Abstract

A unified approach to obtain the characteristics of almost-periodic grating slab waveguides including gain in the waveguide is reported. In this approach the waveguides are divided into short segments, and in each segment the gratings are assumed to be periodic, that is, parameters such as coupling coefficient, grating phase, deviations from the Bragg frequency, and gain in the waveguide are independent of a propagation direction z. Then characteristics of almost-periodic grating slab waveguides can be obtained by multiplying each F matrix of a short segment with the proper grating phase conditions at the interface between two adjacent segments. The appropriateness of this approach is shown for typical aperiodic grating waveguides such as tapered, chirped, and phase-shifted gratings. The results obtained by this method are compared with others and prove to be in good agreement with the results obtained by other methods. In addition to these characteristics, it is shown that the F matrix can be used to obtain the threshold conditions for distributed feedback laser oscillations including reflections from cleaved edges.

© 1987 Optical Society of America

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References

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  1. H. Kogelnik, “Filter Response of Non-Uniform Almost-Periodic Structure,” Bell Syst. Tech. J. 55, 109 (1976).
  2. K. O. Hill, “Aperiodic Distributed-Parameter Waveguides for Integrated Optics,” Appl. Opt. 13, 1853 (1974).
    [CrossRef] [PubMed]
  3. M. Matsuhara, K. O. Hill, “Optical-Waveguide Band-Rejection Filters: Design,” Appl. Opt. 13, 2886 (1974).
    [CrossRef] [PubMed]
  4. M. Matsuhara, K. O. Hill, A. Watanabe, “Optical-Waveguide Filters: Synthesis,” J. Opt. Soc. Am. 65, 804 (1975).
    [CrossRef]
  5. S. H. Kim, C. G. Fostand, “Tunable Narrow-Band Thin Film Waveguide Grating Filters,” IEEE J. Quantum Electron. QE-15, 1405 (1979).
  6. Y. Itaya, T. Matuoka, K. Kuroiwa, T. Ikegami, “Longitudinal Mode Behaviors of 1.5 μm Range GalnAsP/InP Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-20, 230 (1984).
    [CrossRef]
  7. K. Ukata, S. Akiba, K. Sakai, Y. Matsushima, “Effect of Mirror Facets on Lasing Characteristics of Distributed Feedback InGaAsP/InP Laser Diode at 1.5 μm Range,” IEEE J. Quantum Electron. QE-20, 236 (1984).
  8. W. Streifer, R. D. Burham, D. R. Scifres, “Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-11, 154 (1975).
    [CrossRef]
  9. A. Suzuki, T. Tada, “Theory and Experiment on Distributed Feedback Lasers with Chirped Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 236, 532 (1981).
  10. F. Koyama, Y. Suematsu, “Phase-Controlled Active-Distributed-Reflector Laser,” IECE Jpn. OQE84, 67 (1984), in Japanese.
  11. D. Kermisch, “Nonuniform Sinusoidally Modulated Dielectric Gratings,” J. Opt. Soc. of Am. 59, 1409 (1969).
    [CrossRef]
  12. M. Yamada, K. Sakuda, “Adjustable Gain and Bandwidth Light Amplifiers in Terms of Distributed Feedback Structures,” J. Opt. Soc. Am. A4, 69 (1987), to be published.
    [CrossRef]

1984

Y. Itaya, T. Matuoka, K. Kuroiwa, T. Ikegami, “Longitudinal Mode Behaviors of 1.5 μm Range GalnAsP/InP Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-20, 230 (1984).
[CrossRef]

K. Ukata, S. Akiba, K. Sakai, Y. Matsushima, “Effect of Mirror Facets on Lasing Characteristics of Distributed Feedback InGaAsP/InP Laser Diode at 1.5 μm Range,” IEEE J. Quantum Electron. QE-20, 236 (1984).

F. Koyama, Y. Suematsu, “Phase-Controlled Active-Distributed-Reflector Laser,” IECE Jpn. OQE84, 67 (1984), in Japanese.

1981

A. Suzuki, T. Tada, “Theory and Experiment on Distributed Feedback Lasers with Chirped Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 236, 532 (1981).

