Abstract

We describe the theory of a narrowband electrooptic tunable filter based on a Fabry-Perot etalon with distributed Bragg reflectors. The filter can be in either bulk or waveguide form. The input to the filter must be prefiltered to the stop-band of the Bragg mirrors. Once this is accomplished, the etalon possesses a very narrow notch in the Bragg filter stop-band. The notch width is extremely narrow when the Bragg reflectance is high. The location of the notch in the Bragg stop-band is determined by the etalon cavity length and can be tuned by application of an electric field to the electrooptic material comprising the etalon cavity. Absorption in the cavity and Bragg reflectors is included in the theoretical model of the filter. The filter can be constructed from any one of several existing electrooptic organic polymer crystals, if the gratings are made either by partial polymerization of the monomer in crossed-UV beams or by corrugating the surface of the polymer. We show a theoretical example of a notch filter operating at a center wavelength of 1 μm that is 62.75 μm thick, with a notch width of under 1 Å and a transmission of 35%. This type of filter should have applications in high-speed optical modulation and Q-switches for lasers.

© 1986 Optical Society of America

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References

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  1. W. J. Gunning, “Electro-optically Tuned Spectral Filters: A Review,” Opt. Eng. 20, 837 (1981).
    [CrossRef]
  2. J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electro-optic Tunable Filters for Infrared Wavelengths,” IEEE J. Quantum Electron. QE-18, 1253 (1982).
    [CrossRef]
  3. D. M. Pepper, R. L. Abrams, “Narrow Optical Bandpass Filter via Nearly Degenerate Four-Wave Mixing,” Opt. Lett. 3, 212 (1978).
    [CrossRef] [PubMed]
  4. D. A. Pinnow, R. L. Abrams, J. F. Lotspeich, D. M. Henderson, T. K. Plant, R. R. Stephens, C. M. Walker, “An Electro-optic Tunable Filter,” Appl. Phys. Lett. 34, 391 (1979).
    [CrossRef]
  5. P. Yeh, A. Yariv, C-S Hong, “Electromagnetic Propagation in Periodic Stratified Media I: General Theory,” J. Opt. Soc. Am. 67, 423 (1977).
    [CrossRef]
  6. K. H. Richter, W. Guttler, M. Schwoerer, “UV-Holographic Gratings in TS-Diacetylene Single Crystals,” Appl. Phys. A 32, 1 (1983).
    [CrossRef]
  7. A. F. Garito, K. D. Singer, C. C. Teng, in Molecular Optics: Nonlinear Optical Properties of Organic and Polymeric Materials, D. J. Williams, Ed., ACS Symposium Series 233 (American Chemical Society, Washington, DC, 1983).
  8. M. Okuda, K. Onaka, “Bistability of Optical Resonator with Distributed Bragg Reflectors by Using the Kerr Effect,” Jpn. J. Appl. Phys. 16, 769 (1977); “Response of an Optical Resonator with Distributed Bragg Reflectors to Light Pulses,” Jpn. J. Appl. Phys. 17, 1105 (1978).
    [CrossRef]
  9. C. T. Seaton, X. Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).
  10. H. Kogelnik, C. V. Shank, “Coupled-Wave Theory of Distributed Feedback Lasers,” J. Appl. Phys. 43, 2327 (1972).
    [CrossRef]
  11. A. Yariv, M. Nakamura, “Periodic Structures for Integrated Optics,” IEEE J. Quantum Electron. QE-13, 233 (1977).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  13. G. F. Lipscomb, A. F. Garito, R. S. Narang, “An Exceptionally Large Linear Electro-Optic Effect in the Organic Solid MNA,” J. Chem. Phys. 75, 1509 (1981).
    [CrossRef]
  14. See, for example, H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985), p. 56.

