Abstract

We propose the use of a smooth spherical lens in the Christiansen filter cell. The realization of common filtering functions with such a filter configuration is discussed. A systematic design technique based on inverse Fourier cosine transform or inverse Laplace transform is established, with which a desired, prescribed response can be tailored by properly configuring the lens of the filter. Four Christiansen filters centered at 545nm with a full width at half-maximum of 2nm are analyzed in detail.

© 2010 Optical Society of America

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  1. Z. D. Sheng and S. Y. Zhu, “Design of unequal thickness interference filters with quarter wavelength stack,” Semicond. Photon. Technol. 7, 50-55 (2001).
  2. D. Y. Song and J. S. Lee, “Angle-tuned Fabry-Perot etalon filter having Gaussian transmittance curves,” IEEE Photon. Technol. Lett. 12, 1186-1188 (2000).
    [CrossRef]
  3. 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]
  4. 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 DFB laser diode and fiber Bragg grating techniques,” IEEE Photon. Technol. Lett. 12, 1207-1209(2000).
    [CrossRef]
  5. L. R. Chen, D. J. F. Cooper, and P. W. E. Smith, “Transmission filter with multiple flattened passbands based on chirped moire gratings,” IEEE Photon. Technol. Lett. 10, 1283-1285(1998).
    [CrossRef]
  6. U. Wojak, U. Czarnetzki, and H. F. Dobele, “Christiansen filters for the far ultraviolet: an old spectral device in a new light,” Appl. Opt. 26, 4788-4790 (1987).
    [CrossRef] [PubMed]
  7. K. Balasubramanian, M. R. Jacobson, and H. A. Macleod, “New Christiansen filters,” Appl. Opt. 31, 1574-1587 (1992).
    [CrossRef] [PubMed]
  8. N. J. Goddard and A. E. Maturell, “Tunable optical filter for colorimetric applications,” Appl. Opt. 34, 7318-7320 (1995).
    [CrossRef] [PubMed]
  9. R. H. Clarke, “A theory for the Christiansen filter,” Appl. Opt. 7, 861-868 (1968).
    [CrossRef] [PubMed]
  10. J. Li, N. Goddard, and K. Xie, “Christiansen filter realized by an odd smooth cylindrical lens,” J. Opt. Soc. Am. A 27, 100-108 (2010).
    [CrossRef]
  11. J. Li, K. Xie, H. Jiang, and N. Goddard, “Christiansen filters realized with cylindrical lenses of even symmetry,” submitted to J. Opt. Soc. Am. A.

2010

2001

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

2000

D. Y. Song and J. S. Lee, “Angle-tuned Fabry-Perot etalon filter having Gaussian transmittance curves,” IEEE Photon. 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 DFB laser diode and fiber Bragg grating techniques,” IEEE Photon. Technol. Lett. 12, 1207-1209(2000).
[CrossRef]

1998

L. R. Chen, D. J. F. Cooper, and P. W. E. Smith, “Transmission filter with multiple flattened passbands based on chirped moire gratings,” IEEE Photon. 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]

1995

1992

1987

1968

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 DFB laser diode and fiber Bragg grating techniques,” IEEE Photon. Technol. Lett. 12, 1207-1209(2000).
[CrossRef]

Chen, L. R.

L. R. Chen, D. J. F. Cooper, and P. W. E. Smith, “Transmission filter with multiple flattened passbands based on chirped moire gratings,” IEEE Photon. 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 DFB laser diode and fiber Bragg grating techniques,” IEEE Photon. Technol. Lett. 12, 1207-1209(2000).
[CrossRef]

Clarke, R. H.

Cooper, D. J. F.

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

Czarnetzki, U.

Dobele, H. F.

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 DFB laser diode and fiber Bragg grating techniques,” IEEE Photon. Technol. Lett. 12, 1207-1209(2000).
[CrossRef]

Goddard, N.

J. Li, N. Goddard, and K. Xie, “Christiansen filter realized by an odd smooth cylindrical lens,” J. Opt. Soc. Am. A 27, 100-108 (2010).
[CrossRef]

J. Li, K. Xie, H. Jiang, and N. Goddard, “Christiansen filters realized with cylindrical lenses of even symmetry,” submitted to J. Opt. Soc. Am. A.

Goddard, N. J.

Jacobson, M. R.

Jiang, H.

J. Li, K. Xie, H. Jiang, and N. Goddard, “Christiansen filters realized with cylindrical lenses of even symmetry,” submitted to J. Opt. Soc. Am. A.

Lee, J. S.

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

Li, J.

J. Li, N. Goddard, and K. Xie, “Christiansen filter realized by an odd smooth cylindrical lens,” J. Opt. Soc. Am. A 27, 100-108 (2010).
[CrossRef]

J. Li, K. Xie, H. Jiang, and N. Goddard, “Christiansen filters realized with cylindrical lenses of even symmetry,” submitted to J. Opt. Soc. Am. A.

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 DFB laser diode and fiber Bragg grating techniques,” IEEE Photon. 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 DFB laser diode and fiber Bragg grating techniques,” IEEE Photon. Technol. Lett. 12, 1207-1209(2000).
[CrossRef]

Macleod, H. A.

