Abstract

Optical path integration in the time domain is presented as a method of finding the impulse response of dispersive, multiple-scattering systems. Using the particular example of a grating-coupled waveguide filter, we first show how a perturbation solution may be found to the inverse Fourier transform of the governing coupled-mode equations. An interpretation of the successive terms in the solution is then given in terms of path integrals. Finally, we show how analytic approximations to the published (but so far unexplained) impulse responses of such devices are obtained.

© 1992 Optical Society of America

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References

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  1. R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
    [CrossRef]
  2. R. P. Feynman, QED: The Strange Story of Light and Matter (Princeton U. Press, Princeton, N.J., 1985).
  3. M. V. Berry, The Diffraction of Light by Ultrasound (Academic, New York, 1966).
  4. T. C. Poon, A. Korpel, “Feynman diagram approach to acousto-optic scattering in the near Bragg region,” J. Opt. Soc. Am. 71, 1202–1208 (1981).
    [CrossRef]
  5. O. V. Konstantinov, M. M. Panakhov, Yu. F. Romanov, “Electrodynamic perturbation theory for light diffraction from 3-D phase gratings,” Opt. Spectrosc. (USSR) 46, 551–554 (1979).
  6. S. Yu. Karpov, “Higher-order perturbation-theoretic description of light diffraction by volume phase gratings,” Sov. Phys. Tech. Phys. 29, 711–712 (1984).
  7. R. R. A. Syms, “Direct path integral solution for a reflection grating,” Appl. Opt. 27, 29–31 (1988).
    [CrossRef] [PubMed]
  8. R. R. A. Syms, “Path integral formulation of multiple scattering problems in integrated optics,” Appl. Opt. 25, 4402–4412 (1986).
    [CrossRef] [PubMed]
  9. R. R. A. Syms, “Path-integral formulation of nonuniform optical coupled wave problems,” Appl. Opt. 26, 4420–4230 (1987).
    [CrossRef]
  10. C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. 2, 406–412 (Part I); 413–420 (Part II) (1935); Proc. Indian Acad. Sci. 3, 75–84 (Part III), 119–125 (Part IV), 459–465 (Part V) (1936).
  11. H. Kogelnik “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  12. S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719 (1954).
  13. A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  14. L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
    [CrossRef]
  15. R. R. A. Syms, Practical Volume Holography (Oxford U. Press, Oxford, 1990).
  16. R. C. Alferness, L. L. Buhl, “Tunable electro-optic waveguide TE-TM converter/wavelength filter,” Appl. Phys. Lett. 40, 861–862 (1982).
    [CrossRef]
  17. R. C. Alferness, L. L. Buhl, “Polarization independent optical filter using interwaveguide TE-TM conversion,” Appl. Phys. Lett. 39, 131–134 (1981).
    [CrossRef]
  18. D. P. Morgan, “Spatially-variant coupling design for co-directional mode-converting bandpass filters,” in Thin Film Technologies II, J. R. Jacobsson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.652, 139–145 (1986).
  19. R. R. A. Syms, S. Makrimichalou, A. S. Holmes, “High-speed, optical signal processing potential of grating-coupled waveguide filters,” Appl. Opt. 30, 3762–3769 (1991).
    [CrossRef] [PubMed]

1991 (1)

1988 (1)

1987 (1)

R. R. A. Syms, “Path-integral formulation of nonuniform optical coupled wave problems,” Appl. Opt. 26, 4420–4230 (1987).
[CrossRef]

1986 (1)

1984 (1)

S. Yu. Karpov, “Higher-order perturbation-theoretic description of light diffraction by volume phase gratings,” Sov. Phys. Tech. Phys. 29, 711–712 (1984).

