Abstract

We have extended the local normal mode approach of Burns and Milton to derive the condition for adiabatic invariance in graded-index channel waveguides. The results show that in spite of the index-grading and 2-D confinement, the inequality describing the adiabatic condition is basically the same as that for step-index slab waveguides. To verify the predictions of the theory, we have fabricated, using ion exchange in glass, a cross coupler with two asymmetric input channel waveguides and two symmetric output guides. The experimental results show equal output power (3.0 ± 0.1 dB) from the two symmetric guides when either of the two unequal guides is excited, demonstrating excellent agreement with the theory.

© 1990 Optical Society of America

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References

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  1. P. Suchosky, R. V. Ramaswamy, “Design of Single-Mode Step-Tapered Waveguide Sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
    [Crossref]
  2. H. Yajima, “Dielectric Thin Film Optical Branching Waveguide,” Appl. Phys. Lett. 22, 647–649 (1973).
    [Crossref]
  3. H. Yajima, “Coupled Mode Analysis of Dielectric Planar Branching Waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
    [Crossref]
  4. W. K. Burns, A. F. Milton, “Mode Conversion in Planar-Dielectric Separating Waveguides,” IEEE J. Quantum Electron. QE-11, 32–39 (1975).
    [Crossref]
  5. M. Izutsu, A. Enokihara, T. Sueta, “Optical-Waveguide Hybrid Coupler,” Opt. Lett. 7, 549–551 (1982).
    [Crossref] [PubMed]
  6. Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital Optical Switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
    [Crossref]
  7. J. Cook, “Tapered Velocity Coupler,” Bell Syst. Tech. J. 34, 807–822 (1955).
  8. D. Marcuse, “Radiation Losses of Tapered Dielectric Slab Waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).
  9. H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-Loss Small-Mode Passive Waveguides and Near-Adiabatic Tapers in BK7 Glass,” IEEE/OSA J. Lightwave Technol. LT-7, 1590–1596 (1989).
    [Crossref]
  10. R. N. Thurston, “Analysis of Mode Separation in Multichannel Branching Waveguides,” IEEE J. Quantum Electron. QE-23, 1245–1254 (1987).
    [Crossref]

1989 (1)

H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-Loss Small-Mode Passive Waveguides and Near-Adiabatic Tapers in BK7 Glass,” IEEE/OSA J. Lightwave Technol. LT-7, 1590–1596 (1989).
[Crossref]

1987 (3)

R. N. Thurston, “Analysis of Mode Separation in Multichannel Branching Waveguides,” IEEE J. Quantum Electron. QE-23, 1245–1254 (1987).
[Crossref]

P. Suchosky, R. V. Ramaswamy, “Design of Single-Mode Step-Tapered Waveguide Sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[Crossref]

Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital Optical Switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
[Crossref]

1982 (1)

1978 (1)

H. Yajima, “Coupled Mode Analysis of Dielectric Planar Branching Waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
[Crossref]

1975 (1)

W. K. Burns, A. F. Milton, “Mode Conversion in Planar-Dielectric Separating Waveguides,” IEEE J. Quantum Electron. QE-11, 32–39 (1975).
[Crossref]

1973 (1)

H. Yajima, “Dielectric Thin Film Optical Branching Waveguide,” Appl. Phys. Lett. 22, 647–649 (1973).
[Crossref]

1970 (1)

D. Marcuse, “Radiation Losses of Tapered Dielectric Slab Waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).

1955 (1)

J. Cook, “Tapered Velocity Coupler,” Bell Syst. Tech. J. 34, 807–822 (1955).

Baran, J. E.

Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital Optical Switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
[Crossref]

Burns, W. K.

W. K. Burns, A. F. Milton, “Mode Conversion in Planar-Dielectric Separating Waveguides,” IEEE J. Quantum Electron. QE-11, 32–39 (1975).
[Crossref]

Cook, J.

J. Cook, “Tapered Velocity Coupler,” Bell Syst. Tech. J. 34, 807–822 (1955).

Enokihara, A.

Izutsu, M.

Marcuse, D.

D. Marcuse, “Radiation Losses of Tapered Dielectric Slab Waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).

Milton, A. F.

W. K. Burns, A. F. Milton, “Mode Conversion in Planar-Dielectric Separating Waveguides,” IEEE J. Quantum Electron. QE-11, 32–39 (1975).
[Crossref]

Perlmutter, P.

Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital Optical Switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
[Crossref]

Ramaswamy, R. V.

H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-Loss Small-Mode Passive Waveguides and Near-Adiabatic Tapers in BK7 Glass,” IEEE/OSA J. Lightwave Technol. LT-7, 1590–1596 (1989).
[Crossref]

P. Suchosky, R. V. Ramaswamy, “Design of Single-Mode Step-Tapered Waveguide Sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[Crossref]

Silberberg, Y.

Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital Optical Switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
[Crossref]

Srivastava, R.

H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-Loss Small-Mode Passive Waveguides and Near-Adiabatic Tapers in BK7 Glass,” IEEE/OSA J. Lightwave Technol. LT-7, 1590–1596 (1989).
[Crossref]

Suchosky, P.

P. Suchosky, R. V. Ramaswamy, “Design of Single-Mode Step-Tapered Waveguide Sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[Crossref]

Sueta, T.

