Abstract

A rigorous analytic Bloch wave formalism is presented to calculate and analyze the photon lifetime of distributed feedback (DFB) lasers. By expressing the electromagnetic fields in the DFB structure as a superposition of two counterpropagating Bloch waves, the DFB can be considered as a Fabry–Pérot cavity for the Bloch waves. Analytic formulas for the laser threshold condition and the photon lifetime of DFB lasers are then derived, which have the same simple forms and physical insight as those for Fabry–Pérot lasers. Numerical results are presented and interpreted along with comparisons with those obtained from the coupled mode theory and the transfer matrix method.

© 2009 Optical Society of America

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References

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  1. W. Zheng and G. W. Taylor, “Determination of the photon lifetime for DFB lasers,” IEEE J. Quantum Electron. 43, 295-302 (2007).
    [CrossRef]
  2. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2355 (1972).
    [CrossRef]
  3. S. Hansmann, H. Walter, H. Hillmer, and H. Burkhard, “Static and dynamic properties of InGaAsP-InP distributed feedback lasers--a detailed comparison between experiment and theory,” IEEE J. Quantum Electron. 30, 2477-2484 (1994).
    [CrossRef]
  4. L. A. Coldren and S. W. Corzine, “Dynamic effects,” in Diode Lasers and Photonic Integrated Circuits (Wiley, 1995), pp.185-261.
  5. P. St. J. Russell, “Optics of Floquet-Bloch waves in dielectric gratings,” Appl. Phys. B: Photophys. Laser Chem. 39, 231-246 (1986).
    [CrossRef]
  6. A. V. Maslov and C. Z. Ning, “Interpretation of distributed-feedback-laser spectrum using reflection properties of Bloch waves,” J. Appl. Phys. 101, 053117 (2007).
    [CrossRef]
  7. F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555-600 (1928).
  8. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1983).
  9. K. Sakoda, “Eigenmodes of photonic crystals,” in Optical Properties of Photonic Crystals (Springer, 2004), pp. 13-41.
  10. M. G. Davis and R. F. O'Dowd, “A transfer matrix method based large-signal dynamic model for multi-electrode DFB lasers.” IEEE J. Quantum Electron. 30, 2458-2466 (1994).
    [CrossRef]

2007 (2)

W. Zheng and G. W. Taylor, “Determination of the photon lifetime for DFB lasers,” IEEE J. Quantum Electron. 43, 295-302 (2007).
[CrossRef]

A. V. Maslov and C. Z. Ning, “Interpretation of distributed-feedback-laser spectrum using reflection properties of Bloch waves,” J. Appl. Phys. 101, 053117 (2007).
[CrossRef]

1994 (2)

M. G. Davis and R. F. O'Dowd, “A transfer matrix method based large-signal dynamic model for multi-electrode DFB lasers.” IEEE J. Quantum Electron. 30, 2458-2466 (1994).
[CrossRef]

S. Hansmann, H. Walter, H. Hillmer, and H. Burkhard, “Static and dynamic properties of InGaAsP-InP distributed feedback lasers--a detailed comparison between experiment and theory,” IEEE J. Quantum Electron. 30, 2477-2484 (1994).
[CrossRef]

1986 (1)

P. St. J. Russell, “Optics of Floquet-Bloch waves in dielectric gratings,” Appl. Phys. B: Photophys. Laser Chem. 39, 231-246 (1986).
[CrossRef]

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2355 (1972).
[CrossRef]

1928 (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555-600 (1928).

Bloch, F.

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555-600 (1928).

Burkhard, H.

S. Hansmann, H. Walter, H. Hillmer, and H. Burkhard, “Static and dynamic properties of InGaAsP-InP distributed feedback lasers--a detailed comparison between experiment and theory,” IEEE J. Quantum Electron. 30, 2477-2484 (1994).
[CrossRef]

Coldren, L. A.

L. A. Coldren and S. W. Corzine, “Dynamic effects,” in Diode Lasers and Photonic Integrated Circuits (Wiley, 1995), pp.185-261.

Corzine, S. W.

L. A. Coldren and S. W. Corzine, “Dynamic effects,” in Diode Lasers and Photonic Integrated Circuits (Wiley, 1995), pp.185-261.

Davis, M. G.

M. G. Davis and R. F. O'Dowd, “A transfer matrix method based large-signal dynamic model for multi-electrode DFB lasers.” IEEE J. Quantum Electron. 30, 2458-2466 (1994).
[CrossRef]

Hansmann, S.

S. Hansmann, H. Walter, H. Hillmer, and H. Burkhard, “Static and dynamic properties of InGaAsP-InP distributed feedback lasers--a detailed comparison between experiment and theory,” IEEE J. Quantum Electron. 30, 2477-2484 (1994).
[CrossRef]

Hillmer, H.