1985 (1)

C. T. Seaton, X. Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

1983 (1)

K. H. Richter, W. Guttler, M. Schwoerer, “UV-Holographic Gratings in TS-Diacetylene Single Crystals,” Appl. Phys. A 32, 1 (1983).
[CrossRef]

1982 (1)

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electro-optic Tunable Filters for Infrared Wavelengths,” IEEE J. Quantum Electron. QE-18, 1253 (1982).
[CrossRef]

1981 (2)

W. J. Gunning, “Electro-optically Tuned Spectral Filters: A Review,” Opt. Eng. 20, 837 (1981).
[CrossRef]

G. F. Lipscomb, A. F. Garito, R. S. Narang, “An Exceptionally Large Linear Electro-Optic Effect in the Organic Solid MNA,” J. Chem. Phys. 75, 1509 (1981).
[CrossRef]

1979 (1)

D. A. Pinnow, R. L. Abrams, J. F. Lotspeich, D. M. Henderson, T. K. Plant, R. R. Stephens, C. M. Walker, “An Electro-optic Tunable Filter,” Appl. Phys. Lett. 34, 391 (1979).
[CrossRef]

1978 (1)

1977 (3)

P. Yeh, A. Yariv, C-S Hong, “Electromagnetic Propagation in Periodic Stratified Media I: General Theory,” J. Opt. Soc. Am. 67, 423 (1977).
[CrossRef]

M. Okuda, K. Onaka, “Bistability of Optical Resonator with Distributed Bragg Reflectors by Using the Kerr Effect,” Jpn. J. Appl. Phys. 16, 769 (1977); “Response of an Optical Resonator with Distributed Bragg Reflectors to Light Pulses,” Jpn. J. Appl. Phys. 17, 1105 (1978).
[CrossRef]

A. Yariv, M. Nakamura, “Periodic Structures for Integrated Optics,” IEEE J. Quantum Electron. QE-13, 233 (1977).
[CrossRef]

1972 (1)

H. Kogelnik, C. V. Shank, “Coupled-Wave Theory of Distributed Feedback Lasers,” J. Appl. Phys. 43, 2327 (1972).
[CrossRef]

Abrams, R. L.

D. A. Pinnow, R. L. Abrams, J. F. Lotspeich, D. M. Henderson, T. K. Plant, R. R. Stephens, C. M. Walker, “An Electro-optic Tunable Filter,” Appl. Phys. Lett. 34, 391 (1979).
[CrossRef]

D. M. Pepper, R. L. Abrams, “Narrow Optical Bandpass Filter via Nearly Degenerate Four-Wave Mixing,” Opt. Lett. 3, 212 (1978).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Garito, A. F.

G. F. Lipscomb, A. F. Garito, R. S. Narang, “An Exceptionally Large Linear Electro-Optic Effect in the Organic Solid MNA,” J. Chem. Phys. 75, 1509 (1981).
[CrossRef]

A. F. Garito, K. D. Singer, C. C. Teng, in Molecular Optics: Nonlinear Optical Properties of Organic and Polymeric Materials, D. J. Williams, Ed., ACS Symposium Series 233 (American Chemical Society, Washington, DC, 1983).

Gibbs, H. M.

See, for example, H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985), p. 56.

Gunning, W. J.

W. J. Gunning, “Electro-optically Tuned Spectral Filters: A Review,” Opt. Eng. 20, 837 (1981).
[CrossRef]

Guttler, W.

K. H. Richter, W. Guttler, M. Schwoerer, “UV-Holographic Gratings in TS-Diacetylene Single Crystals,” Appl. Phys. A 32, 1 (1983).
[CrossRef]

Henderson, D. M.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electro-optic Tunable Filters for Infrared Wavelengths,” IEEE J. Quantum Electron. QE-18, 1253 (1982).
[CrossRef]

D. A. Pinnow, R. L. Abrams, J. F. Lotspeich, D. M. Henderson, T. K. Plant, R. R. Stephens, C. M. Walker, “An Electro-optic Tunable Filter,” Appl. Phys. Lett. 34, 391 (1979).
[CrossRef]

Hong, C-S

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-Wave Theory of Distributed Feedback Lasers,” J. Appl. Phys. 43, 2327 (1972).
[CrossRef]

Lipscomb, G. F.

