Abstract

The Christiansen filter that is realized by odd smooth cylindrical lenses is analyzed in detail. Several popular filtering functions are discussed. The corresponding lens profile functions are obtained by an inverse scattering theory, which enables the filter to synthesize a desired prescribed response function. This kind of Christiansen filter has a passband narrower than that of the traditional Christiansen filter. Three Christiansen filters centered at 545nm with full width at half-maximum of 2nm are synthesized, and the approach to a better suppression of halos from the main transmission peak of the filters is presented in a systematic way.

© 2009 Optical Society of America

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References

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  1. S. Li, K. S. Chiang, A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “A novel tunable all-optical incoherent negative tap fiber-optic transversal filter based on a DFB laser diode and fiber Bragg grating techniques,” IEEE Photonics Technol. Lett. 12, 1207-1209 (2000).
    [CrossRef]
  2. L. R. Chen, D. J. F. Cooper, and P. W. E. Smith, “Transmission filter with multiplex flattened passbands based on chirped moiré gratings,” IEEE Photonics Technol. Lett. 10, 1283-1285 (1998).
    [CrossRef]
  3. P. Yeh, “Zero crossing birefringent filters,” Opt. Commun. 35, 15-19 (1980).
    [CrossRef]
  4. Y. Ohman, “A new monochromator,” Nature 141, 157-158 (1938).
    [CrossRef]
  5. Z. D. Sheng and S. Y. Zhu, “Design of unequal-thickness interference filters with quarter-wavelength stack,” Semiconductor Photon Technol. 7, 50-55 (2001).
  6. D. Y. Song and J. S. Lee, “Angle-tuned Fabry-Perot filter having Gaussian transmittance curves,” IEEE Photonics Technol. Lett. 12, 1186-1188 (2000).
    [CrossRef]
  7. H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, 1989).
  8. F. Bakhti and P. Sansonetti, “Design and realization of multiple quarter-wave phase-shifts UV-written bandpass filters in optical fibers,” J. Lightwave Technol. 15, 1433-1437 (1997).
    [CrossRef]
  9. C. Christiansen, “Untersuchungen uber die optischen Eigenschaften fein vertheilter Korper; erste mitteileng,” Ann. Phys. 23, 298 (1884).
    [CrossRef]
  10. U. Wojak, U. Czarnetzki, and H. F. Dobele, “Christiansen filters for the far ultraviolet: an old spectral device in a new light,” Opt. Lett. 26, 4788-4790 (1987).
  11. A. E. Maturell, “A tunable optical filter for use in the visible range,” M.Sc. dissertation, (University of Manchester, 1994).
  12. K. Balasubramanian, M. R. Jacobson, and H. A. Macleod, “New Christiansen filters,” Appl. Opt. 31, 1574-1587 (1992).
    [CrossRef] [PubMed]
  13. N. J. Goddard and A. E. Maturell, “Tunable optical filter for colorimetric applications,” Appl. Opt. 34, 7318-7320 (1995).
    [CrossRef] [PubMed]
  14. G. L. Fischer, R. W. Boyd, T. R. Moore, and J. E. Sipe, “Nonlinear-optical Christiansen filter as an optical power limiter,” Opt. Lett. 21, 1643-1645 (1996).
    [CrossRef] [PubMed]
  15. A. K. Varshneya, M. C. Loo, and T. F. Soules, “Glass inhomogeneity measurement using the Shelyubskii method,” J. Acoust. Soc. Am. 68, 380-385 (1985).
  16. K. Xie, J. Li, and N. Goddard, “Christiansen filter realized with cylindrical lens,” J. Opt. A, Pure Appl. Opt. (to be published).

2001

Z. D. Sheng and S. Y. Zhu, “Design of unequal-thickness interference filters with quarter-wavelength stack,” Semiconductor Photon Technol. 7, 50-55 (2001).