S. Hansmann, H. Walter, H. Hillmer, and H. Burkhard, “Static and dynamic properties of InGaAsP-InP distributed feedback lasers--a detailed comparison between experiment and theory,” IEEE J. Quantum Electron. 30, 2477-2484 (1994).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2355 (1972).
[CrossRef]

Maslov, A. V.

A. V. Maslov and C. Z. Ning, “Interpretation of distributed-feedback-laser spectrum using reflection properties of Bloch waves,” J. Appl. Phys. 101, 053117 (2007).
[CrossRef]

Ning, C. Z.

A. V. Maslov and C. Z. Ning, “Interpretation of distributed-feedback-laser spectrum using reflection properties of Bloch waves,” J. Appl. Phys. 101, 053117 (2007).
[CrossRef]

O'Dowd, R. F.

M. G. Davis and R. F. O'Dowd, “A transfer matrix method based large-signal dynamic model for multi-electrode DFB lasers.” IEEE J. Quantum Electron. 30, 2458-2466 (1994).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell, “Optics of Floquet-Bloch waves in dielectric gratings,” Appl. Phys. B: Photophys. Laser Chem. 39, 231-246 (1986).
[CrossRef]

Sakoda, K.

K. Sakoda, “Eigenmodes of photonic crystals,” in Optical Properties of Photonic Crystals (Springer, 2004), pp. 13-41.

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2355 (1972).
[CrossRef]

Taylor, G. W.

W. Zheng and G. W. Taylor, “Determination of the photon lifetime for DFB lasers,” IEEE J. Quantum Electron. 43, 295-302 (2007).
[CrossRef]

Walter, H.

S. Hansmann, H. Walter, H. Hillmer, and H. Burkhard, “Static and dynamic properties of InGaAsP-InP distributed feedback lasers--a detailed comparison between experiment and theory,” IEEE J. Quantum Electron. 30, 2477-2484 (1994).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1983).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1983).

Zheng, W.

W. Zheng and G. W. Taylor, “Determination of the photon lifetime for DFB lasers,” IEEE J. Quantum Electron. 43, 295-302 (2007).
[CrossRef]

Appl. Phys. B: Photophys. Laser Chem. (1)

P. St. J. Russell, “Optics of Floquet-Bloch waves in dielectric gratings,” Appl. Phys. B: Photophys. Laser Chem. 39, 231-246 (1986).
[CrossRef]

IEEE J. Quantum Electron. (3)

W. Zheng and G. W. Taylor, “Determination of the photon lifetime for DFB lasers,” IEEE J. Quantum Electron. 43, 295-302 (2007).
[CrossRef]

S. Hansmann, H. Walter, H. Hillmer, and H. Burkhard, “Static and dynamic properties of InGaAsP-InP distributed feedback lasers--a detailed comparison between experiment and theory,” IEEE J. Quantum Electron. 30, 2477-2484 (1994).
[CrossRef]

M. G. Davis and R. F. O'Dowd, “A transfer matrix method based large-signal dynamic model for multi-electrode DFB lasers.” IEEE J. Quantum Electron. 30, 2458-2466 (1994).
[CrossRef]

J. Appl. Phys. (2)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2355 (1972).
[CrossRef]

A. V. Maslov and C. Z. Ning, “Interpretation of distributed-feedback-laser spectrum using reflection properties of Bloch waves,” J. Appl. Phys. 101, 053117 (2007).
[CrossRef]

Z. Phys. (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555-600 (1928).

Other (3)

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1983).

K. Sakoda, “Eigenmodes of photonic crystals,” in Optical Properties of Photonic Crystals (Springer, 2004), pp. 13-41.

L. A. Coldren and S. W. Corzine, “Dynamic effects,” in Diode Lasers and Photonic Integrated Circuits (Wiley, 1995), pp.185-261.

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Figures (14)

Fig. 1
Fig. 1

Dispersion relation for light propagation in a periodic structure ( d 1 = d 2 = 120     nm , n 1 = 3.0 , n 2 = 3.6 ). The wave vector becomes a complex number when its frequency is in the stop bands.

Fig. 2
Fig. 2

(a) Reflection of a plane wave at the surface of a semi-infinite periodic structure; (b) reflection of a Bloch wave at the boundary of a periodic structure.

Fig. 3
Fig. 3

(a) Reflectivity as a function of the Bloch wave vector in the semi-infinite periodic structure; (b) reflectivity as a function of the wavelength in the free space.

Fig. 4
Fig. 4

Reflection of a Bloch wave at the boundaries of a finite periodic structure.

Fig. 5
Fig. 5

Resonant Bloch wave vectors of a 12-period structure ( d 1 = d 2 = 120     nm , n 1 = 3.0 , n 2 = 3.6 ) in the first Brillouin zone.

Fig. 6
Fig. 6

Comparison between the group velocity and the energy velocity in the vicinity of the stop band.