G. F. Lipscomb, A. F. Garito, R. S. Narang, “An Exceptionally Large Linear Electro-Optic Effect in the Organic Solid MNA,” J. Chem. Phys. 75, 1509 (1981).
[CrossRef]

Lotspeich, J. F.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electro-optic Tunable Filters for Infrared Wavelengths,” IEEE J. Quantum Electron. QE-18, 1253 (1982).
[CrossRef]

D. A. Pinnow, R. L. Abrams, J. F. Lotspeich, D. M. Henderson, T. K. Plant, R. R. Stephens, C. M. Walker, “An Electro-optic Tunable Filter,” Appl. Phys. Lett. 34, 391 (1979).
[CrossRef]

Mai, X.

C. T. Seaton, X. Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

Nakamura, M.

A. Yariv, M. Nakamura, “Periodic Structures for Integrated Optics,” IEEE J. Quantum Electron. QE-13, 233 (1977).
[CrossRef]

Narang, R. S.

G. F. Lipscomb, A. F. Garito, R. S. Narang, “An Exceptionally Large Linear Electro-Optic Effect in the Organic Solid MNA,” J. Chem. Phys. 75, 1509 (1981).
[CrossRef]

Okuda, M.

M. Okuda, K. Onaka, “Bistability of Optical Resonator with Distributed Bragg Reflectors by Using the Kerr Effect,” Jpn. J. Appl. Phys. 16, 769 (1977); “Response of an Optical Resonator with Distributed Bragg Reflectors to Light Pulses,” Jpn. J. Appl. Phys. 17, 1105 (1978).
[CrossRef]

Onaka, K.

M. Okuda, K. Onaka, “Bistability of Optical Resonator with Distributed Bragg Reflectors by Using the Kerr Effect,” Jpn. J. Appl. Phys. 16, 769 (1977); “Response of an Optical Resonator with Distributed Bragg Reflectors to Light Pulses,” Jpn. J. Appl. Phys. 17, 1105 (1978).
[CrossRef]

Pepper, D. M.

Pinnow, D. A.

D. A. Pinnow, R. L. Abrams, J. F. Lotspeich, D. M. Henderson, T. K. Plant, R. R. Stephens, C. M. Walker, “An Electro-optic Tunable Filter,” Appl. Phys. Lett. 34, 391 (1979).
[CrossRef]

Plant, T. K.

D. A. Pinnow, R. L. Abrams, J. F. Lotspeich, D. M. Henderson, T. K. Plant, R. R. Stephens, C. M. Walker, “An Electro-optic Tunable Filter,” Appl. Phys. Lett. 34, 391 (1979).
[CrossRef]

Richter, K. H.

K. H. Richter, W. Guttler, M. Schwoerer, “UV-Holographic Gratings in TS-Diacetylene Single Crystals,” Appl. Phys. A 32, 1 (1983).
[CrossRef]

Schwoerer, M.

K. H. Richter, W. Guttler, M. Schwoerer, “UV-Holographic Gratings in TS-Diacetylene Single Crystals,” Appl. Phys. A 32, 1 (1983).
[CrossRef]

Seaton, C. T.

C. T. Seaton, X. Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-Wave Theory of Distributed Feedback Lasers,” J. Appl. Phys. 43, 2327 (1972).
[CrossRef]

Singer, K. D.

A. F. Garito, K. D. Singer, C. C. Teng, in Molecular Optics: Nonlinear Optical Properties of Organic and Polymeric Materials, D. J. Williams, Ed., ACS Symposium Series 233 (American Chemical Society, Washington, DC, 1983).

Stegeman, G. I.

C. T. Seaton, X. Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

Stephens, R. R.

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electro-optic Tunable Filters for Infrared Wavelengths,” IEEE J. Quantum Electron. QE-18, 1253 (1982).
[CrossRef]

D. A. Pinnow, R. L. Abrams, J. F. Lotspeich, D. M. Henderson, T. K. Plant, R. R. Stephens, C. M. Walker, “An Electro-optic Tunable Filter,” Appl. Phys. Lett. 34, 391 (1979).
[CrossRef]

Teng, C. C.