Thurston, R. N.

R. N. Thurston, “Analysis of Mode Separation in Multichannel Branching Waveguides,” IEEE J. Quantum Electron. QE-23, 1245–1254 (1987).
[Crossref]

Yajima, H.

H. Yajima, “Coupled Mode Analysis of Dielectric Planar Branching Waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
[Crossref]

H. Yajima, “Dielectric Thin Film Optical Branching Waveguide,” Appl. Phys. Lett. 22, 647–649 (1973).
[Crossref]

Zhenguang, H.

H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-Loss Small-Mode Passive Waveguides and Near-Adiabatic Tapers in BK7 Glass,” IEEE/OSA J. Lightwave Technol. LT-7, 1590–1596 (1989).
[Crossref]

Appl. Phys. Lett. (2)

Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital Optical Switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
[Crossref]

H. Yajima, “Dielectric Thin Film Optical Branching Waveguide,” Appl. Phys. Lett. 22, 647–649 (1973).
[Crossref]

Bell Syst. Tech. J. (2)

J. Cook, “Tapered Velocity Coupler,” Bell Syst. Tech. J. 34, 807–822 (1955).

D. Marcuse, “Radiation Losses of Tapered Dielectric Slab Waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).

IEEE J. Quantum Electron. (4)

H. Yajima, “Coupled Mode Analysis of Dielectric Planar Branching Waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
[Crossref]

W. K. Burns, A. F. Milton, “Mode Conversion in Planar-Dielectric Separating Waveguides,” IEEE J. Quantum Electron. QE-11, 32–39 (1975).
[Crossref]

R. N. Thurston, “Analysis of Mode Separation in Multichannel Branching Waveguides,” IEEE J. Quantum Electron. QE-23, 1245–1254 (1987).
[Crossref]

P. Suchosky, R. V. Ramaswamy, “Design of Single-Mode Step-Tapered Waveguide Sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[Crossref]

IEEE/OSA J. Lightwave Technol. (1)

H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-Loss Small-Mode Passive Waveguides and Near-Adiabatic Tapers in BK7 Glass,” IEEE/OSA J. Lightwave Technol. LT-7, 1590–1596 (1989).
[Crossref]

Opt. Lett. (1)

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Figures (6)

Fig. 1
Fig. 1

Illustration of the step transitions used to model the structure.

Fig. 2
Fig. 2

Schematics of the index profile used for calculation of local normal modes. The graded-index profile is broken down into a series of r step-index slabs.

Fig. 3
Fig. 3

Output amplitude ratio vs device angle when only the fundamental local normal mode is excited at the input; i denotes the fundamental mode and j denotes the first-order mode.

Fig. 4
Fig. 4

Comparison of the amplitude coefficient of the converted mode for step-index and graded-index structures.

Fig. 5
Fig. 5

Optical waveguide cross coupler.

Fig. 6
Fig. 6

Comparison of the theoretical and experimental intensity profiles at the output of the device when either of the inputs A or B is excited.

Equations (25)

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e y i = E t ( z ) ɛ i ( x , z ) exp [ - i α i ( z ) ] ,
α i ( z ) = β i z + ϕ i ;
A j 1 = i = 0 m c i j A i 0 cos ( α i 0 - α j 1 ) ,
c i j = 2 β i 0 β j 1 ( β j 0 + β j 1 ) ( β j 0 + β i 1 β i 0 + β i 1 ) I i 0 , j 1 I i 0 , i 0 I j 1 , j 1 ,
tan α j 1 = i = 0 m c i j A i 0 sin α i 0 i = 0 m c i j A i 0 cos α i 0 .
I γ , δ = ɛ γ ɛ δ d x .
α i 0 = β i L + α i 1 ( from the previous calculation point ) ,
e y = E ( x ) exp ( - i β z ) ,
d 2 E d x 2 + ( k o 2 n 2 - β 2 ) E = 0
N = β k o .
κ i = β 2 - k o 2 n i 2
d 2 E i d x 2 + k o 2 ( n i 2 - N 2 ) E i = 0 ,
E i = { a i cos ( u i ) + b i sin ( u i ) if N n i , a i cosh ( u i ) + b i sinh ( u i ) if N > n i ,
u 1 = κ 1 ( x 1 - x ) and u i = κ i ( x - x i - 1 ) ,             i = 2 , 3 , , r .
E 1 exp ( - u 1 ) a 1 = - b 1 ,
E r exp ( - u r ) a r = - b r .
a 1 = a 2 and a 1 κ 1 = b 2 κ 2 .
a i + 1 = { a i cos ( u i ) + b i sin ( u i ) if N n i , a i cosh ( u i ) + b i sinh ( u i ) if N > n i ,
b i + 1 κ i + 1 = { - a i κ i sin ( u i ) + b i κ i cos ( u i ) if N n i , a i κ i sinh ( u i ) + b i κ i cosh ( u i ) if N > n i .
N j + 1 = N j - f ( N j ) f ( N j ) / N j .
Δ β 0.43 θ γ ¯ ,
γ ¯ = β ¯ 2 - ( k o n s ) 2 ,
Δ β γ ¯ = Δ N N 2 - n s 2 = 5.34 × 10 - 3 .
Δ β > 3 θ γ ¯ .
power division = - 10 log P D P C + P D ( dB ) ,

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