1982 (1)

R. C. Alferness, L. L. Buhl, “Tunable electro-optic waveguide TE-TM converter/wavelength filter,” Appl. Phys. Lett. 40, 861–862 (1982).
[CrossRef]

1981 (2)

R. C. Alferness, L. L. Buhl, “Polarization independent optical filter using interwaveguide TE-TM conversion,” Appl. Phys. Lett. 39, 131–134 (1981).
[CrossRef]

T. C. Poon, A. Korpel, “Feynman diagram approach to acousto-optic scattering in the near Bragg region,” J. Opt. Soc. Am. 71, 1202–1208 (1981).
[CrossRef]

1979 (1)

O. V. Konstantinov, M. M. Panakhov, Yu. F. Romanov, “Electrodynamic perturbation theory for light diffraction from 3-D phase gratings,” Opt. Spectrosc. (USSR) 46, 551–554 (1979).

1977 (1)

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

1973 (1)

A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

1969 (1)

H. Kogelnik “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1954 (1)

S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719 (1954).

1948 (1)

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[CrossRef]

1935 (1)

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. 2, 406–412 (Part I); 413–420 (Part II) (1935); Proc. Indian Acad. Sci. 3, 75–84 (Part III), 119–125 (Part IV), 459–465 (Part V) (1936).

Alferness, R. C.

R. C. Alferness, L. L. Buhl, “Tunable electro-optic waveguide TE-TM converter/wavelength filter,” Appl. Phys. Lett. 40, 861–862 (1982).
[CrossRef]

R. C. Alferness, L. L. Buhl, “Polarization independent optical filter using interwaveguide TE-TM conversion,” Appl. Phys. Lett. 39, 131–134 (1981).
[CrossRef]

Berry, M. V.

M. V. Berry, The Diffraction of Light by Ultrasound (Academic, New York, 1966).

Buhl, L. L.

R. C. Alferness, L. L. Buhl, “Tunable electro-optic waveguide TE-TM converter/wavelength filter,” Appl. Phys. Lett. 40, 861–862 (1982).
[CrossRef]

R. C. Alferness, L. L. Buhl, “Polarization independent optical filter using interwaveguide TE-TM conversion,” Appl. Phys. Lett. 39, 131–134 (1981).
[CrossRef]

Feynman, R. P.

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[CrossRef]

R. P. Feynman, QED: The Strange Story of Light and Matter (Princeton U. Press, Princeton, N.J., 1985).

Holmes, A. S.

Karpov, S. Yu.

S. Yu. Karpov, “Higher-order perturbation-theoretic description of light diffraction by volume phase gratings,” Sov. Phys. Tech. Phys. 29, 711–712 (1984).

Kogelnik, H.

H. Kogelnik “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Konstantinov, O. V.

O. V. Konstantinov, M. M. Panakhov, Yu. F. Romanov, “Electrodynamic perturbation theory for light diffraction from 3-D phase gratings,” Opt. Spectrosc. (USSR) 46, 551–554 (1979).

Korpel, A.

Makrimichalou, S.

Miller, S. E.

S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719 (1954).

Morgan, D. P.

D. P. Morgan, “Spatially-variant coupling design for co-directional mode-converting bandpass filters,” in Thin Film Technologies II, J. R. Jacobsson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.652, 139–145 (1986).

Nath, N. S. N.

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. 2, 406–412 (Part I); 413–420 (Part II) (1935); Proc. Indian Acad. Sci. 3, 75–84 (Part III), 119–125 (Part IV), 459–465 (Part V) (1936).

Panakhov, M. M.

O. V. Konstantinov, M. M. Panakhov, Yu. F. Romanov, “Electrodynamic perturbation theory for light diffraction from 3-D phase gratings,” Opt. Spectrosc. (USSR) 46, 551–554 (1979).

Poon, T. C.

Raman, C. V.

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. 2, 406–412 (Part I); 413–420 (Part II) (1935); Proc. Indian Acad. Sci. 3, 75–84 (Part III), 119–125 (Part IV), 459–465 (Part V) (1936).

Romanov, Yu. F.

O. V. Konstantinov, M. M. Panakhov, Yu. F. Romanov, “Electrodynamic perturbation theory for light diffraction from 3-D phase gratings,” Opt. Spectrosc. (USSR) 46, 551–554 (1979).

Solymar, L.

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

Syms, R. R. A.

Yariv, A.