2000

D. Y. Song and J. S. Lee, “Angle-tuned Fabry-Perot filter having Gaussian transmittance curves,” IEEE Photonics Technol. Lett. 12, 1186-1188 (2000).
[CrossRef]

S. Li, K. S. Chiang, A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “A novel tunable all-optical incoherent negative tap fiber-optic transversal filter based on a DFB laser diode and fiber Bragg grating techniques,” IEEE Photonics Technol. Lett. 12, 1207-1209 (2000).
[CrossRef]

1998

L. R. Chen, D. J. F. Cooper, and P. W. E. Smith, “Transmission filter with multiplex flattened passbands based on chirped moiré gratings,” IEEE Photonics Technol. Lett. 10, 1283-1285 (1998).
[CrossRef]

1997

F. Bakhti and P. Sansonetti, “Design and realization of multiple quarter-wave phase-shifts UV-written bandpass filters in optical fibers,” J. Lightwave Technol. 15, 1433-1437 (1997).
[CrossRef]

1996

1995

1992

1987

U. Wojak, U. Czarnetzki, and H. F. Dobele, “Christiansen filters for the far ultraviolet: an old spectral device in a new light,” Opt. Lett. 26, 4788-4790 (1987).

1985

A. K. Varshneya, M. C. Loo, and T. F. Soules, “Glass inhomogeneity measurement using the Shelyubskii method,” J. Acoust. Soc. Am. 68, 380-385 (1985).

1980

P. Yeh, “Zero crossing birefringent filters,” Opt. Commun. 35, 15-19 (1980).
[CrossRef]

1938

Y. Ohman, “A new monochromator,” Nature 141, 157-158 (1938).
[CrossRef]

1884

C. Christiansen, “Untersuchungen uber die optischen Eigenschaften fein vertheilter Korper; erste mitteileng,” Ann. Phys. 23, 298 (1884).
[CrossRef]

Bakhti, F.

F. Bakhti and P. Sansonetti, “Design and realization of multiple quarter-wave phase-shifts UV-written bandpass filters in optical fibers,” J. Lightwave Technol. 15, 1433-1437 (1997).
[CrossRef]

Balasubramanian, K.

Bennion, I.

S. Li, K. S. Chiang, A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “A novel tunable all-optical incoherent negative tap fiber-optic transversal filter based on a DFB laser diode and fiber Bragg grating techniques,” IEEE Photonics Technol. Lett. 12, 1207-1209 (2000).
[CrossRef]

Boyd, R. W.

Chen, L. R.

L. R. Chen, D. J. F. Cooper, and P. W. E. Smith, “Transmission filter with multiplex flattened passbands based on chirped moiré gratings,” IEEE Photonics Technol. Lett. 10, 1283-1285 (1998).
[CrossRef]

Chiang, K. S.

S. Li, K. S. Chiang, A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “A novel tunable all-optical incoherent negative tap fiber-optic transversal filter based on a DFB laser diode and fiber Bragg grating techniques,” IEEE Photonics Technol. Lett. 12, 1207-1209 (2000).
[CrossRef]

Christiansen, C.

C. Christiansen, “Untersuchungen uber die optischen Eigenschaften fein vertheilter Korper; erste mitteileng,” Ann. Phys. 23, 298 (1884).
[CrossRef]

Cooper, D. J. F.

L. R. Chen, D. J. F. Cooper, and P. W. E. Smith, “Transmission filter with multiplex flattened passbands based on chirped moiré gratings,” IEEE Photonics Technol. Lett. 10, 1283-1285 (1998).
[CrossRef]

Czarnetzki, U.

U. Wojak, U. Czarnetzki, and H. F. Dobele, “Christiansen filters for the far ultraviolet: an old spectral device in a new light,” Opt. Lett. 26, 4788-4790 (1987).

Dobele, H. F.

U. Wojak, U. Czarnetzki, and H. F. Dobele, “Christiansen filters for the far ultraviolet: an old spectral device in a new light,” Opt. Lett. 26, 4788-4790 (1987).

Fischer, G. L.

Gambling, A.

S. Li, K. S. Chiang, A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “A novel tunable all-optical incoherent negative tap fiber-optic transversal filter based on a DFB laser diode and fiber Bragg grating techniques,” IEEE Photonics Technol. Lett. 12, 1207-1209 (2000).
[CrossRef]

Goddard, N.

K. Xie, J. Li, and N. Goddard, “Christiansen filter realized with cylindrical lens,” J. Opt. A, Pure Appl. Opt. (to be published).

