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Mode separation and selectivity of Bragg-reflector lasers
Volume 23, number 3
OPTICS COMMUNICATIONS
December 1977
MODE SEPARATION AND SELECTIVITY OF BRAGG-REFLECTOR LASERS T. TSUKADA and C.L. TANG * Central Research Laboratory, Hitachi, Ltd., Kokubun/i, Tokyo 185, Japan Received 23 March 1977 Mode separation and selectivity of Bragg-reflector lasers are studied. The characteristic equation including the effect of the external reflectors is derived by using the coupled-mode formalism. From this equation, the dependence of the mode separation and the selectivity on the reflector length is calculated. Criteria for choosing the active region length and the reflector length to achieve low threshold and good mode selectivity are developed.
Distributed feedback (DFB)lasers [1] and Braggreflector lasers [2,3] have received much interest recently. In these devices, feedback is provided by backward Bragg scattering in a periodically perturbed waveguide. In Bragg-reflector lasers, the unperturbed active regions are separated from the corrugated reflector regions resulting in ease o f fabrication [4,5] and flexibility in choosing the parameters o f the reflector regions independently from those o f the pumping region. Mode characteristics of these Bragg-reflector lasers have not been discussed in detail before. In this paper, we report on the longitudinal mode separation and selectivity o f these lasers. The schematic diagram of our analytical model is shown in fig. 1 in which the change o f refractive index is indicated. The corrugated reflector regions with external reflectors can be analyzed following the theoretical analyses o f Kogelnik and Shank [1], and Streifer, Burnham, and Scifres [6]. In the corrugated regions, the periodic variation o f the refractive index n(z) and/or loss a(z) may be expressed as follows:
n(z) = n + nl cos(2/30z + ~2),
(la)
a(z) = ot + OtlCOS(2/30z + g2),
(lb)
where/3 o is the wave number associated with the Bragg wavelength XO and is expressed by 130 = 7r/A =- 27rn/X0. A and ~2 are the spatial period and the phase o f the corrugation, respectively. * School of Electrical Engineering, Cornell University, Ithaca, N.Y. 14853, U.S.A.
~,
Br~gretlaetor I
~---
-(l~" LI)I2
LII2
--
Aet~
-~"
--012
region
.~12-~.--
0
Brogg reflector2 P2
012 - ~
- - I-2/2
#17'
(.gi'L2ll2 Z
Fig. 1. Diagram showing the change of refractive index of Braggreflector lasers. The refractive index of the active region is constant and is not shown in order to show the phase of the grating Denoting the reflection coefficient at z = - ( l +L 1)/2 and z = (l + L 2 ) / 2 as~ 1 and j52, respectively, and neglecting the reflections at z = -+//2 we obtain the following equation which describes the behavior of Braggreflector lasers: exp(-2aa/) exp (2j(,6 a -/30)I)
= [I i=1,2
--jK + Pi {'yc°th(?Li/2) - (o~ + j6)} "ycoth(?Li/2 ) +o~ +jc5 +jKpi
'
(2)
where aaC~0 ) is the gain coefficient in the active medium,/3 a is 27rna/X = naW/C , and n a is the refractive index o f the active region. K is the coupling coefficient given by = 7rnl/X 0 + jotl/2.
(3)
6 is the parameter corresponding to the frequency shift from the Bragg frequency 323
Volume 23, number 3
OPTICS COMMUNICATIONS
: n(co - coo)/C.
18
(4) o~
3' is described by the dispersion relation .),2 = (0~ +jS)2 +K2,
December 1977
~<1 14
(5)
~
I
and Pl = 'bleJS2e-J~°(l+L')' 02 = 'b2e-JS2e-J3°(l+L2)"
0.6
Values o f a a and 8 calculated from (2) give us the threshold and mode characteristics of these lasers. We first consider the simple case of lasers with Bragg reflectors of equal length and without external reflectors. Putting L 1 = L 2 = L and P l = '02 = 0, we obtain the threshold and resonant frequencies as follows:
aal = - l n RB,
0 B + mTr = (21/L)(6L/2) + el,
n)co/c,
(7)
(8)
a n d m is an integer (m = 0, +1, +2 . . . . ). e is the basic parameter which differentiates Bragg-reflector lasers from DFB lasers. In index-coupled DFB lasers, it is known [ 1 - 3 ] that there is no oscillation at the Bragg frequency, co B. However, in Bragg-reflector lasers, the Bragg condition can be satisfied when (see, for example, eq. (10) of ref. [7])
el = (m - 51) n .