A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (3)

R. C. Alferness, L. L. Buhl, “Tunable electro-optic waveguide TE-TM converter/wavelength filter,” Appl. Phys. Lett. 40, 861–862 (1982).
[CrossRef]

R. C. Alferness, L. L. Buhl, “Polarization independent optical filter using interwaveguide TE-TM conversion,” Appl. Phys. Lett. 39, 131–134 (1981).
[CrossRef]

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

Bell Syst. Tech. J. (2)

H. Kogelnik “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719 (1954).

IEEE J. Quantum Electron. (1)

A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Spectrosc. (USSR) (1)

O. V. Konstantinov, M. M. Panakhov, Yu. F. Romanov, “Electrodynamic perturbation theory for light diffraction from 3-D phase gratings,” Opt. Spectrosc. (USSR) 46, 551–554 (1979).

Proc. Indian Acad. Sci. (1)

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. 2, 406–412 (Part I); 413–420 (Part II) (1935); Proc. Indian Acad. Sci. 3, 75–84 (Part III), 119–125 (Part IV), 459–465 (Part V) (1936).

Rev. Mod. Phys. (1)

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

S. Yu. Karpov, “Higher-order perturbation-theoretic description of light diffraction by volume phase gratings,” Sov. Phys. Tech. Phys. 29, 711–712 (1984).

Other (4)

R. P. Feynman, QED: The Strange Story of Light and Matter (Princeton U. Press, Princeton, N.J., 1985).

M. V. Berry, The Diffraction of Light by Ultrasound (Academic, New York, 1966).

R. R. A. Syms, Practical Volume Holography (Oxford U. Press, Oxford, 1990).

D. P. Morgan, “Spatially-variant coupling design for co-directional mode-converting bandpass filters,” in Thin Film Technologies II, J. R. Jacobsson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.652, 139–145 (1986).

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Figures (6)

Fig. 1
Fig. 1

Routes to the impulse response of an optical coupled-mode device: (a) conventional routes and (b) path integration in the time domain.

Fig. 2
Fig. 2

(a) a TE–TM mode converter filter, (b) interwaveguide mode converter filter, and (c) phase-matching diagram for grating-assisted coupling between codirectional modes.

Fig. 3
Fig. 3

Plots of the first few functions I n ′ = I n /(−jα) n

Fig. 4
Fig. 4

Diagrammatic representation of low-order path integrals.

Fig. 5
Fig. 5

Plots of |B2(τ, t)| for 1, κL = π/4; 2, π/2; 3, 3π/4; and 4, π. Plots were computed with a six-term series.

Fig. 6
Fig. 6

Sketches of two envelopes B1{τ, tT1} and B2{τ, tT2}.

Tables (1)

Tables Icon

Table 1 Some Low-Order Path Integrals

Equations (27)