Maturell, A. E.

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,” Semicond. Photon. Technol. 7, 50-55 (2001).

Smith, P. W. E.

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

Song, D. Y.

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

Wojak, U.

Xie, K.

J. Li, N. Goddard, and K. Xie, “Christiansen filter realized by an odd smooth cylindrical lens,” J. Opt. Soc. Am. A 27, 100-108 (2010).
[CrossRef]

J. Li, K. Xie, H. Jiang, and N. Goddard, “Christiansen filters realized with cylindrical lenses of even symmetry,” submitted to J. Opt. Soc. Am. A.

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 DFB laser diode and fiber Bragg grating techniques,” IEEE Photon. 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,” Semicond. Photon. Technol. 7, 50-55 (2001).

Appl. Opt.

IEEE Photon. Technol. Lett.

D. Y. Song and J. S. Lee, “Angle-tuned Fabry-Perot etalon filter having Gaussian transmittance curves,” IEEE Photon. 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 DFB laser diode and fiber Bragg grating techniques,” IEEE Photon. Technol. Lett. 12, 1207-1209(2000).
[CrossRef]

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

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]

J. Opt. Soc. Am. A

Semicond. Photon. Technol.

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

Other

J. Li, K. Xie, H. Jiang, and N. Goddard, “Christiansen filters realized with cylindrical lenses of even symmetry,” submitted to J. Opt. Soc. Am. A.

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Figures (8)

Fig. 1
Fig. 1

(a) Diagram of the experimental optical setup, (b) variation of refractive indices of solid ( n s ) and liquid ( n 1 ) with wavelength.

Fig. 2
Fig. 2

Liquid profile functions of the first four all-real-roots polynomial spherical filters. The lenses are defined by the area above the curves: solid curve, N = 1 ; dotted curve, N = 2 ; dashed-dotted curve, N = 3 ; dashed curve N = 4 .

Fig. 3
Fig. 3

Liquid profile functions of the spherical Gaussian filters. The lenses are defined by the area above the curves: solid curve, B = 0 ; dotted curve, B = 1.5 ; dashed-dotted curve, B = 3.0 ; dashed curve, B = 4.5 .

Fig. 4
Fig. 4

Liquid profile functions of the first three spherical filters of the sinc family. The lenses are defined by the area above the curves: solid curve, N = 1 ; dashed-dotted curve, N = 2 ; dashed curve, N = 3 .

Fig. 5
Fig. 5

(a) Transmittance of the second-order odd separable sinc filter, (b) insertion loss of the second-order odd separable sinc filter.

Fig. 6
Fig. 6

Cross section of the second-order spherical sinc filter on the plane cut through the origin.

Fig. 7
Fig. 7

Cross section of the spherical Gaussian filter on the plane cut through the origin for B = 3 .

Fig. 8
Fig. 8

(a) Transmittance of the spherical Gaussian filter for B = 3 : solid curve, truncated lens; dashed curve: untruncated lens. The two curves are overlapped everywhere in this scale. (b) insertion loss of the spherical Gaussian filter for B = 3 : solid curve, truncated lens; dashed curve, ideal lens.

Equations (60)