1979

S. H. Kim, C. G. Fostand, “Tunable Narrow-Band Thin Film Waveguide Grating Filters,” IEEE J. Quantum Electron. QE-15, 1405 (1979).

1976

H. Kogelnik, “Filter Response of Non-Uniform Almost-Periodic Structure,” Bell Syst. Tech. J. 55, 109 (1976).

1975

W. Streifer, R. D. Burham, D. R. Scifres, “Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-11, 154 (1975).
[CrossRef]

M. Matsuhara, K. O. Hill, A. Watanabe, “Optical-Waveguide Filters: Synthesis,” J. Opt. Soc. Am. 65, 804 (1975).
[CrossRef]

1974

1969

D. Kermisch, “Nonuniform Sinusoidally Modulated Dielectric Gratings,” J. Opt. Soc. of Am. 59, 1409 (1969).
[CrossRef]

Akiba, S.

K. Ukata, S. Akiba, K. Sakai, Y. Matsushima, “Effect of Mirror Facets on Lasing Characteristics of Distributed Feedback InGaAsP/InP Laser Diode at 1.5 μm Range,” IEEE J. Quantum Electron. QE-20, 236 (1984).

Burham, R. D.

W. Streifer, R. D. Burham, D. R. Scifres, “Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-11, 154 (1975).
[CrossRef]

Fostand, C. G.

S. H. Kim, C. G. Fostand, “Tunable Narrow-Band Thin Film Waveguide Grating Filters,” IEEE J. Quantum Electron. QE-15, 1405 (1979).

Hill, K. O.

Ikegami, T.

Y. Itaya, T. Matuoka, K. Kuroiwa, T. Ikegami, “Longitudinal Mode Behaviors of 1.5 μm Range GalnAsP/InP Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-20, 230 (1984).
[CrossRef]

Itaya, Y.

Y. Itaya, T. Matuoka, K. Kuroiwa, T. Ikegami, “Longitudinal Mode Behaviors of 1.5 μm Range GalnAsP/InP Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-20, 230 (1984).
[CrossRef]

Kermisch, D.

D. Kermisch, “Nonuniform Sinusoidally Modulated Dielectric Gratings,” J. Opt. Soc. of Am. 59, 1409 (1969).
[CrossRef]

Kim, S. H.

S. H. Kim, C. G. Fostand, “Tunable Narrow-Band Thin Film Waveguide Grating Filters,” IEEE J. Quantum Electron. QE-15, 1405 (1979).

Kogelnik, H.

H. Kogelnik, “Filter Response of Non-Uniform Almost-Periodic Structure,” Bell Syst. Tech. J. 55, 109 (1976).

Koyama, F.

F. Koyama, Y. Suematsu, “Phase-Controlled Active-Distributed-Reflector Laser,” IECE Jpn. OQE84, 67 (1984), in Japanese.

Kuroiwa, K.

Y. Itaya, T. Matuoka, K. Kuroiwa, T. Ikegami, “Longitudinal Mode Behaviors of 1.5 μm Range GalnAsP/InP Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-20, 230 (1984).
[CrossRef]

Matsuhara, M.

Matsushima, Y.

K. Ukata, S. Akiba, K. Sakai, Y. Matsushima, “Effect of Mirror Facets on Lasing Characteristics of Distributed Feedback InGaAsP/InP Laser Diode at 1.5 μm Range,” IEEE J. Quantum Electron. QE-20, 236 (1984).

Matuoka, T.

Y. Itaya, T. Matuoka, K. Kuroiwa, T. Ikegami, “Longitudinal Mode Behaviors of 1.5 μm Range GalnAsP/InP Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-20, 230 (1984).
[CrossRef]

Sakai, K.

K. Ukata, S. Akiba, K. Sakai, Y. Matsushima, “Effect of Mirror Facets on Lasing Characteristics of Distributed Feedback InGaAsP/InP Laser Diode at 1.5 μm Range,” IEEE J. Quantum Electron. QE-20, 236 (1984).