Fig. 7
Fig. 7

Transmission spectrum of a DFB structure.

Fig. 8
Fig. 8

(a) Amplitude and (b) phase of the eigenfunction f + ( z ) for different wavelengths on the longer wavelength side of the stop band corresponding to resonant modes of a 1100-period DFB structure with m = 1 , m = 10 , and m = 500 .

Fig. 9
Fig. 9

(a) Amplitude and (b) phase of the eigenfunction f + ( z ) for different wavelengths on the shorter wavelength side of the stop band corresponding to resonant modes of a 1100-period DFB structure with m = 1 , m = 10 , and m = 500 .

Fig. 10
Fig. 10

Field distribution at three different wavelengths in a DFB structure corresponding to points 0, 1, and 2 indicated in Fig. 6.

Fig. 11
Fig. 11

Photon lifetime of a DFB structure calculated with the CWM and TMM and Bloch wave formalism along with a Fabry–Perot cavity for comparison.

Fig. 12
Fig. 12

(a) Resonant wavelength and the corresponding Bloch wave vector (in units of π D ) as a function of the length of the DFB structure; (b) reflectivity of the Bloch wave and the group velocity as a function of the wavelength. The arrows indicate the direction of increasing length of the DFB structure.

Fig. 13
Fig. 13

Photon lifetime of a DFB structure with different internal propagation loss values, with comparison of results from the CWM and TMM and Bloch wave formalism.

Fig. 14
Fig. 14

Schematic drawing of a periodic waveguide structure.

Equations (51)