(9)
Since the reflectivity at the Bragg frequency has the highest value, the lowest threshold is realized here. When el is equal to rnTr (see, for example, eq. (12) of ref. [7] ), the resonant modes are symmetrically placed with respect to cob without any resonance at co B . This situation is the same as that for index-coupled DFB lasers. When el deviates from these values, the resonant modes become asymmetrical with respect to co B and with respect to the lowest-threshold mode. We calculated the mode separation as a function of the reflector length, assuming the Bragg-mode operation or the oscillation frequency is at the Bragg-frequency. The Bragg condition (9) reduces to 324
, 2
:5
4
(LI2)12 Fig. 2. Mode separation as a function of the reflector length. I, II, and llI correspond to the cases: I) Kl = 0.5, al = 1, II) Kl = 1, cd = 0.5, and III) Kl = 0.5, el = 0.25, respectively. 1 was assumed to be 500 ,*m. The broken lines show the approximation when (L/2)/l is small.
1 = (m -- ~) nA/(n a - n).
and e is a parameter associated with the refractive index difference between the active region and the reflector region defined as: e = (n a -
I
(6a,b)
where R B and 0 B are defined as follows:
RB eJ0B ~ 3,coth(TL/2) + a + j S '
"0
(10)
The adjustment of the active region length to satisfy (10) is possible by knowing beforehand, possibly experimentally, the refractive index difference between the active region and the reflector region. For ( n - n a ) / n ~ 10 -4 and A = 0.12/am, I becomes 600/am. The calculated mode separation between the Bragg mode and the next mode as a function of the reflector length L / 2 is shown in fig. 2. When L / 2 is small, it can be shown from eq. (7) that the mode separation AX can be approximated as follows: AX = X 2/2neffleff ,
(11)
where kdn] neff = n 1 -- n--dX-]' /eft :- l + L/2. This means that the mode separation of the Bragg-reflector laser is approximately the same as that of a Fabry-P6rot laser of length l + L/2 in this limit; or, as far as the mode separation is concerned, the total laser length is extended by half the length of the Bragg-reflectors. As the L / 2 is increased, AX approaches a constant value. Therefore, further increase of L/2 does not change much the mode separation. The threshold in the Bragg mode operation (aa/)0 and the corresponding mode selectivity were also calculated as a function of the reflector length, and are shown in figs. 3(a) and (b). The mode selectivity was evaluated by calculating the ratio r of the t h r e s h o l d (Otal)l of the next mode to that of the Bragg-frequency mode (aal)o. These figures give
Volume 23, number 3
OPTICS COMMUNICATIONS
December 1977 1.8
°~ o A
g
/ .,<
,0 1.4
2
,.oI 0
I
J
J
2
5
\\N
06 o
I
2
3
4
fL/2)/£
( L/2I/L
(a)
Fig. 4. Mode separation of the Bragg reflector laser with one external reflector (cleaved facet of GaAs) and one Braggreflector. The broken line is an approximation for small values of (L/2)/l. The cases 1, II, and III are the same as in fig. 2.
5
,,.9
"=3 I=1
1.5 o u i:1
I
.5 O
0
~
~
I
2
3
4
0
i
0
I
2
_r
3
4
(L/2)/~ (L/2)/2
(b)
(a)
Fig. 3. a) Threshold of Bragg-reflector lasers operating in the Bragg-mode as a function of the reflector length, b) The ratio of the threshold of the mode next to the Bragg mode to that of the Bragg mode as a function of the reflector length. The cases I, II, and III are the same as in fig. 2.
