Abstract

The characteristics of light propagating near the axis of a birefringent filter are studied. A generalized formulation to describe the nearly-off-axis transmissivity of a Solc birefringent filter is derived. On this basis, the polarization conoscopic figures of Solc filters with different numbers of birefringent plates are simulated. Furthermore the variation of spectral transmission with angle of incidence is analyzed, and the field-of-view transmissivity and the spectral transmissivity averaged with respect to the spread of incident light are given. Primary experiments for verification are also demonstrated.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Evans, “The birefringent filter,” J. Opt. Soc. Am. 3, 229–242 (1949).
    [CrossRef]
  2. I. Solc, “Birefringent chain filters,” J. Opt. Soc. Am. 6, 621–625 (1965).
    [CrossRef]
  3. S. E. Harris, E. O. Amman, I. C. Chang, “Optical network synthesis using birefringent crystals. I. Synthesis of lossless networks of equal-length crystals,” J. Opt. Soc. Am. 10, 1267–1279 (1964).
    [CrossRef]
  4. E. O. Ammann, “Optical network synthesis using birefringent crystals,” J. Opt. Soc. Am. 56, 952–955 (1966).
    [CrossRef]
  5. E. O. Ammann, “Synthesis of optical birefringent network,” in Progress in Optics, Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 123–177.
  6. W. J. Carlsen, C. F. Buhrer, “Flat passband birefringent wavelength-division multiplexers,” Electron. Lett. 3, 106–107 (1987).
    [CrossRef]
  7. P. Melman, W. J. Carlsen, B. Foley, “Tunable birefringent wavelength-division multiplexer/demultiplexer,” Electron. Lett. 21, 635–635 (1985).
    [CrossRef]
  8. S. V. Kartalopoulos, Introduction to DWDM Technology: Data in a Rainbow (Wiley-IEEE Press, New York, 2000).
  9. A. M. Title, W. J. Rosenberg, “Improvements in birefrin-gent filters. 5: Field of view effects,” Appl. Opt. 18, 3443–3456 (1979).
    [CrossRef] [PubMed]
  10. A. L. Bloom, “Modes of a laser resonator containing tilted birefringent plates,” J. Opt. Soc. Am. 4, 447–452 (1974).
    [CrossRef]
  11. D. R. Preuss, J. L. Gole, “Three-stage birefringent filter tuning smoothly over the visible region: theoretical treatment and experimental design,” Appl. Opt. 19, 702–710 (1980).
    [CrossRef] [PubMed]
  12. G. Holtom, O. Teschke, “Design of a birefringent filter for high-power dye lasers,” IEEE J. Quantum Electron. QE-10, 577–579 (1974).
    [CrossRef]
  13. S. Zhu, “Birefringent filter with tilted optic axis for tuning dye lasers: theory and design,” Appl. Opt. 29, 410–415 (1990).
    [CrossRef] [PubMed]
  14. F. Ortwein, J. Mental, E. Schmidt, “A birefringent filter as a tunning element for a multilane He–Ne laser,” J. Phys. D 22, 488–491 (1989).
    [CrossRef]
  15. E. Schmidt, T. Mentael, K.-H. Krahn, “Three-color He–Se+ laser with optimized output power,” Appl. Opt. 25, 1383–1388 (1986).
    [CrossRef]
  16. X. Wang, J. Yao, “Transmitted and tuning characteristics of birefringent filters,” Appl. Opt. 31, 4505–4508 (1992).
    [CrossRef] [PubMed]
  17. J. Mentel, E. Schmidt, T. Mavrudis, “Birefringent filter with arbitrary orientation of the optic axis: an analysis of improved accuracy,” Appl. Opt. 31, 5022–5029 (1992).
    [CrossRef] [PubMed]
  18. Z. Shao, “Precise and versatile formula for birefringent filters,” Appl. Opt. 35, 4147–4151 (1996).
    [CrossRef] [PubMed]
  19. A Yariv, P Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 152.
  20. P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 4, 423–438 (1977).
    [CrossRef]

1996 (1)

1992 (2)

1990 (1)

1989 (1)

F. Ortwein, J. Mental, E. Schmidt, “A birefringent filter as a tunning element for a multilane He–Ne laser,” J. Phys. D 22, 488–491 (1989).
[CrossRef]