Goddard, N. J.

Jacobson, M. R.

Lee, J. S.

D. Y. Song and J. S. Lee, “Angle-tuned Fabry-Perot filter having Gaussian transmittance curves,” IEEE Photonics Technol. Lett. 12, 1186-1188 (2000).
[CrossRef]

Li, J.

K. Xie, J. Li, and N. Goddard, “Christiansen filter realized with cylindrical lens,” J. Opt. A, Pure Appl. Opt. (to be published).

Li, S.

S. Li, K. S. Chiang, A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “A novel tunable all-optical incoherent negative tap fiber-optic transversal filter based on a DFB laser diode and fiber Bragg grating techniques,” IEEE Photonics Technol. Lett. 12, 1207-1209 (2000).
[CrossRef]

Liu, Y.

S. Li, K. S. Chiang, A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “A novel tunable all-optical incoherent negative tap fiber-optic transversal filter based on a DFB laser diode and fiber Bragg grating techniques,” IEEE Photonics Technol. Lett. 12, 1207-1209 (2000).
[CrossRef]

Loo, M. C.

A. K. Varshneya, M. C. Loo, and T. F. Soules, “Glass inhomogeneity measurement using the Shelyubskii method,” J. Acoust. Soc. Am. 68, 380-385 (1985).

Macleod, H. A.

Maturell, A. E.

N. J. Goddard and A. E. Maturell, “Tunable optical filter for colorimetric applications,” Appl. Opt. 34, 7318-7320 (1995).
[CrossRef] [PubMed]

A. E. Maturell, “A tunable optical filter for use in the visible range,” M.Sc. dissertation, (University of Manchester, 1994).

Moore, T. R.

Ohman, Y.

Y. Ohman, “A new monochromator,” Nature 141, 157-158 (1938).
[CrossRef]

Sansonetti, P.

F. Bakhti and P. Sansonetti, “Design and realization of multiple quarter-wave phase-shifts UV-written bandpass filters in optical fibers,” J. Lightwave Technol. 15, 1433-1437 (1997).
[CrossRef]

Sheng, Z. D.

Z. D. Sheng and S. Y. Zhu, “Design of unequal-thickness interference filters with quarter-wavelength stack,” Semiconductor Photon Technol. 7, 50-55 (2001).

Sipe, J. E.

Smith, P. W. E.

L. R. Chen, D. J. F. Cooper, and P. W. E. Smith, “Transmission filter with multiplex flattened passbands based on chirped moiré gratings,” IEEE Photonics Technol. Lett. 10, 1283-1285 (1998).
[CrossRef]

Song, D. Y.

D. Y. Song and J. S. Lee, “Angle-tuned Fabry-Perot filter having Gaussian transmittance curves,” IEEE Photonics Technol. Lett. 12, 1186-1188 (2000).
[CrossRef]

Soules, T. F.

A. K. Varshneya, M. C. Loo, and T. F. Soules, “Glass inhomogeneity measurement using the Shelyubskii method,” J. Acoust. Soc. Am. 68, 380-385 (1985).

Varshneya, A. K.

A. K. Varshneya, M. C. Loo, and T. F. Soules, “Glass inhomogeneity measurement using the Shelyubskii method,” J. Acoust. Soc. Am. 68, 380-385 (1985).

Wojak, U.

U. Wojak, U. Czarnetzki, and H. F. Dobele, “Christiansen filters for the far ultraviolet: an old spectral device in a new light,” Opt. Lett. 26, 4788-4790 (1987).

Xie, K.

K. Xie, J. Li, and N. Goddard, “Christiansen filter realized with cylindrical lens,” J. Opt. A, Pure Appl. Opt. (to be published).

Yeh, P.

P. Yeh, “Zero crossing birefringent filters,” Opt. Commun. 35, 15-19 (1980).
[CrossRef]

Zhang, L.

S. Li, K. S. Chiang, A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “A novel tunable all-optical incoherent negative tap fiber-optic transversal filter based on a DFB laser diode and fiber Bragg grating techniques,” IEEE Photonics Technol. Lett. 12, 1207-1209 (2000).
[CrossRef]

Zhu, S. Y.

