Abstract

We present an efficient method for designing birefringent filters comprising a number of birefringent sections with equal length and arbitrary orientation between two polarizers and for producing a specified spectral response in transmission. The method uses a digital filter design algorithm (i.e., the Remez algorithm) to determine an optimal polynomial approximation to obtain a specified finite impulse response, and a layer-peeling algorithm to calculate the filter structure parameters. The design procedure is demonstrated for a 14-section bandpass filter with sidelobes below -40 dB. The influence of errors in length and orientation of the birefringent sections on the filter’s spectral response is also discussed.

© 2002 Optical Society of America

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References

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  1. B. Lyot, “Optical apparatus with wide field using interference of polarized light,” C. R. Acad. Sci. 197, 1593 (1933).
  2. I. Solc, “Birefringent chain filters,” J. Opt. Soc. Am. 55, 621–625 (1965).
    [CrossRef]
  3. C. S. Brown, J. F. Kuhl, M. A. El, “Polarization-maintaining fiber birefringent filter for WDM,” SPIE998Components for Fiber Optic Applications and Coherent Lightwave Communications, 270–275 (1988).
  4. Q. Zou, “A high-performance, liquid-crystal based optical tunable birefringent filter,” Conference Digest. 2000 Conference on Lasers and Electro-Optics Europe, (Optical Society of America, Washington, D.C., 2000) pp. 132.
  5. J. Minowa, Y. Fujii, “High performance bandpass filter for WDM transmission,” Appl. Opt. 23, 193–194 (1984).
    [CrossRef]
  6. N. Takato, T. Komonato, A. Sugita, K. Jinguji, H. Toba, M. Kawachi, “Silica-based integrated optic Mach-Zender multi/demultiplexer family with channel spacing of 0.01–250 nm,” IEEE J. Sel. Areas Commun. 8, 1120–1127 (1990).
    [CrossRef]
  7. A. d’Alessandro, D. A. Smith, J. E. Baran, “Multichannel operation of an integrated acoustooptic wavelength routing switch for WDM system,” IEEE Photon. Technol. Lett. 6, 390–393 (1994).
    [CrossRef]
  8. A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
    [CrossRef]
  9. J. W. Evans, “Solc birefringent filter,” J. Opt. Soc. Am. 48, 142–145 (1958).
    [CrossRef]
  10. S. E. Harris, E. O. Ammann, I. C. Chang, “Optical network synthesis using birefringent crystals: I. Synthesis of lossless networks of equal-length crystals,” J. Opt. Soc. Am. 54, 1267–1279 (1964).
    [CrossRef]
  11. I. Schur, “Uber Potentzreihen, die im Innern des Einheitskreises Beschrankt Sind,” J. fur die Reine und Angewandte Mathematik 147, 205–232 (1917). (English translation reprinted in Operator Theory: Advances and Applications 18, 31–60 (1986).
  12. J. Pauly, P. Le Roux, Q. Nishimura, A. Macovski, “Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm,” IEEE Trans. Med. Imaging 10, 53–65 (1991).
    [CrossRef]
  13. T. W. Parks, C. S. Burrus, Digital Filter Design, (Wiley, New York, 1987).
  14. D. M. Etter, Engineering Problem Solving with matlab, (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  15. R. J. Pegis, “An exact design method for multiplayer dielectric films,” J. Opt. Soc. Am. 51, 1255–1264 (1961).
    [CrossRef]
  16. L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

1999 (1)

A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
[CrossRef]

1994 (1)

A. d’Alessandro, D. A. Smith, J. E. Baran, “Multichannel operation of an integrated acoustooptic wavelength routing switch for WDM system,” IEEE Photon. Technol. Lett. 6, 390–393 (1994).
[CrossRef]

1991 (1)

J. Pauly, P. Le Roux, Q. Nishimura, A. Macovski, “Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm,” IEEE Trans. Med. Imaging 10, 53–65 (1991).
[CrossRef]

1990 (1)

N. Takato, T. Komonato, A. Sugita, K. Jinguji, H. Toba, M. Kawachi, “Silica-based integrated optic Mach-Zender multi/demultiplexer family with channel spacing of 0.01–250 nm,” IEEE J. Sel. Areas Commun. 8, 1120–1127 (1990).
[CrossRef]

1984 (1)

1965 (1)

1964 (1)

1961 (1)

1958 (1)

1933 (1)

B. Lyot, “Optical apparatus with wide field using interference of polarized light,” C. R. Acad. Sci. 197, 1593 (1933).

1917 (1)

I. Schur, “Uber Potentzreihen, die im Innern des Einheitskreises Beschrankt Sind,” J. fur die Reine und Angewandte Mathematik 147, 205–232 (1917). (English translation reprinted in Operator Theory: Advances and Applications 18, 31–60 (1986).

Ammann, E. O.

Azaña, J.

A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
[CrossRef]

Baran, J. E.

A. d’Alessandro, D. A. Smith, J. E. Baran, “Multichannel operation of an integrated acoustooptic wavelength routing switch for WDM system,” IEEE Photon. Technol. Lett. 6, 390–393 (1994).
[CrossRef]

Brown, C. S.