1987 (1)

W. J. Carlsen, C. F. Buhrer, “Flat passband birefringent wavelength-division multiplexers,” Electron. Lett. 3, 106–107 (1987).
[CrossRef]

1986 (1)

1985 (1)

P. Melman, W. J. Carlsen, B. Foley, “Tunable birefringent wavelength-division multiplexer/demultiplexer,” Electron. Lett. 21, 635–635 (1985).
[CrossRef]

1980 (1)

1979 (1)

1977 (1)

P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 4, 423–438 (1977).
[CrossRef]

1974 (2)

G. Holtom, O. Teschke, “Design of a birefringent filter for high-power dye lasers,” IEEE J. Quantum Electron. QE-10, 577–579 (1974).
[CrossRef]

A. L. Bloom, “Modes of a laser resonator containing tilted birefringent plates,” J. Opt. Soc. Am. 4, 447–452 (1974).
[CrossRef]

1966 (1)

1965 (1)

I. Solc, “Birefringent chain filters,” J. Opt. Soc. Am. 6, 621–625 (1965).
[CrossRef]

1964 (1)

S. E. Harris, E. O. Amman, I. C. Chang, “Optical network synthesis using birefringent crystals. I. Synthesis of lossless networks of equal-length crystals,” J. Opt. Soc. Am. 10, 1267–1279 (1964).
[CrossRef]

1949 (1)

J. W. Evans, “The birefringent filter,” J. Opt. Soc. Am. 3, 229–242 (1949).
[CrossRef]

Amman, E. O.

S. E. Harris, E. O. Amman, I. C. Chang, “Optical network synthesis using birefringent crystals. I. Synthesis of lossless networks of equal-length crystals,” J. Opt. Soc. Am. 10, 1267–1279 (1964).
[CrossRef]

Ammann, E. O.

E. O. Ammann, “Optical network synthesis using birefringent crystals,” J. Opt. Soc. Am. 56, 952–955 (1966).
[CrossRef]

E. O. Ammann, “Synthesis of optical birefringent network,” in Progress in Optics, Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 123–177.

Bloom, A. L.

A. L. Bloom, “Modes of a laser resonator containing tilted birefringent plates,” J. Opt. Soc. Am. 4, 447–452 (1974).
[CrossRef]

Buhrer, C. F.

W. J. Carlsen, C. F. Buhrer, “Flat passband birefringent wavelength-division multiplexers,” Electron. Lett. 3, 106–107 (1987).
[CrossRef]

Carlsen, W. J.

W. J. Carlsen, C. F. Buhrer, “Flat passband birefringent wavelength-division multiplexers,” Electron. Lett. 3, 106–107 (1987).
[CrossRef]

P. Melman, W. J. Carlsen, B. Foley, “Tunable birefringent wavelength-division multiplexer/demultiplexer,” Electron. Lett. 21, 635–635 (1985).
[CrossRef]

Chang, I. C.

S. E. Harris, E. O. Amman, I. C. Chang, “Optical network synthesis using birefringent crystals. I. Synthesis of lossless networks of equal-length crystals,” J. Opt. Soc. Am. 10, 1267–1279 (1964).
[CrossRef]

Evans, J. W.

J. W. Evans, “The birefringent filter,” J. Opt. Soc. Am. 3, 229–242 (1949).
[CrossRef]

Foley, B.

P. Melman, W. J. Carlsen, B. Foley, “Tunable birefringent wavelength-division multiplexer/demultiplexer,” Electron. Lett. 21, 635–635 (1985).
[CrossRef]

Gole, J. L.

Harris, S. E.

S. E. Harris, E. O. Amman, I. C. Chang, “Optical network synthesis using birefringent crystals. I. Synthesis of lossless networks of equal-length crystals,” J. Opt. Soc. Am. 10, 1267–1279 (1964).
[CrossRef]

Holtom, G.

G. Holtom, O. Teschke, “Design of a birefringent filter for high-power dye lasers,” IEEE J. Quantum Electron. QE-10, 577–579 (1974).
[CrossRef]

Hong, C. S.

P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 4, 423–438 (1977).
[CrossRef]

Kartalopoulos, S. V.