Z. D. Sheng and S. Y. Zhu, “Design of unequal-thickness interference filters with quarter-wavelength stack,” Semiconductor Photon Technol. 7, 50-55 (2001).

Ann. Phys.

C. Christiansen, “Untersuchungen uber die optischen Eigenschaften fein vertheilter Korper; erste mitteileng,” Ann. Phys. 23, 298 (1884).
[CrossRef]

Appl. Opt.

IEEE Photonics Technol. Lett.

D. Y. Song and J. S. Lee, “Angle-tuned Fabry-Perot filter having Gaussian transmittance curves,” IEEE Photonics Technol. Lett. 12, 1186-1188 (2000).
[CrossRef]

S. Li, K. S. Chiang, A. Gambling, Y. Liu, L. Zhang, and I. Bennion, “A novel tunable all-optical incoherent negative tap fiber-optic transversal filter based on a DFB laser diode and fiber Bragg grating techniques,” IEEE Photonics Technol. Lett. 12, 1207-1209 (2000).
[CrossRef]

L. R. Chen, D. J. F. Cooper, and P. W. E. Smith, “Transmission filter with multiplex flattened passbands based on chirped moiré gratings,” IEEE Photonics Technol. Lett. 10, 1283-1285 (1998).
[CrossRef]

J. Acoust. Soc. Am.

A. K. Varshneya, M. C. Loo, and T. F. Soules, “Glass inhomogeneity measurement using the Shelyubskii method,” J. Acoust. Soc. Am. 68, 380-385 (1985).

J. Lightwave Technol.

F. Bakhti and P. Sansonetti, “Design and realization of multiple quarter-wave phase-shifts UV-written bandpass filters in optical fibers,” J. Lightwave Technol. 15, 1433-1437 (1997).
[CrossRef]

Nature

Y. Ohman, “A new monochromator,” Nature 141, 157-158 (1938).
[CrossRef]

Opt. Commun.

P. Yeh, “Zero crossing birefringent filters,” Opt. Commun. 35, 15-19 (1980).
[CrossRef]

Opt. Lett.

U. Wojak, U. Czarnetzki, and H. F. Dobele, “Christiansen filters for the far ultraviolet: an old spectral device in a new light,” Opt. Lett. 26, 4788-4790 (1987).

G. L. Fischer, R. W. Boyd, T. R. Moore, and J. E. Sipe, “Nonlinear-optical Christiansen filter as an optical power limiter,” Opt. Lett. 21, 1643-1645 (1996).
[CrossRef] [PubMed]

Semiconductor Photon Technol.

Z. D. Sheng and S. Y. Zhu, “Design of unequal-thickness interference filters with quarter-wavelength stack,” Semiconductor Photon Technol. 7, 50-55 (2001).

Other

A. E. Maturell, “A tunable optical filter for use in the visible range,” M.Sc. dissertation, (University of Manchester, 1994).

H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, 1989).

K. Xie, J. Li, and N. Goddard, “Christiansen filter realized with cylindrical lens,” J. Opt. A, Pure Appl. Opt. (to be published).

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Figures (7)

Fig. 1
Fig. 1

Diagram of the experimental optical setup.

Fig. 2
Fig. 2

Variation of refractive indices of solid ( n s ) and liquid ( n l ) with wavelength.

Fig. 3
Fig. 3

Transmittance of the first-order sinc Christiansen filter.

Fig. 4
Fig. 4

Shape of the lens of the composite Gaussian-sinc Christiansen filter.

Fig. 5
Fig. 5

(a) Transmittance of the composite Gaussian-sinc Christiansen filter. Solid curve, truncated lens; dotted curve, untruncated lens. The two curves are overlapped everywhere in this scale. (b) Insertion losses of the composite Gaussian-sinc Christiansen filter. Solid curve, truncated lens; dotted curve, untruncated lens.

Fig. 6
Fig. 6

Shape of the lens of the Gaussian Christiansen filter.

Fig. 7
Fig. 7

(a) Transmittance of the Gaussian Christiansen filter. Solid curve, truncated lens; dotted curve, untruncated lens. The two curves are overlapped everywhere in this scale. (b) Insertion loss of the Gaussian Christiansen filter. Solid curve, truncated lens; dotted curve, untruncated lens.