C. S. Brown, J. F. Kuhl, M. A. El, “Polarization-maintaining fiber birefringent filter for WDM,” SPIE998Components for Fiber Optic Applications and Coherent Lightwave Communications, 270–275 (1988).

Burrus, C. S.

T. W. Parks, C. S. Burrus, Digital Filter Design, (Wiley, New York, 1987).

Carballar, A.

A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
[CrossRef]

Chang, I. C.

d’Alessandro, A.

A. d’Alessandro, D. A. Smith, J. E. Baran, “Multichannel operation of an integrated acoustooptic wavelength routing switch for WDM system,” IEEE Photon. Technol. Lett. 6, 390–393 (1994).
[CrossRef]

El, M. A.

C. S. Brown, J. F. Kuhl, M. A. El, “Polarization-maintaining fiber birefringent filter for WDM,” SPIE998Components for Fiber Optic Applications and Coherent Lightwave Communications, 270–275 (1988).

Etter, D. M.

D. M. Etter, Engineering Problem Solving with matlab, (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Evans, J. W.

Fujii, Y.

Gold, B.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Harris, S. E.

Jinguji, K.

N. Takato, T. Komonato, A. Sugita, K. Jinguji, H. Toba, M. Kawachi, “Silica-based integrated optic Mach-Zender multi/demultiplexer family with channel spacing of 0.01–250 nm,” IEEE J. Sel. Areas Commun. 8, 1120–1127 (1990).
[CrossRef]

Kawachi, M.

N. Takato, T. Komonato, A. Sugita, K. Jinguji, H. Toba, M. Kawachi, “Silica-based integrated optic Mach-Zender multi/demultiplexer family with channel spacing of 0.01–250 nm,” IEEE J. Sel. Areas Commun. 8, 1120–1127 (1990).
[CrossRef]

Komonato, T.

N. Takato, T. Komonato, A. Sugita, K. Jinguji, H. Toba, M. Kawachi, “Silica-based integrated optic Mach-Zender multi/demultiplexer family with channel spacing of 0.01–250 nm,” IEEE J. Sel. Areas Commun. 8, 1120–1127 (1990).
[CrossRef]

Kuhl, J. F.

C. S. Brown, J. F. Kuhl, M. A. El, “Polarization-maintaining fiber birefringent filter for WDM,” SPIE998Components for Fiber Optic Applications and Coherent Lightwave Communications, 270–275 (1988).

Le Roux, P.

J. Pauly, P. Le Roux, Q. Nishimura, A. Macovski, “Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm,” IEEE Trans. Med. Imaging 10, 53–65 (1991).
[CrossRef]

Lyot, B.

B. Lyot, “Optical apparatus with wide field using interference of polarized light,” C. R. Acad. Sci. 197, 1593 (1933).

Macovski, A.

J. Pauly, P. Le Roux, Q. Nishimura, A. Macovski, “Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm,” IEEE Trans. Med. Imaging 10, 53–65 (1991).
[CrossRef]

Minowa, J.

Muriel, M. A.

A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
[CrossRef]

Nishimura, Q.

J. Pauly, P. Le Roux, Q. Nishimura, A. Macovski, “Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm,” IEEE Trans. Med. Imaging 10, 53–65 (1991).
[CrossRef]

Parks, T. W.

T. W. Parks, C. S. Burrus, Digital Filter Design, (Wiley, New York, 1987).

Pauly, J.

J. Pauly, P. Le Roux, Q. Nishimura, A. Macovski, “Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm,” IEEE Trans. Med. Imaging 10, 53–65 (1991).
[CrossRef]

Pegis, R. J.

Rabiner, L. R.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Schur, I.

I. Schur, “Uber Potentzreihen, die im Innern des Einheitskreises Beschrankt Sind,” J. fur die Reine und Angewandte Mathematik 147, 205–232 (1917). (English translation reprinted in Operator Theory: Advances and Applications 18, 31–60 (1986).

Smith, D. A.

A. d’Alessandro, D. A. Smith, J. E. Baran, “Multichannel operation of an integrated acoustooptic wavelength routing switch for WDM system,” IEEE Photon. Technol. Lett. 6, 390–393 (1994).
[CrossRef]

Solc, I.

Sugita, A.

N. Takato, T. Komonato, A. Sugita, K. Jinguji, H. Toba, M. Kawachi, “Silica-based integrated optic Mach-Zender multi/demultiplexer family with channel spacing of 0.01–250 nm,” IEEE J. Sel. Areas Commun. 8, 1120–1127 (1990).
[CrossRef]

Takato, N.

N. Takato, T. Komonato, A. Sugita, K. Jinguji, H. Toba, M. Kawachi, “Silica-based integrated optic Mach-Zender multi/demultiplexer family with channel spacing of 0.01–250 nm,” IEEE J. Sel. Areas Commun. 8, 1120–1127 (1990).
[CrossRef]

Toba, H.

N. Takato, T. Komonato, A. Sugita, K. Jinguji, H. Toba, M. Kawachi, “Silica-based integrated optic Mach-Zender multi/demultiplexer family with channel spacing of 0.01–250 nm,” IEEE J. Sel. Areas Commun. 8, 1120–1127 (1990).
[CrossRef]

Zou, Q.