Sakuda, K.

M. Yamada, K. Sakuda, “Adjustable Gain and Bandwidth Light Amplifiers in Terms of Distributed Feedback Structures,” J. Opt. Soc. Am. A4, 69 (1987), to be published.
[CrossRef]

Scifres, D. R.

W. Streifer, R. D. Burham, D. R. Scifres, “Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-11, 154 (1975).
[CrossRef]

Streifer, W.

W. Streifer, R. D. Burham, D. R. Scifres, “Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-11, 154 (1975).
[CrossRef]

Suematsu, Y.

F. Koyama, Y. Suematsu, “Phase-Controlled Active-Distributed-Reflector Laser,” IECE Jpn. OQE84, 67 (1984), in Japanese.

Suzuki, A.

A. Suzuki, T. Tada, “Theory and Experiment on Distributed Feedback Lasers with Chirped Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 236, 532 (1981).

Tada, T.

A. Suzuki, T. Tada, “Theory and Experiment on Distributed Feedback Lasers with Chirped Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 236, 532 (1981).

Ukata, K.

K. Ukata, S. Akiba, K. Sakai, Y. Matsushima, “Effect of Mirror Facets on Lasing Characteristics of Distributed Feedback InGaAsP/InP Laser Diode at 1.5 μm Range,” IEEE J. Quantum Electron. QE-20, 236 (1984).

Watanabe, A.

Yamada, M.

M. Yamada, K. Sakuda, “Adjustable Gain and Bandwidth Light Amplifiers in Terms of Distributed Feedback Structures,” J. Opt. Soc. Am. A4, 69 (1987), to be published.
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, “Filter Response of Non-Uniform Almost-Periodic Structure,” Bell Syst. Tech. J. 55, 109 (1976).

IECE Jpn.

F. Koyama, Y. Suematsu, “Phase-Controlled Active-Distributed-Reflector Laser,” IECE Jpn. OQE84, 67 (1984), in Japanese.

IEEE J. Quantum Electron.

S. H. Kim, C. G. Fostand, “Tunable Narrow-Band Thin Film Waveguide Grating Filters,” IEEE J. Quantum Electron. QE-15, 1405 (1979).

Y. Itaya, T. Matuoka, K. Kuroiwa, T. Ikegami, “Longitudinal Mode Behaviors of 1.5 μm Range GalnAsP/InP Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-20, 230 (1984).
[CrossRef]

K. Ukata, S. Akiba, K. Sakai, Y. Matsushima, “Effect of Mirror Facets on Lasing Characteristics of Distributed Feedback InGaAsP/InP Laser Diode at 1.5 μm Range,” IEEE J. Quantum Electron. QE-20, 236 (1984).

W. Streifer, R. D. Burham, D. R. Scifres, “Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-11, 154 (1975).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. of Am.

D. Kermisch, “Nonuniform Sinusoidally Modulated Dielectric Gratings,” J. Opt. Soc. of Am. 59, 1409 (1969).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

A. Suzuki, T. Tada, “Theory and Experiment on Distributed Feedback Lasers with Chirped Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 236, 532 (1981).

Other

M. Yamada, K. Sakuda, “Adjustable Gain and Bandwidth Light Amplifiers in Terms of Distributed Feedback Structures,” J. Opt. Soc. Am. A4, 69 (1987), to be published.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Definition of grating phase for the first-order grating.

Fig. 2
Fig. 2

Schematic diagram of a periodic distributed feedback waveguide.

Fig. 3
Fig. 3

Corrugations of finite length L.

Fig. 4
Fig. 4

Schematic diagram of N divisions of an almost-periodic distributed feedback waveguide and parameters in the kth segment: k = 1,2,…, N, i.e., κk is a coupling coefficient, βBk is the Bragg frequency, ϕk is a grating phase, gk is the gain, and E A k and E B k are forward and backward electric field amplitudes, respectively.

Fig. 5
Fig. 5

Some examples of aperiodic distributed feedback waveguides, i.e., (a) tapered grating, (b) chirped grating, and (c) phase-shifted grating.