Equations (33)

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Ξ ( s ) = k 0 2 π a 2 a 2 E ( x ) exp ( i k 0 s x ) d x ,
Ξ ( 0 ) = k 0 2 π a 2 a 2 E 0 exp ( i φ 0 i δ φ ) d x .
t ( Δ ) = 1 2 1 2 exp [ i Δ f ( x ) ] d x ,
t ( Δ ) = 2 0 1 2 cos [ Δ f ( x ) ] d x .
t ( Δ ) = 2 0 T ( f ) cos ( Δ f ) d f ,
T ( f ) = 1 π 0 t ( Δ ) cos ( Δ f ) d Δ .
x ( f ) = 0 f T ( ς ) d ς .
Γ = | t ( Δ ) | 2 = 1 1 + Δ 2 N ,
t ( Δ ) = 1 1 + Δ 2 N .
T ( f ) = 1 π K 0 ( f ) ,
t ( Δ ) = 1 1 + Δ 2 N .
T ( f ) = 1 2 e f .
f ( x ) = ln ( 1 2 x ) .
x ( f ) = 1 π 0 f sin c ( ς ) d ς = 1 π S i ( f ) .
t ( Δ ) = exp ( Δ 2 ) ,
T ( f ) = 1 2 π exp ( f 2 4 ) .
x ( f ) = 1 2 π 0 f exp ( ς 2 4 ) d ς = 1 2 erf ( f 2 ) .
t ( Δ ) = sinc N ( a 0 Δ N ) ,
T ( f ) = 1 2 a 0 Π ( f a 0 ) .
T ( f ) = 1 2 a 0 Λ ( f 2 a 0 ) ,
f ( x ) = 2 a 0 ( 1 1 2 x )
T ( f ) = { 3 3 8 a 0 ( 1 f 2 a 0 2 ) , 0 f < a 0 3 3 3 8 a 0 ( 1 2 a 0 2 f 2 3 a 0 f + 3 2 ) , a 0 3 f < 3 a 0 0 , 3 a 0 f } .
x ( f ) = { 3 8 ( f a 0 ) 3 + 3 3 8 f a 0 , 0 f a 0 < 3 3 3 16 ( f a 0 ) 3 9 16 ( f a 0 ) 2 + 9 3 16 f a 0 1 16 , 3 3 f a 0 < 3 1 2 , 3 f a 0 } .
t ( Δ ) = S i N ( b 0 Δ N ) ( b 0 Δ N ) N ,
T ( f ) = { 1 2 b 0 ln f b 0 , 0 f b 0 1 0 f b 0 > 1 } .
x ( f ) = f 2 b 0 ( 1 ln f b 0 ) .
t ( Δ ) = sin c ( π Δ Δ 0 ) exp ( Δ 2 w 0 2 ) ,
w 0 = Δ c ln 1 2 [ 2 sin c ( π Δ c Δ 0 ) ] .
T ( f ) = w 0 Δ 0 4 π 3 2 Π ( Δ 0 x π ) exp [ w 0 2 ( f x ) 2 4 ] d x = Δ 0 4 π [ erf ( w 0 f 2 + π w 0 2 Δ 0 ) erf ( w 0 f 2 π w 0 2 Δ 0 ) ] .
x ( f ) = Δ 0 2 π w 0 [ ( w 0 f 2 + π w 0 2 Δ 0 ) erf ( w 0 f 2 + π w 0 2 Δ 0 ) ( w 0 f 2 π w 0 2 Δ 0 ) erf ( w 0 f 2 π w 0 2 Δ 0 ) ] + Δ 0 2 π 3 2 w 0 { exp [ ( w 0 f 2 + π w 0 2 Δ 0 ) 2 ] exp [ ( w 0 f 2 π w 0 2 Δ 0 ) 2 ] } .
n l = 1.454 + 0.05 exp ( 1.7 λ ( nm ) 400 400 ) ,
n s = 1.456 + 0.036 exp ( λ ( nm ) 400 400 ) .
l ( x ) l ( 0 ) = l 0 f ( x ) = 2 a 0 l 0 x = 4.85 x

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