S. V. Kartalopoulos, Introduction to DWDM Technology: Data in a Rainbow (Wiley-IEEE Press, New York, 2000).

Krahn, K.-H.

Mavrudis, T.

Melman, P.

P. Melman, W. J. Carlsen, B. Foley, “Tunable birefringent wavelength-division multiplexer/demultiplexer,” Electron. Lett. 21, 635–635 (1985).
[CrossRef]

Mentael, T.

Mental, J.

F. Ortwein, J. Mental, E. Schmidt, “A birefringent filter as a tunning element for a multilane He–Ne laser,” J. Phys. D 22, 488–491 (1989).
[CrossRef]

Mentel, J.

Ortwein, F.

F. Ortwein, J. Mental, E. Schmidt, “A birefringent filter as a tunning element for a multilane He–Ne laser,” J. Phys. D 22, 488–491 (1989).
[CrossRef]

Preuss, D. R.

Rosenberg, W. J.

Schmidt, E.

Shao, Z.

Solc, I.

I. Solc, “Birefringent chain filters,” J. Opt. Soc. Am. 6, 621–625 (1965).
[CrossRef]

Teschke, O.

G. Holtom, O. Teschke, “Design of a birefringent filter for high-power dye lasers,” IEEE J. Quantum Electron. QE-10, 577–579 (1974).
[CrossRef]

Title, A. M.

Wang, X.

Yao, J.

Yariv, A

A Yariv, P Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 152.

Yariv, A.

P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 4, 423–438 (1977).
[CrossRef]

Yeh, P

A Yariv, P Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 152.

Yeh, P.

P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 4, 423–438 (1977).
[CrossRef]

Zhu, S.

Appl. Opt. (7)

Electron. Lett. (2)

W. J. Carlsen, C. F. Buhrer, “Flat passband birefringent wavelength-division multiplexers,” Electron. Lett. 3, 106–107 (1987).
[CrossRef]

P. Melman, W. J. Carlsen, B. Foley, “Tunable birefringent wavelength-division multiplexer/demultiplexer,” Electron. Lett. 21, 635–635 (1985).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Holtom, O. Teschke, “Design of a birefringent filter for high-power dye lasers,” IEEE J. Quantum Electron. QE-10, 577–579 (1974).
[CrossRef]

J. Opt. Soc. Am. (6)

J. W. Evans, “The birefringent filter,” J. Opt. Soc. Am. 3, 229–242 (1949).
[CrossRef]

I. Solc, “Birefringent chain filters,” J. Opt. Soc. Am. 6, 621–625 (1965).
[CrossRef]

S. E. Harris, E. O. Amman, I. C. Chang, “Optical network synthesis using birefringent crystals. I. Synthesis of lossless networks of equal-length crystals,” J. Opt. Soc. Am. 10, 1267–1279 (1964).
[CrossRef]

A. L. Bloom, “Modes of a laser resonator containing tilted birefringent plates,” J. Opt. Soc. Am. 4, 447–452 (1974).
[CrossRef]

P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 4, 423–438 (1977).
[CrossRef]

E. O. Ammann, “Optical network synthesis using birefringent crystals,” J. Opt. Soc. Am. 56, 952–955 (1966).
[CrossRef]

J. Phys. D (1)

F. Ortwein, J. Mental, E. Schmidt, “A birefringent filter as a tunning element for a multilane He–Ne laser,” J. Phys. D 22, 488–491 (1989).
[CrossRef]

Other (3)

A Yariv, P Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 152.

S. V. Kartalopoulos, Introduction to DWDM Technology: Data in a Rainbow (Wiley-IEEE Press, New York, 2000).

E. O. Ammann, “Synthesis of optical birefringent network,” in Progress in Optics, Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 123–177.

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 (9)

Fig. 1
Fig. 1

Configuration of four-stage folded Solc birefringent filter.

Fig. 2
Fig. 2

Off-axis ray in the crystal coordinate system.

Fig. 3
Fig. 3

Coordinates for a multistage birefringent filter.

Fig. 4
Fig. 4

Calculated spectral transmission functions of a four-stage Solc filter: (a) ϕ = 0 ° , (b) ϕ = 90 ° .