A. F. Garito, K. D. Singer, C. C. Teng, in Molecular Optics: Nonlinear Optical Properties of Organic and Polymeric Materials, D. J. Williams, Ed., ACS Symposium Series 233 (American Chemical Society, Washington, DC, 1983).

Walker, C. M.

D. A. Pinnow, R. L. Abrams, J. F. Lotspeich, D. M. Henderson, T. K. Plant, R. R. Stephens, C. M. Walker, “An Electro-optic Tunable Filter,” Appl. Phys. Lett. 34, 391 (1979).
[CrossRef]

Winful, H. G.

C. T. Seaton, X. Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Yariv, A.

P. Yeh, A. Yariv, C-S Hong, “Electromagnetic Propagation in Periodic Stratified Media I: General Theory,” J. Opt. Soc. Am. 67, 423 (1977).
[CrossRef]

A. Yariv, M. Nakamura, “Periodic Structures for Integrated Optics,” IEEE J. Quantum Electron. QE-13, 233 (1977).
[CrossRef]

Yeh, P.

Appl. Phys. A (1)

K. H. Richter, W. Guttler, M. Schwoerer, “UV-Holographic Gratings in TS-Diacetylene Single Crystals,” Appl. Phys. A 32, 1 (1983).
[CrossRef]

Appl. Phys. Lett. (1)

D. A. Pinnow, R. L. Abrams, J. F. Lotspeich, D. M. Henderson, T. K. Plant, R. R. Stephens, C. M. Walker, “An Electro-optic Tunable Filter,” Appl. Phys. Lett. 34, 391 (1979).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. F. Lotspeich, R. R. Stephens, D. M. Henderson, “Electro-optic Tunable Filters for Infrared Wavelengths,” IEEE J. Quantum Electron. QE-18, 1253 (1982).
[CrossRef]

A. Yariv, M. Nakamura, “Periodic Structures for Integrated Optics,” IEEE J. Quantum Electron. QE-13, 233 (1977).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik, C. V. Shank, “Coupled-Wave Theory of Distributed Feedback Lasers,” J. Appl. Phys. 43, 2327 (1972).
[CrossRef]

J. Chem. Phys. (1)

G. F. Lipscomb, A. F. Garito, R. S. Narang, “An Exceptionally Large Linear Electro-Optic Effect in the Organic Solid MNA,” J. Chem. Phys. 75, 1509 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

M. Okuda, K. Onaka, “Bistability of Optical Resonator with Distributed Bragg Reflectors by Using the Kerr Effect,” Jpn. J. Appl. Phys. 16, 769 (1977); “Response of an Optical Resonator with Distributed Bragg Reflectors to Light Pulses,” Jpn. J. Appl. Phys. 17, 1105 (1978).
[CrossRef]

Opt. Eng. (2)

C. T. Seaton, X. Mai, G. I. Stegeman, H. G. Winful, “Nonlinear Guided Wave Applications,” Opt. Eng. 24, 593 (1985).

W. J. Gunning, “Electro-optically Tuned Spectral Filters: A Review,” Opt. Eng. 20, 837 (1981).
[CrossRef]

Opt. Lett. (1)

Other (3)

A. F. Garito, K. D. Singer, C. C. Teng, in Molecular Optics: Nonlinear Optical Properties of Organic and Polymeric Materials, D. J. Williams, Ed., ACS Symposium Series 233 (American Chemical Society, Washington, DC, 1983).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

See, for example, H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985), p. 56.

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Figures (8)

Fig. 1
Fig. 1

Schematic of a bulk EOTF based on a Fabry-Perot etalon made from an electrooptic material surrounded by Bragg reflectors. The reflectors of length L have a sinusoidal refractive index with modulation depth n1/n0 and period d0. The mirrors exhibit the greatest scattering at wavelengths λ = λ0 where λ0 = 2d0. The cavity has length Δ. Electrodes are attached on the ends for longitudinal modulation and on the top and bottom of the cavity for transverse modulation.