Q. Zou, “A high-performance, liquid-crystal based optical tunable birefringent filter,” Conference Digest. 2000 Conference on Lasers and Electro-Optics Europe, (Optical Society of America, Washington, D.C., 2000) pp. 132.

Appl. Opt. (1)

C. R. Acad. Sci. (1)

B. Lyot, “Optical apparatus with wide field using interference of polarized light,” C. R. Acad. Sci. 197, 1593 (1933).

IEEE J. Sel. Areas Commun. (1)

N. Takato, T. Komonato, A. Sugita, K. Jinguji, H. Toba, M. Kawachi, “Silica-based integrated optic Mach-Zender multi/demultiplexer family with channel spacing of 0.01–250 nm,” IEEE J. Sel. Areas Commun. 8, 1120–1127 (1990).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

A. d’Alessandro, D. A. Smith, J. E. Baran, “Multichannel operation of an integrated acoustooptic wavelength routing switch for WDM system,” IEEE Photon. Technol. Lett. 6, 390–393 (1994).
[CrossRef]

A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
[CrossRef]

IEEE Trans. Med. Imaging (1)

J. Pauly, P. Le Roux, Q. Nishimura, A. Macovski, “Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm,” IEEE Trans. Med. Imaging 10, 53–65 (1991).
[CrossRef]

J. fur die Reine und Angewandte Mathematik (1)

I. Schur, “Uber Potentzreihen, die im Innern des Einheitskreises Beschrankt Sind,” J. fur die Reine und Angewandte Mathematik 147, 205–232 (1917). (English translation reprinted in Operator Theory: Advances and Applications 18, 31–60 (1986).

J. Opt. Soc. Am. (4)

Other (5)

C. S. Brown, J. F. Kuhl, M. A. El, “Polarization-maintaining fiber birefringent filter for WDM,” SPIE998Components for Fiber Optic Applications and Coherent Lightwave Communications, 270–275 (1988).

Q. Zou, “A high-performance, liquid-crystal based optical tunable birefringent filter,” Conference Digest. 2000 Conference on Lasers and Electro-Optics Europe, (Optical Society of America, Washington, D.C., 2000) pp. 132.

T. W. Parks, C. S. Burrus, Digital Filter Design, (Wiley, New York, 1987).

D. M. Etter, Engineering Problem Solving with matlab, (Prentice-Hall, Englewood Cliffs, N.J., 1993).

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

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Figures (5)

Fig. 1
Fig. 1

Birefringent filter configuration.

Fig. 2
Fig. 2

Performance of the 14-section birefringent filter (a) Normalized specified response. The solid wave represents the desired response and the dotted wave is the approximation by Remez algorithm in matlab. (b) Designed response.

Fig. 3
Fig. 3

Ideal spectral response of a lowpass filter.

Fig. 4
Fig. 4

Response of the 44-section birefringent filter.

Fig. 5
Fig. 5

Filter spectral response as length and angle errors are introduced. Ideal response is represented by a solid wave and the response with errors by a dotted wave: (a) ΔL ≤ 10-5 m, (b) Δφ ≤ 10-2 rad.

Tables (1)

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Table 1 Polynomial Coefficients and Orientation Angles

Equations (19)

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EA1EA2=RφpMφi, ηPxEx0Ey0.
Rφp=cos φpsin φp-sin φpcos φp.
Px=1000.
RφpMφi, η= RφpR-φnRηRφn  R-φ2RηRφ2R-φ1RηRφ1,= Rφp-φnRηRφn-φn-1RηRφ2-φ1RηRφ1,= RθnRηRθn-1Rη  Rθ1RηRθ0,=QnQn-1  Q1Q0.
Rθi=cos θisin θi-sin θicos θi,
Rη=ejη/200e-jη/2.
RφpMφi, η=αn-βn*βnαn*,Qi=ci-di*dici*,
αn-βn*βnαn*= cn-dn*dncn*cn-1-dn-1*dn-1cn-1*  c1-d1*d1c1*c0-d0*d0c0*,
αiβi= ci-di*dici*αi-1βi-1.
Qi=RθiRη=z1/2Ci-SiSiCi100z-1
αiβi=z1/2Ci-SiSiCi100z-1αi-1βi-1.
AiBi= Ci-Siz-1SiCiz-1Ai-1Bi-1.
A0B0= cos θ1-sin θ1, A1B1= C1 cos θ1+S1 sin θ1z-1S1 cos θ1-C1 sin θ1z-1,  AnBn= i=0n aiz-ii=0n biz-i.
Ai-1Bi-1= CiSi-SizCizAiBi= CiAi+SiBi-SiAi+CiBiz.
CiAi,i+SiBi,i=0,-SiAi,0+ CiBi,0=0.
θi=tg-1Ai,iBi,i=tg-1-Bi,0Ai,0,
φp=θn+φn, φn=θn-1+φn-1,  φ2=θ1+φ1,φ1=θ0.
EA1EA2= αn-βn*βnαn*10=αnβn,
|Anz|2+|Bnz|2=1.

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