Fig. 5
Fig. 5

Simulated conoscopic interference figures of folded, Solc, birefringent filters of (a) single-stage, (b) two-stage, (c) three-stage, (d) four-stage, (e) six-stage, and (f) ten-stage configuration.

Fig. 6
Fig. 6

Calculated average view-field transmission and experimental results for a Solc single-stage filter versus the incline angle.

Fig. 7
Fig. 7

Calculated average view-field transmissions of Solc folded filters versus the incident angles for (a) single-stage, (b) two-stage, (c) three-stage, and (d) four-stage configuration.

Fig. 8
Fig. 8

Calculated average spectral transmissions of Solc folded filters of (a) single-stage, (b) two-stage, (c) three-stage, and (d) four-stage configuration.

Fig. 9
Fig. 9

Experimental conoscopic figure for a Solc single-stage filter.

Equations (35)

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

E o ( θ ,   φ ) = M ( θ ,   ϕ ) E i ,
M ( θ ,   ϕ ) = P 2 M N ( θ ,   ϕ ,   ρ N ) × M N - 1 ( θ ,   ϕ ,   ρ N - 1 ) M n ( θ ,   ϕ ,   ρ n ) × M 1 ( θ ,   ϕ ,   ρ 1 ) P 1 ,
T ( θ ,   ϕ ) = | M ( θ ,   ϕ ) | 2 .
Δ ( θ ,   ϕ ) = 2 π λ   d n e 1 - sin 2   θ   sin 2   ϕ n e 2 - sin 2   θ   cos 2   ϕ n o 2 1 / 2 - n o 1 - sin 2   θ n o 2 1 / 2 .
Δ 0 = 2 π d ( n e - n o ) λ .
Δ ( θ ,   ϕ ) = Δ 0 1 + sin 2   θ sin 2   ϕ 2 n o n e - cos 2   ϕ 2 n o 2 .
Δ ( θ ,   ϕ ,   ± ρ ) = 2 π λ   d n e 1 - sin 2   θ   sin 2 ( ϕ ρ ) n e 2 - sin 2   θ   cos 2 ( ϕ ρ ) n o 2 1 / 2 - n o 1 - sin 2   θ n o 2 1 / 2 .
M ( θ ,   ϕ ,   ± ρ ) = cos   ρ sin   ρ ± sin   ρ cos   ρ   exp [ - i Δ ( θ ,   ϕ ,   ± ρ ) / 2 ] 0 0 exp [ + i Δ ( θ ,   ϕ ,   ± ρ ) / 2 ]   cos   ρ ± sin   ρ sin   ρ cos   ρ = cos   Δ ( θ ,   ϕ ,   ± ρ ) 2 - i   cos   2 ρ   sin   Δ ( θ ,   ϕ ,   ± ρ ) 2 - i   sin   2 ρ   sin   Δ ( θ ,   ϕ ,   ± ρ ) 2 - i   sin   2 ρ   sin   Δ ( θ ,   ϕ ,   ± ρ ) 2 cos   Δ ( θ ,   ϕ ,   ± ρ ) 2 + i   cos   2 ρ   sin   Δ ( θ ,   ϕ ,   ± ρ ) 2 .
M ( θ ,   ϕ ) = 0 0 0 1 M 2 m ( θ ,   ϕ ,   - ρ ) M 2 m - 1 ( θ ,   ϕ ,   + ρ ) × M 2 ( θ ,   ϕ ,   - ρ ) M 1 ( θ ,   ϕ ,   + ρ ) × 1 0 0 0 = 0 0 0 1 × [ M 2 ( θ ,   ϕ ,   - ρ ) M 1 ( θ ,   ϕ ,   + ρ ) ] m 1 0 0 0 .
M x ( θ ,   ϕ ) = 0 ,