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E ( x , y ) = E 0 exp [ i ϕ 0 i k 0 ( l ( x , y ) n s + ( L l ( x , y ) ) n l ) ] ,
Ξ ( s 1 , s 2 ) = ( k 0 2 π ) 2 b / 2 b / 2 a / 2 a / 2 E ( x , y ) exp [ i k 0 ( s 1 x + s 2 y ) ] d x d y ,
Ξ ( 0 , 0 ) = ( k 0 2 π ) 2 b / 2 b / 2 a / 2 a / 2 E 0 exp [ i ϕ 0 i k 0 ( l ( x , y ) n s + ( L l ( x , y ) ) n l ) ] d x d y .
( k 0 2 π ) 2 A E 0 exp [ i ϕ 0 i k 0 L n l ] ,
t = 1 A b / 2 b / 2 a / 2 a / 2 exp [ i k 0 l ( x , y ) ( n s n l ) ] d x d y ,
t ( Δ ) = 1 A b / 2 b / 2 a / 2 a / 2 exp [ i Δ f ( x , y ) ] d x d y ,
t ( Δ ) = ( sin Δ Δ ) 2 .
t ( Δ ) = 1 / 2 1 / 2 exp [ i Δ f x ( x ) ] d x 1 / 2 1 / 2 exp [ i Δ f y ( y ) ] d y ,
t ( Δ ) = [ 2 0 1 / 2 cos ( Δ f ) d x ] 2
t ( Δ ) = [ 2 0 1 / 2 exp ( i Δ f ) d x ] 2
t ( Δ ) = [ 2 0 T ( f ) cos ( Δ f ) d f ] 2 .
t ( Δ ) = exp ( 2 Δ 2 ) ,
T ( f ) = 1 2 π exp ( f 2 4 ) .
x = 1 2 erf ( f x 2 ) ; y = 1 2 erf ( f y 2 ) .
t ( Δ ) = sinc 2 N ( a 0 Δ N ) ,
t ( Δ ) = sinc 4 ( a 0 Δ 2 ) .
T ( f ) = 1 2 a 0 Λ ( f 2 a 0 ) ,
t ( s ) = [ 0 T ( F ) exp ( s F ) d F ] 2 ,
t ( s ) = s N 2 N ( s + s N ) 2 N ,
s N = 1 2 N 1 .
T ( F ) = s N N ( N 1 ) ! F N 1 exp ( s N F ) .
x ( F ) = 1 2 1 2 exp ( s N F ) k = 0 N 1 1 ( N 1 k ) ! s N N 1 k F N 1 k .
T ( F ) = 1 π exp ( F 2 4 ) .
T ( F ) = 1 ζ π exp ( ( F B ) 2 4 ) ,
t ( s ) = exp ( 2 s B ) exp ( 2 s 2 ) ζ 2 [ 1 + erf ( s + B 2 ) ] 2 .
t ( s ) = ( 1 exp ( 2 a 0 s N ) ) 2 N / ( 2 a 0 s N ) 2 N ,
t ( s ) = [ 1 exp ( 2 a 0 s ) ] 2 ( 2 a 0 s ) 2 .
T ( F ) = 1 2 a 0 Π ( F 2 a 0 ) ,
t ( s ) = ( 1 exp ( 2 a 0 s 2 ) ) 4 / ( 2 a 0 s 2 ) 4 .
T ( F ) = 1 2 a 0 Λ ( F 2 a 0 1 ) ,
x ( F ) = { 1 8 a 0 2 F 2 0 F 2 a 0 < 1 1 2 + F 2 a 0 1 4 ( F 2 a 0 ) 2 F 2 a 0 < 2
t ( Δ ) = 2 0 1 exp ( i Δ f ) r d r .
t ( s ) = 0 T ( F ) exp ( s F ) d F .
T ( F ) = 1 2 π i σ i σ + i t ( s ) exp ( s F ) d s ( F 0 , σ 0 ) .
r 2 ( F ) = 0 F T ( ς ) d ς .
t ( s ) = s N N ( s + s N ) N ,
s N = 1 2 N 1 .
| t ( s c ) | s c = i 2 = 1 2 .
T ( F ) = s N N ( N 1 ) ! F N 1 exp ( s N F ) .
r 2 ( F ) = 1 exp ( s N F ) k = 0 N 1 1 ( N 1 k ) ! s N N 1 k F N 1 k .
r 2 ( F ) = 1 ( s 2 F + 1 ) exp ( s 2 F )
r 2 ( F ) = 1 ( 1 2 s 3 2 F 2 + s 3 F + 1 ) exp ( s 3 F )
r 2 ( F ) = 1 ( 1 6 s 4 3 F 3 + 1 2 s 4 2 F 2 + s 4 F + 1 ) exp ( s 4 F ) .
T ( F ) = 1 ζ π exp ( ( F B ) 2 4 ) ,
t ( s ) = exp ( s B ) exp ( s 2 ) ζ [ 1 + erf ( s + B 2 ) ] .
r 2 ( F ) = 1 ζ [ erf ( B 2 ) + erf ( F B 2 ) ] .
lim r 0 d F d r = 2 ζ π lim F 0 [ erf ( B 2 ) + erf ( F B 2 ) ] / exp ( B 2 4 ) = 0 ,
t ( s ) = ( 1 exp ( 2 a 0 s N ) ) N / ( 2 a 0 s N ) N ,
t ( s ) = 1 exp ( 2 a 0 s ) 2 a 0 s .
T ( F ) = 1 2 a 0 Π ( F 2 a 0 ) ,
t ( s ) = ( 1 exp ( 2 a 0 s 2 ) ) 2 / ( 2 a 0 s 2 ) 2 ,
T ( F ) = 1 2 a 0 Λ ( F 2 a 0 1 ) ,
r = { 1 2 ( F a 0 ) 0 F a 0 < 2 1 + 2 F a 0 1 4 ( F a 0 ) 2 F a 0 < 2 2
t ( s ) = ( 1 exp ( 2 a 0 s 3 ) ) 3 / ( 2 a 0 s 3 ) 3 .
T ( F ) = { 1 2 b ( F b ) 2 0 F b < 1 3 4 b 1 b ( F b 3 2 ) 2 1 F b < 2 1 2 b ( 3 F b ) 2 2 F b < 3 ,
b = 2 a 0 3 .
r 2 = { 1 6 ( F b ) 3 0 F b < 1 1 2 3 2 F b + 3 2 ( F b ) 2 1 3 ( F b ) 3 1 F b < 2 21 6 + 9 2 F b 3 2 ( F b ) 2 + 1 6 ( F b ) 3 2 F b < 3 .
T ( f ) = 1 2 a 0 Π ( f a 0 ) ,
T ( F ) = 1 2 a 0 Π ( F 2 a 0 ) ,
t ( s ) = exp ( 3 s ) exp ( s 2 ) 1.966 [ 1 + erf ( 1.5 s ) ] .

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