Fig. 2
Fig. 2

Transmission function for a Bragg mirror with coupling constant κL = 1, 2, and 4, length L = 25 μm, and resonant wavelength λ0 = 1 μm. The minimum transmission occurring at λ = λ0 is given by TB(min) = sech2(κL) and decreases with increasing modulation depth. The width of the Bragg stop-band, where transmission returns to unity, is given by λ+ − λ, where λ ± - 1 = λ 0 - 1 ( 1 / 2 π ) ( π 2 / L 2 ) + κ 2 and increases with increasing modulation depth.

Fig. 3
Fig. 3

Transmission of the EOTF for Bragg mirrors with κL = 1 and 2. Absorption effects have been excluded. The cavity length Δ = 12.75 μm, and L = 25 μm. The free spectral range of each etalon depends on κL, but in both cases it is slightly less than the halfwidth of the Bragg stop-band.

Fig. 4
Fig. 4

Transmission of the EOTF for Bragg mirrors with κL = 1 and 2. Absorption effects have been excluded. The cavity length is Δ = 5.25 μm and L = 25 μm. The free spectral range of each etalon is larger than the halfwidth of the Bragg stop-band, and the edges of the transmission function are given by the edges of the corresponding Bragg mirror stop-band.

Fig. 5
Fig. 5

Transmission of the EOTF for Bragg mirrors with κL = 1 and 2. Absorption effects have been excluded. The cavity length Δ = 25.25 μm and L = 25 μm. Several notches now appear in the Bragg stop-band for both filters because the free spectral ranges are smaller than the Bragg stop-band halfwidth.

Fig. 6
Fig. 6

Transmission of the EOTF for the same device as in Fig. 3 with κL = 2 except a voltage has been applied to the cavity to tune the notch away from Δβ = 0. The shift corresponds to an electrooptic-induced phase shift of 0.855 rad, i.e., a voltage V = 0.272Vπ, where Vπ is the halfwave voltage of the cavity material. Larger voltages induce larger phase shifts and can tune the notch to any desired location within the stop-band.

Fig. 7
Fig. 7

Transmission of the EOTF for Bragg mirrors with κL = 2 and 4. The loss coefficients are αB = αC = 0.25 cm−1. The cavity length Δ = 12.75 μm, and the mirror length L = 25 μm. Despite loss, the transmission is still very high for κL = 2. For κL = 4, the peak cavity finesse = 4.7 × 103, so the small absorption length αCΔ ≈ 3.19 × 10−4 can still produce a 65% reduction in the peak intensity.

Fig. 8
Fig. 8

Close-up view of the transmission of the EOTF illustrated in Fig. 7 for κL = 2 and 4. The notch widths are seen to be ~20 and 1 Å, respectively. Transmission is over 95% for κL = 2 and is ~35% for κL = 4. Reduction of transmission for higher finesse etalons is expected due to multiple reflections in the device.

Equations (13)

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r ( Δ β ) = i κ sinh δ L δ cosh δ L + i Δ β sinh δ L ,
T B = κ 2 - Δ β 2 κ 2 cosh 2 δ L - Δ β 2 .
r ( Δ β ) = r ( Δ β ) exp [ i ψ ( Δ β ) ] ,
r ( Δ β ) = κ sinh δ L κ 2 cosh 2 δ L - Δ β 2 ,
tan ψ ( Δ β ) = δ Δ β coth δ L .
τ = ( 1 + 4 T B 2 { I m [ r exp ( i ϕ F . P ) ] } 2 ) - 1 ,
ϕ = ϕ F . P + ψ = Δ ( Δ β + γ ) + ψ ( Δ β ) + β 0 Δ
τ = ( 1 + F sin 2 ϕ ) - 1 ,
F = 4 κ 2 δ 4 sinh 2 δ L ( κ 2 cosh 2 δ L - Δ β 2 ) .
ϕ = Δ ( Δ β + γ ) + [ ψ ( Δ β ) - ψ ( 0 ) ]
τ = ( A + F sin 2 ϕ ) - 1 ,
R B = i k sinh Γ L 2 ( α B + i Δ β ) sinh Γ L + Γ cosh Γ L 2 ,
T B = Γ 2 ( α B + i Δ β ) sinh Γ L + Γ cosh Γ L 2 ,

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