Fig. 6
Fig. 6

Characteristics of reflection coefficient |R|2 vs normalized frequency ΔβL for three different types of aperiodic distributed feedback waveguide: (a) linear tapered grating, (b) linear chirped grating, and (c) phase-shifted grating.

Fig. 7
Fig. 7

Threshold characteristics for the DFB laser, threshold gain gthL vs normalized frequency ΔβL.

Equations (28)

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dA / dz = κ exp [ i ( 2 Δ β z ϕ ) ] B + gA , dB / dz = κ exp [ i ( 2 Δ β z ϕ ) ] A gB ,
Δ β = β β B = β M π / Λ ,
E A ( z ) = A ( z ) exp ( i β z ) , E B ( z ) = B ( z ) exp ( + i β z ) ,
E A ( z ) = [ c 1 exp ( Γ 1 z ) + c 2 exp ( Γ 2 z ) ] exp [ ( g i β ) z ] , E B ( z ) = { exp [ i ( 2 Δ β z ϕ ) ] / κ } [ c 1 Γ 1 exp ( Γ 1 z ) + c 2 Γ 2 × exp ( Γ 2 z ) ] exp [ ( g i β ) z ] ,
Δ β = Δ β + ig , Γ 1 = i Δ β γ , Γ 2 = i Δ β + γ ,
( E A ( 0 ) E B ( 0 ) ) = [ F ] ( E A ( L ) E B ( L ) ) .
F 11 = [ cosh ( γ L ) + i Δ β L sinh ( γ L ) / ( γ L ) ] exp ( i β B L ) , F 12 = κ L sinh ( γ L ) exp [ i ( β B L + ϕ ) ] / ( γ L ) , F 21 = κ L sinh ( γ L ) exp [ i ( β B L + ϕ ) ] / ( γ L ) , F 22 = [ cosh ( γ L ) i Δ β L sinh ( γ L ) / ( γ L ) ] exp [ i ( β B L ) ] .
| F | = F 11 F 22 F 12 F 21 = 1 .
Δ n 2 ( x , y ) = 2 a 1 ( x ) cos ( 2 π z / Λ ) ,
F [ Δ n 2 ( x , z ) ] = 4 π a 1 ( x ) [ δ ( S 2 π / Λ ) + δ ( S 2 π / Λ ) ] ,
Δ n 2 ( x , z ) = { 2 a 1 ( x ) cos ( 2 π z / Λ ) | z | L / 2 , 0 | z | > L / 2 ,
F [ Δ n 2 ( x , z ) ] = 2 { sin [ ( S + 2 π / Λ ) L / 2 ] / [ S + 2 π / Λ ] + sin [ ( S 2 π / Λ ) L / 2 ] / [ S 2 π / Λ ] } a 1 ( x ) .
Λ k L k ,
[ F k ] = [ F ( κ k , Δ β k , L k , g k , ϕ k ) ] .
[ F ] = Π k = 1 N [ F k ] .
κ ( z ) = κ 0 [ 1 + T a ( z ) ] ,
ϕ k = ϕ k 1 + 2 β B L k 1 .
β B ( z ) = β B O + β Bs ( z ) ,
Λ s ( z ) Λ 0 = β Bs ( z ) β BO ,
ϕ s = 2 β Bs ( ζ ) d ζ .
ϕ k = ϕ k 1 + 2 β B L k 1 .
Δ ϕ k = ϕ k ϕ k 1 2 β B L k 1 .
T = 1 / F 11 ,
R = F 21 / F 11 .
κ ( z ) = κ 0 [ 1 + T ( z L / 2 ) / L ] ,
β B ( z ) = β BO + 2 V ( z L / 2 ) / L 2 ,
[ F R ] = 1 / { ( 1 r 1 ) ( 1 r 2 ) } 1 / 2 ( 1 , ( r 1 ) 1 / 2 ( r 1 ) 1 / 2 , 1 ) [ F ] ( 1 , ( r 2 ) 1 / 2 ( r 1 ) 1 / 2 , 1 ) .
F R 11 = 0 .

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