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n ( z ) = { n 1 , 0 z < d 1 n 2 , d 1 z < D ,
n ( z + D ) = n ( z ) ,
E = E ( z ) e i k z ,
E ( z + D ) = E ( z ) .
cos ( k D ) = cos ( k 1 d 1 ) cos ( k 2 d 2 ) 1 2 ( n 2 n 1 + n 1 n 2 ) sin ( k 1 d 1 ) sin ( k 2 d 2 ) ,
k μ = ω c 0 n μ , μ = 1 , 2 ,
E = a f ( z ) e i k z ,
f ( z + D ) = f ( z ) .
E = a f + ( z ) e i k z + b f ( z ) e i k z ,
{ 1 + r = t f + ( z 0 ) i k 0 ( 1 r ) = t [ f + ( z 0 ) + i k f + ( z 0 ) ] } ,
{ r = i k 0 f + ( z 0 ) f + ( z 0 ) i k f + ( z 0 ) f + ( z 0 ) + i k 0 f + ( z 0 ) + i k f + ( z 0 ) , t = 1 + r f + ( z 0 ) , , }
{ f + ( z 0 ) + r f ( z 0 ) = t , f + ( z 0 ) + i k f + ( z 0 ) + r [ f ( z 0 ) i k f ( z 0 ) ] = i k 0 t . }
{ r = f + ( z 0 ) + i k f + ( z 0 ) i k 0 f + ( z 0 ) f ( z 0 ) + i k f ( z 0 ) + i k 0 f ( z 0 ) , t = f + ( z 0 ) + r f ( z 0 ) . }
r = f + ( 0 ) + i k f + ( 0 ) i k 0 f + ( 0 ) f ( 0 ) + i k f ( 0 ) + i k 0 f ( 0 ) = β 1 + β 1 .
r = r 0 + t 0 r 2 t 1 e i 2 k N D + t 0 r 2 2 r 1 t 1 e i 4 k N D + t 0 r 2 3 r 1 2 t 1 e i 6 k N D + = r 0 + t 0 r 2 t 1 e i 2 k N D 1 r 1 r 2 e i 2 k N D ,
{ r 1 = f ( 0 ) + i k f ( 0 ) i k L f ( 0 ) f + ( 0 ) + i k f + ( 0 ) + i k L f + ( 0 ) = α 1 α 1 + , r 2 = f + ( 0 ) + i k f + ( 0 ) i k R f + ( 0 ) f ( 0 ) + i k f ( 0 ) + i k R f ( 0 ) = β 1 + β 1 , }
1 r 1 r 2 e i 2 k N D = 0.
2 k N D + φ 1 + φ 2 = 2 ( N m ) π ,
ε ( t ) = ε 0 exp ( t τ p ) .
ε 0 exp ( 2 L v e τ p ) = ε 0 | r 1 r 2 | 2 exp ( 2 α L ) ,
v e = S w ,
1 τ FP = v g α + v g L ln ( 1 | r 1 r 2 | ) .
1 τ DFB = v e α eff + v e L ln ( 1 | r 1 r 2 | ) ,
v g = d ω d k .
v g = sin ( k D ) D 1 2 [ a sin ( b ) + c sin ( d ) ] + 1 4 ( n 2 n 1 + n 1 n 2 ) [ a sin ( b ) c sin ( d ) ] ,
{ a = n 1 d 1 c 0 + n 2 d 2 c 0 , b = k 1 d 1 + k 2 d 2 , c = n 1 d 1 c 0 n 2 d 2 c 0 , d = k 1 d 1 k 2 d 2 , }
{ × E = i ω μ H , × H = i ω ε E . }
2 E ( z , t ) z 2 + ω 2 n 2 c 2 E ( z , t ) = 0 ,
E ( z , t ) = { [ A 1 ( m ) e i k 1 ( z m D ) + B 1 ( m ) e i k 1 ( z m D ) ] e i ω t , m D z < m D + d 1 , [ A 2 ( m ) e i k 2 ( z m D d 1 ) + B 2 ( m ) e i k 2 ( z m D d 1 ) ] e i ω t , mD + d 1 z < ( m + 1 ) D , }
( A μ ( m ) B μ ( m ) ) , μ = 1 , 2 .
A 1 ( m ) e i k 1 d 1 + B 1 ( m ) e i k 1 d 1 = A 2 ( m ) + B 2 ( m ) ,
A 1 ( m ) e i k 1 d 1 B 1 ( m ) e i k 1 d 1 = n 2 n 1 ( A 2 ( m ) B 2 ( m ) ) ,
A 2 ( m ) e i k 2 d 2 + B 2 ( m ) e i k 2 d 2 = A 1 ( m + 1 ) + B 1 ( m + 1 ) ,
A 2 ( m ) e i k 2 d 2 B 2 ( m ) e i k 2 d 2 = n 1 n 2 ( A 1 ( m + 1 ) B 1 ( m + 1 ) ) .
( 1 1 1 1 ) ( A 2 ( m ) B 2 ( m ) ) = ( e i k 1 d 1 e i k 1 d 1 1 Z e i k 1 d 1 1 Z e i k 1 d 1 ) ( A 1 ( m ) B 1 ( m ) ) ,
( 1 1 1 1 ) ( A 1 ( m + 1 ) B 1 ( m + 1 ) ) = ( e i k 2 d 2 e i k 2 d 2 Z e i k 2 d 2 Z e i k 2 d 2 ) ( A 2 ( m ) B 2 ( m ) ) ,
( A 1 ( m + 1 ) B 1 ( m + 1 ) ) = T ( A 1 ( m ) B 1 ( m ) ) = ( T 11 T 12 T 21 T 22 ) ( A 1 ( m ) B 1 ( m ) ) ,
T 11 = T 22 * = [ cos ( k 2 d 2 ) + i 2 ( Z + 1 Z ) sin ( k 2 d 2 ) ] e i k 1 d 1 ,
T 12 = T 21 * = i 2 ( Z 1 Z ) sin ( k 2 d 2 ) e i k 1 d 1 ,
Γ = S 1 T S = ( e i k D 0 0 e i k D ) ,
S = ( 1 T 12 e i k D T 11 e i k D T 11 T 12 1 ) .
( A 1 ( m ) B 1 ( m ) ) = S ( a ( m ) b ( m ) ) .
( a ( m ) b ( m ) ) = ( e i k D 0 0 e i k D ) ( a ( m 1 ) b ( m 1 ) ) = Γ ( a ( m 1 ) b ( m 1 ) ) = Γ m ( a ( 0 ) b ( 0 ) ) .
( A 1 ( m ) B 1 ( m ) ) = a ( 0 ) e i m k D ( α 1 + β 1 + ) + b ( 0 ) e i k m D ( α 1 β 1 ) ,
( α 1 + β 1 + ) = ( 1 e i k D T 11 T 12 ) ,
( α 1 β 1 ) = ( T 12 e i k D T 11 1 ) .
( A 2 ( m ) B 2 ( m ) ) = a ( 0 ) e i m k D ( α 2 + β 2 + ) + b ( 0 ) e i k m D ( α 2 β 2 ) ,
α 2 ± = 1 2 ( 1 + 1 Z ) e i k 1 d 1 α 1 ± + 1 2 ( 1 1 Z ) e i k 1 d 1 β 1 ± ,
β 2 ± = 1 2 ( 1 1 Z ) e i k 1 d 1 α 1 ± + 1 2 ( 1 + 1 Z ) e i k 1 d 1 β 1 ± .
f ± ( z ) = { e i k ( z m D ) [ α 1 ± e i k 1 ( z m D ) + β 1 ± e i k 1 ( z m D ) ] , mD z < mD + d 1 , e i k ( z m D ) [ α 2 ± e i k 2 ( z m D d 1 ) + β 2 ± e i k 2 ( z m D d 1 ) ] , mD + d 1 z < ( m + 1 ) D , }
E ( z , t ) = [ a f + ( z ) e i k z + b f ( z ) e i k z ] e i ω t .

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