann, M. J.

J. W. Simmons, M. J. Guttmann, States, Waves and Photons: A Modern Introduction to Light (Addison-Wesley, Reading, Mass., 1970).

Haelterman, M.

J. Garcia-Mateos, F. Canal, M. Haelterman, “Passive fiber ring flip-flop memory based on polarization dynamics,” Opt. Commun. 137, 427–436 (1997).
[CrossRef]

Hauge, P. S.

Izutsu, M.

M. Izutsu, S. Shikama, T. Sueta, “Integrated optical SSB modulator/frequency shifter,” IEEE J. Quantum Electron. QE-17, 2225–2229 (1981).
[CrossRef]

Jellison, G. E.

Kaminow, I. P.

Kulminskii, A. M.

A. P. Voitovich, A. M. Kulminskii, V. N. Severikov, “Nonlinear dynamics of laser system at intracavity modulation of polarization,” Opt. Commun. 126, 152–166 (1996).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), p. 363.

Marathay, A. S.

May, A. D.

A. D. May, I. Stephan, “Polarization dynamics in quasi-isotropic lasers,” in Coherence and Quantum Optics, J. H. Eberly, ed. (Plenum, New York, 1990).

Melling, A.

F. Durst, A. Melling, A. Whitelaw, Principles and Practice of Laser-Doppler Velocimetry (Academic, London, 1981).

Moss, B. C.

L. E. Drain, B. C. Moss, “The frequency shifting of laser light by electro-optic techniques,” Opto-Electron. 4, 429–439 (1972).
[CrossRef]

Peters, C. J.

Pinard, J.

P. Connes, D. H. Tuan, J. Pinard, “Décomposition de raises spectrales par modulation en haute fréquence,” J. Phys. Radium 23, 173–183 (1962).
[CrossRef]

Severikov, V. N.

A. P. Voitovich, A. M. Kulminskii, V. N. Severikov, “Nonlinear dynamics of laser system at intracavity modulation of polarization,” Opt. Commun. 126, 152–166 (1996).
[CrossRef]

Shikama, S.

M. Izutsu, S. Shikama, T. Sueta, “Integrated optical SSB modulator/frequency shifter,” IEEE J. Quantum Electron. QE-17, 2225–2229 (1981).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

Simmons, J. W.

J. W. Simmons, M. J. Guttmann, States, Waves and Photons: A Modern Introduction to Light (Addison-Wesley, Reading, Mass., 1970).

Stallard, I. W.

I. W. Stallard et al., “Electro-optic frequency translators and their applications in coherent optical fiber systems,” Br. Telecom. J. 4, 16–22 (1986).

Stephan, I.

A. D. May, I. Stephan, “Polarization dynamics in quasi-isotropic lasers,” in Coherence and Quantum Optics, J. H. Eberly, ed. (Plenum, New York, 1990).

Sueta, T.

M. Izutsu, S. Shikama, T. Sueta, “Integrated optical SSB modulator/frequency shifter,” IEEE J. Quantum Electron. QE-17, 2225–2229 (1981).
[CrossRef]

Svirina, L. P.

L. P. Svirina, V. G. Gudelev, Yu. P. Zhurik, “Spontaneous pulsations in gas class-A lasers with weakly anisotropic cavities,” Phys. Rev. A 56, 5053–5064 (1997).
[CrossRef]

Tuan, D. H.

P. Connes, D. H. Tuan, J. Pinard, “Décomposition de raises spectrales par modulation en haute fréquence,” J. Phys. Radium 23, 173–183 (1962).
[CrossRef]

Tudor, T.

T. Tudor, “The evolution of light polarization state along a KDP electro-optical modulator,” Optik 109, 27–34 (1998).

T. Tudor, I. Vinkler, “Time-varying coherency matrices and spectral coherency matrices,” Pure Appl. Opt. 7, 1451–1457 (1998).
[CrossRef]

Turner, E. H.

Vinkler, I.

T. Tudor, I. Vinkler, “Time-varying coherency matrices and spectral coherency matrices,” Pure Appl. Opt. 7, 1451–1457 (1998).
[CrossRef]

Voitovich, A. P.

A. P. Voitovich, A. M. Kulminskii, V. N. Severikov, “Nonlinear dynamics of laser system at intracavity modulation of polarization,” Opt. Commun. 126, 152–166 (1996).
[CrossRef]

Whitelaw, A.

F. Durst, A. Melling, A. Whitelaw, Principles and Practice of Laser-Doppler Velocimetry (Academic, London, 1981).

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), p. 363.

Zhurik, Yu. P.

L. P. Svirina, V. G. Gudelev, Yu. P. Zhurik, “Spontaneous pulsations in gas class-A lasers with weakly anisotropic cavities,” Phys. Rev. A 56, 5053–5064 (1997).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. F. Buhrer, D. Baird, E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Br. Telecom. J. (1)

I. W. Stallard et al., “Electro-optic frequency translators and their applications in coherent optical fiber systems,” Br. Telecom. J. 4, 16–22 (1986).