a g o o d i n d i c a t i o n o f h o w t o c h o o s e t h e Bragg-reflector l e n g t h . F o r case II o f fig. 3, for e x a m p l e , L / 2 ~ 2l is t h e o p t i m u m l e n g t h f r o m the s t a n d p o i n t o f t h e m o d e selectivity. As t o t h e case III, t h e ratio s h o w s a r a t h e r comp l e x b e h a v i o r . H o w e v e r , it will be a good choice also to use t h e r e f l e c t o r l e n g t h L/2 ~ 2l. F r o m the standp o i n t o f t h e low t h r e s h o l d , it is also a g o o d choice as seen f r o m fig. 3(a). A b s o r p t i o n in t h e r e f l e c t o r regions greatly degrades t h e m o d e selectivity. W h e n al = 5 ( a = 100 cm -1 for l = 5 0 0 / a m ) a n d tel = 1, for e x a m p l e , we o b t a i n a m a x i m u m r o f 1.05 w i t h 'C~al ~ 2.3. This m e a n s a p o o r m o d e selectivity. N e x t we discuss t h e case o f a G a A s laser w i t h one
o
g
g v
0
t
2
3
4
(L/2)/1
(b) Fig. 5. a) Threshold of the Bragg-mode as a function of the reflector length. The Bragg-reflector has one external reflector (cleaved facet of GaAs) and one Bragg reflector, b) The threshold ratio r of the same laser. I, 1I and III correspond to the same cases as in fig. 2. 325
Volume 23, number 3
OPTICS COMMUNICATIONS
cleaved facet and one Bragg-reflector without any external reflector. Putting 01 -- 0, L 2 = 0, and/)2 = 0.565 e - j a e -j&l, we obtain from (2) O~al= - 0 . 5 In 0.565 - 0.5 In RBI,
(12a)
0131/2 + rnTr = 61 + (-3,
(12b)
where ® = el + (~0/2)1 + rr/2 + g2/2, and RB1 and 0B1 are obtained from eq. ( 7 ) b y replacingL/2 by L 1 / 2 . The procedure to find the resonant frequencies and the corresponding thresholds is basically the same as before. However, tile difference lies in that O is a completely undetermined value. A change o f l by a fraction of one micron will cause a phase shift o f 2rr in ®, which will change the lasing characteristics o f Bragg reflector lasers. The wavelength separation is shown in fig. 4 for the case of Bragg operation. When the reflector length is small, lef t. can be approximated as leff = l + L / 4 .
(13)
Therefore, again the length of the laser is extended by half the reflector length. This is generally true as long as the reflector length is small compared with the active region. The mode selectivity is shown in fig. 5. Because the exlernal reflector is not spectrally selec-
326
December 1977
rive, the threshold ratio is smaller than that of the twoBragg-reflector case. In conclusion, the mode characteristics of Bragg reflector lasers have been studied. The results indicate that low threshold and the high mode selectivity can be achieved by a proper choice o f the active region length and the Bragg reflector length.
References [ 1] H. Kogelnik and C.V. Shank, J. Appl. Phys. 43 (1972) 2327. [21 C.L. Tang, Laser source considerations in integrated optics, Short Course in Integrated Optics, U.C. Santa Barbara, California (1973); and in: Introduction to integrated opiics, ed. M.K. Barnoski (Plenum Press, New York, 1974) p. 490. [31 S. Wang, IEEE J. Quantum Electron. QE-10 (1974) 413. [4] F.K. Reinhart, R.A. Logan and C.V. Shank, Appl. Phys. Lett. 27 (1975) 45. [5 ] W.-T. Tsang and S. Wang, Appl. Phys. Lett. 28 (1976) 596. [6] W. Streifer, R.D. Burnham and D.R. Scifres, IEEE J. Quantum Electron QE-11 (1975) 154. [7 ] T. Tsukada and C.L. Tang, IEEE J. Quantum Electron., to be published.
OPTICS COMMUNICATIONS
December 1977
MODE SEPARATION AND SELECTIVITY OF BRAGG-REFLECTOR LASERS T. TSUKADA and C.L. TANG * Central Research Laboratory, Hitachi, Ltd., Kokubun/i, Tokyo 185, Japan Received 23 March 1977 Mode separation and selectivity of Bragg-reflector lasers are studied. The characteristic equation including the effect of the external reflectors is derived by using the coupled-mode formalism. From this equation, the dependence of the mode separation and the selectivity on the reflector length is calculated. Criteria for choosing the active region length and the reflector length to achieve low threshold and good mode selectivity are developed.