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K = 2 π / Λ = β 10 - β 20 .
E ( x , y , z , ω ) = A 1 ( z , ω ) E T 1 ( x , y ) exp [ - j β 1 ( ω ) z ] + A 2 ( z , ω ) E T 2 ( x , y ) exp [ - j β 2 ( ω ) z ] .
dA 1 / d z + j κ exp [ - j Δ β z ] A 2 = 0 , dA 2 / d z + j κ exp [ + j Δ β z ] A 1 = 0 ,
Δ β = K - ( β 1 - β 2 ) .
E ( x , y , z , t ) = a 1 ( z , t ) E T 1 ( x , y ) + a 2 ( z , t ) E T 2 ( x , y ) .
β 1 = β 10 + γ 1 ( ω - ω 0 ) , β 2 = β 20 + γ 2 ( ω - ω 0 ) .
E = A 1 exp [ - j γ 1 ( ω - ω 0 ) z ] exp ( - j β 10 z ) E T 1 + A 2 exp [ - j γ 2 ( ω - ω 0 ) z ] exp ( - j β 20 z ) E T 2 .
a i ( z , t ) = exp ( - j β i 0 z ) F - 1 { [ A i ( z , ω ) ] × exp [ - j γ i ( ω - ω 0 ) ] z }             for i = 1 , 2.
a i ( z , t ) = exp [ j ( ω 0 t - β i 0 z ) ] B i [ z , t - T i ] ,
Δ β = ( γ 2 - γ 1 ) ( ω - ω 0 ) .
d B 1 ( z , ω ) / d z + j κ exp [ - j ( γ 2 - γ 1 ) z ω ] B 2 ( z , ω ) = 0 , d B 2 ( z , ω ) / d z + j κ exp [ + j ( γ 2 - γ 1 ) z ω ] B 1 ( z , ω ) = 0.
dB 1 ( z , t ) / d z + j κ B 2 [ z , t - ( γ 2 - γ 1 ) z ] = 0 , dB 2 ( z , t ) / d z + j κ B 1 [ z , t + ( γ 2 - γ 1 ) z ] = 0.
B 1 ( z , t ) = δ ( t ) - j κ 0 z B 2 [ z 1 , t - ( γ 2 - γ 1 ) z 1 ] d z 1 , B 2 ( z , t ) = - j κ 0 z B 1 [ z 1 , t + ( γ 2 - γ 1 ) z 1 ] d z 1 .
B 1 ( τ , t ) = δ ( t ) - j α 0 τ B 2 ( τ 1 , t - τ 1 ) d τ 1 , B 2 ( τ , t ) = - j α 0 τ B 1 ( τ 1 , t + τ 1 ) d τ 1 ,
B 1 ( τ , t ) = I 0 ( τ , t ) + I 2 ( τ , t ) + I 4 ( τ , t ) + , B 2 ( τ , t ) = I 1 ( τ , t ) + I 3 ( τ , t ) + I 5 ( τ , t ) + .
I 0 ( τ , t ) = δ ( t ) , I n ( τ , t ) = - j α 0 τ I n - 1 ( τ 1 , t + τ 1 ) d τ 1 ( n odd ) . I n ( τ , t ) = - j α 0 τ I n - 1 ( τ 1 , t - τ 1 ) d τ 1 ( n even ) .
I 0 = δ ( t ) I 1 = - j α 0 τ δ ( t + τ 1 ) d τ 1 I 2 = ( - j α ) 2 0 τ 0 τ 1 δ [ t - ( τ 1 - τ 2 ) ] d τ 2 d τ 1 I 3 = ( - j α ) 3 0 τ 0 τ 1 0 τ 2 δ [ t + ( τ 1 - τ 2 + τ 3 ) ] d τ 3 d τ 2 d τ 1
- δ ( τ 1 ) d τ 1 = 1.
0 τ 1 δ [ t - ( τ 1 - τ 2 ) ] d τ 2 = 1             for 0 t τ 1 .
0 τ 0 τ 1 δ [ t - ( τ 1 - τ 2 ) ] d τ 2 d τ 1 = t τ d τ 1 = ( τ - t )             for 0 t τ .
0 τ 2 δ [ t + ( τ 1 - τ 2 + τ 3 ) ] d τ 3 = 1             for - τ 1 t - ( τ 1 - τ 2 ) ,
0 τ 1 0 τ 2 δ [ t + ( τ 1 - τ 2 + τ 3 ) ] d τ 3 d τ 2 = t + τ 1 τ 1 d τ 2 = - t             for - τ 1 t 0.
0 τ 0 τ 1 0 τ 2 δ [ t + ( τ 1 - τ 2 + τ 3 ) ] d τ 3 d τ 2 d τ 1 = - t τ - t d τ 1 = - t ( t + τ )             for - τ t 0.
B 1 ( τ , t ) = δ ( t ) - α 2 ( τ - t ) + α 4 t ( τ - t ) 2 / 2 ,             0 t τ , B 2 ( τ , t ) = - j [ α + α 3 t ( τ + t ) + ] ,             - τ t 0.
B 2 ( τ , t ) = - j α { 1 + ( α τ ) 2 ( t / τ ) [ 1 + ( t / τ ) ] + } .
B 2 ( τ , t ) = - j α { 1 + ( α τ ) 2 [ ( t + τ ) 2 - 1 / 4 ] } ,             - τ / 2 t τ / 2.
κ ( z ) = κ 0 { 1 - ( κ 0 L ) 2 [ ( z / L ) 2 - 1 / 4 ] } ,             - L / 2 z L / 2 ,

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