M y ( θ ,   ϕ ) = sin   4 ρ   sin   Δ ( + ρ ) 2   sin   Δ ( - ρ ) 2 + i   sin   2 ρ   sin   Δ ( + ρ ) - Δ ( - ρ ) 2   sin   mK Λ sin   K Λ ,
K Λ = cos - 1 cos   Δ ( + ρ ) 2   cos   Δ ( - ρ ) 2 - cos   4 ρ   sin   Δ ( + ρ ) 2   sin   Δ ( - ρ ) 2 .
M ( θ ,   ϕ ) = 0 0 0 1 M 2 m + 1 ( θ ,   ϕ ,   + ρ ) M 2 m ( θ ,   ϕ ,   - ρ ) × M 2 m - 1 ( θ ,   ϕ ,   + ρ ) M 2 ( θ ,   ϕ ,   - ρ ) × M 1 ( θ ,   ϕ ,   + ρ ) 1 0 0 0 = 0 0 0 1 M 2 m + 1 ( θ ,   ϕ ,   + ρ ) × [ M 2 ( θ ,   ϕ ,   - ρ ) M 1 ( θ ,   ϕ ,   + ρ ) ] m 1 0 0 0 .
M x ( θ ,   ϕ ) = 0 ,
M y ( θ ,   ϕ ) = i sin   2 ρ   cos   Δ ( + ρ ) 2   sin   Δ ( - ρ ) - Δ ( + ρ ) 2 + sin   4 ρ   cos   2 ρ   sin 2   Δ ( + ρ ) 2   sin   Δ ( - ρ ) 2 × sin   mK Λ sin   K Λ - sin   2 ρ   sin   Δ ( + ρ ) 2   cos   mK Λ .
T ( θ ,   ϕ ) = | M ( θ ,   ϕ ) | 2 = sin   4 ρ   sin   Δ ( + ρ ) 2   sin   Δ ( - ρ ) 2 2 + sin   2 ρ   sin   Δ ( + ρ ) - Δ ( - ρ ) 2 2 × sin   mK Λ sin   K Λ 2 .
T ( θ ,   ϕ ) = | M ( θ ,   ϕ ) | 2 = sin   2 ρ   cos   Δ ( + ρ ) 2   sin   Δ ( - ρ ) - Δ ( + ρ ) 2 + sin   4 ρ   cos   2 ρ   sin 2   Δ ( + ρ ) 2   sin   Δ ( - ρ ) 2 × sin   mK Λ sin   K Λ - sin   2 ρ   sin   Δ ( + ρ ) 2   cos   mK Λ 2 .
K Λ = π - 2 χ .
T = | tan   2 ρ   cos   χ ( sin   N χ / sin   χ ) | 2 ,
cos   χ = cos   2 ρ   sin   1 2   Δ 0 .
T ( θ ,   λ ) = f { Δ ( θ ,   λ ) } .
Δ x = Δ 0 1 + sin 2   θ sin 2   ρ 2 n o n e - cos 2   ρ 2 n o 2 .
δ λ ( ϕ = 0 ) λ = Δ x - Δ 0 Δ 0 = - sin 2   θ cos 2   ρ 2 n o 2 - sin 2   ρ 2 n o n e .
Δ y = Δ 0 1 + sin 2   θ cos 2   ρ 2 n o n e - sin 2   ρ 2 n o 2 .
δ λ ( ϕ = π / 2 ) λ = Δ y - Δ 0 Δ 0 = - sin 2   θ cos 2   ρ 2 n o n e - sin 2   ρ 2 n o 2 .
tan 2   ρ = tan 2   π / 4 N tan 2   π / 8 < n o / n e < n e / n o .
tan 2   ρ = tan 2   π / 4 N tan 2   π / 8 < n e / n o < n o / n e
T ( θ ,   ϕ ) = 1 2 [ 1 - cos   Δ ( θ ,   ϕ ) ] .
x 2 = f 2   tan 2   θ   cos 2   ϕ f 2   sin 2   θ   cos 2   ϕ ,
y 2 = f 2   tan 2   θ   sin 2   ϕ f 2   sin 2   θ   sin 2   ϕ .
x 2 / 2 n o f 2 ( Δ 0 - Δ ) Δ 0   n o - y 2 / 2 n o f 2 ( Δ 0 - Δ ) Δ 0   n e = 1 .
y = ± ( n e / n o ) 1 / 2 x .
Φ in ( θ 0 ) = 0 2 π 0 θ 0 A 2   sin   θ d θ d ϕ ,
Φ out = 0 2 π 0 θ 0 A 2 I ( θ ,   ϕ ) sin   θ d θ d ϕ .
T ( λ ,   θ 0 ) = Φ out ( λ ,   θ 0 ) Φ in ( λ ,   θ 0 ) .

Metrics