IEEE J. Quantum Electron. (2)

M. Izutsu, S. Shikama, T. Sueta, “Integrated optical SSB modulator/frequency shifter,” IEEE J. Quantum Electron. QE-17, 2225–2229 (1981).
[CrossRef]

J. P. Campbell, “Rotating waveplate optical frequency shifting in lithium niobate,” IEEE J. Quantum Electron. QE-7, 450–457 (1971).
[CrossRef]

J. Lightwave Technol. (1)

A. Linke, ed., Special issue on coherent communication, J. Lightwave Technol. LT-8(3) (1990).

J. Opt. Soc. Am. (2)

J. Phys. Radium (1)

P. Connes, D. H. Tuan, J. Pinard, “Décomposition de raises spectrales par modulation en haute fréquence,” J. Phys. Radium 23, 173–183 (1962).
[CrossRef]

Opt. Commun. (2)

J. Garcia-Mateos, F. Canal, M. Haelterman, “Passive fiber ring flip-flop memory based on polarization dynamics,” Opt. Commun. 137, 427–436 (1997).
[CrossRef]

A. P. Voitovich, A. M. Kulminskii, V. N. Severikov, “Nonlinear dynamics of laser system at intracavity modulation of polarization,” Opt. Commun. 126, 152–166 (1996).
[CrossRef]

Optik (2)

T. Tudor, “The evolution of light polarization state along a KDP electro-optical modulator,” Optik 109, 27–34 (1998).

J. Ben Uri, “Polarization and interference in optics: Part II,” Optik 47, 405–420 (1977).

Opto-Electron. (1)

L. E. Drain, B. C. Moss, “The frequency shifting of laser light by electro-optic techniques,” Opto-Electron. 4, 429–439 (1972).
[CrossRef]

Phys. Rev. A (1)

L. P. Svirina, V. G. Gudelev, Yu. P. Zhurik, “Spontaneous pulsations in gas class-A lasers with weakly anisotropic cavities,” Phys. Rev. A 56, 5053–5064 (1997).
[CrossRef]

Pure Appl. Opt. (1)

T. Tudor, I. Vinkler, “Time-varying coherency matrices and spectral coherency matrices,” Pure Appl. Opt. 7, 1451–1457 (1998).
[CrossRef]

Rev. Sci. Instrum. (1)

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
[CrossRef]

Other (9)

P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, Oxford, UK, 1947), Preface, p. VIII.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), p. 363.

C. Brosseau, Fundamentals of Polarized Light. A Statistical Optics Approach (Wiley, New York, 1998), p. 93.

A. D. May, I. Stephan, “Polarization dynamics in quasi-isotropic lasers,” in Coherence and Quantum Optics, J. H. Eberly, ed. (Plenum, New York, 1990).

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

J. W. Simmons, M. J. Guttmann, States, Waves and Photons: A Modern Introduction to Light (Addison-Wesley, Reading, Mass., 1970).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1996).

F. Durst, A. Melling, A. Whitelaw, Principles and Practice of Laser-Doppler Velocimetry (Academic, London, 1981).

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Equations (21)

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M0=exp(iϕ)00exp(-iϕ),
M=R(-θ)M0R(θ),
R(θ)=cos θsin θ-sin θcos θ.
M=R(-Ωt)M0R(Ωt),
R(Ωt)=cos Ωtsin Ωt-sin Ωtcos Ωt.
M(ϕ, Ω, t)=exp(iϕ)cos2 Ωt+exp(-iϕ)sin2 Ωt[exp(iϕ)-exp(-iϕ)]sin Ωt cos Ωt[exp(iϕ)-exp(-iϕ)]sin Ωt cos Ωtexp(iϕ)sin2 Ωt+exp(-iϕ)cos2 Ωt=cos ϕ+i sin ϕ cos 2Ωti sin ϕ sin 2Ωti sin ϕ sin 2Ωtcos ϕ-i sin ϕ cos 2Ωt=cos ϕ+i 12 sin ϕ[exp(i2Ωt)+exp(-i2Ωt)]12 sin ϕ[exp(i2Ωt)-exp(-i2Ωt)]12 sin ϕ[exp(i2Ωt)-exp(-i2Ωt)]cos ϕ-i 12 sin ϕ[exp(i2Ωt)+exp(-i2Ωt)].
M(ϕ, Ω, t)=cos ϕ1001+12 sin ϕi11-iexp(i2Ωt)+12 sin ϕi-1-1-iexp(-i2Ωt).
|Eout(t)=M(ϕ, Ω, t)|Ein.
|Ein=a1iexp(iωt).
|Eout(t)=a cos ϕ1iexp(iωt)+ia sin ϕ1-i×exp[i(ω+2Ω)t].
|Eout(t)=ai1exp[i(ω+2Ω)t].
B1=12 i11-i,B2=12 i-1-1-i
|RL|=12 1i[1i]=12 1ii-1=-i2 i-1-1-i=-iB2,
|LR|=12 1-i[1-i]=12 1-i-i-1=-i2 i11-i=-iB1.
M(ϕ, Ω, t)=cos ϕ·I+i sin ϕ|LR|exp(i2Ωt)+i sin ϕ|RL|exp(-i2Ωt).
|Eout=M(ϕ, Ω, t)|Ein=M(ϕ, Ω, t)|R=cos ϕ I|Rexp(iωt)+i sin ϕ|LR|R×exp[i(ω+2Ω)t]+i sin ϕ|RL|R×exp[i(ω-2Ω)t]=cos ϕ|Rexp(i2ωt)+i sin ϕ|L×exp[i(ω+2Ω)t].
|Eout=M(ϕ, Ω, t)|Ein=M(ϕ, Ω, t)|L=cos ϕ I|Lexp(i2ωt)+i sin ϕ|LR|L×exp[i(ω+2Ω)t]+i sin ϕ|RL|L×exp[i(ω-2Ω)t]=cos ϕ|Lexp(i2ωt)+i sin ϕ|R×exp[i(ω-2Ω)t].
|Eout=i|Lexp[i(ω+2Ω)t],
|Eout=i|Rexp[i(ω-2Ω)t],
|Eout=M(ϕ, Ω, t)|Ein=cos ϕ I|Eexp(i2ωt)+i sin ϕ|LR|E×exp[i(ω+2Ω)t]+i sin ϕ|RL|E×exp[i(ω-2Ω)t].
|Eout=|LR|Eexp[i(ω+2Ω)t]+|RL|E×exp[i(ω-2Ω)t].

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