Distributed feedback (DFB)lasers [1] and Braggreflector lasers [2,3] have received much interest recently. In these devices, feedback is provided by backward Bragg scattering in a periodically perturbed waveguide. In Bragg-reflector lasers, the unperturbed active regions are separated from the corrugated reflector regions resulting in ease o f fabrication [4,5] and flexibility in choosing the parameters o f the reflector regions independently from those o f the pumping region. Mode characteristics of these Bragg-reflector lasers have not been discussed in detail before. In this paper, we report on the longitudinal mode separation and selectivity o f these lasers. The schematic diagram of our analytical model is shown in fig. 1 in which the change o f refractive index is indicated. The corrugated reflector regions with external reflectors can be analyzed following the theoretical analyses o f Kogelnik and Shank [1], and Streifer, Burnham, and Scifres [6]. In the corrugated regions, the periodic variation o f the refractive index n(z) and/or loss a(z) may be expressed as follows:
n(z) = n + nl cos(2/30z + ~2),
(la)
a(z) = ot + OtlCOS(2/30z + g2),
(lb)
where/3 o is the wave number associated with the Bragg wavelength XO and is expressed by 130 = 7r/A =- 27rn/X0. A and ~2 are the spatial period and the phase o f the corrugation, respectively. * School of Electrical Engineering, Cornell University, Ithaca, N.Y. 14853, U.S.A.
~,
Br~gretlaetor I
~---
-(l~" LI)I2
LII2
--
Aet~
-~"
--012
region
.~12-~.--
0
Brogg reflector2 P2
012 - ~
- - I-2/2
#17'
(.gi'L2ll2 Z
Fig. 1. Diagram showing the change of refractive index of Braggreflector lasers. The refractive index of the active region is constant and is not shown in order to show the phase of the grating Denoting the reflection coefficient at z = - ( l +L 1)/2 and z = (l + L 2 ) / 2 as~ 1 and j52, respectively, and neglecting the reflections at z = -+//2 we obtain the following equation which describes the behavior of Braggreflector lasers: exp(-2aa/) exp (2j(,6 a -/30)I)
= [I i=1,2
--jK + Pi {'yc°th(?Li/2) - (o~ + j6)} "ycoth(?Li/2 ) +o~ +jc5 +jKpi
'
(2)
where aaC~0 ) is the gain coefficient in the active medium,/3 a is 27rna/X = naW/C , and n a is the refractive index o f the active region. K is the coupling coefficient given by = 7rnl/X 0 + jotl/2.
(3)
6 is the parameter corresponding to the frequency shift from the Bragg frequency 323
Volume 23, number 3
OPTICS COMMUNICATIONS
: n(co - coo)/C.
18
(4) o~
3' is described by the dispersion relation .),2 = (0~ +jS)2 +K2,
December 1977
~<1 14
(5)
~
I
and Pl = 'bleJS2e-J~°(l+L')' 02 = 'b2e-JS2e-J3°(l+L2)"
0.6
Values o f a a and 8 calculated from (2) give us the threshold and mode characteristics of these lasers. We first consider the simple case of lasers with Bragg reflectors of equal length and without external reflectors. Putting L 1 = L 2 = L and P l = '02 = 0, we obtain the threshold and resonant frequencies as follows:
aal = - l n RB,
0 B + mTr = (21/L)(6L/2) + el,
n)co/c,
(7)
(8)
a n d m is an integer (m = 0, +1, +2 . . . . ). e is the basic parameter which differentiates Bragg-reflector lasers from DFB lasers. In index-coupled DFB lasers, it is known [ 1 - 3 ] that there is no oscillation at the Bragg frequency, co B. However, in Bragg-reflector lasers, the Bragg condition can be satisfied when (see, for example, eq. (10) of ref. [7])
el = (m - 51) n .
(9)
Since the reflectivity at the Bragg frequency has the highest value, the lowest threshold is realized here. When el is equal to rnTr (see, for example, eq. (12) of ref. [7] ), the resonant modes are symmetrically placed with respect to cob without any resonance at co B . This situation is the same as that for index-coupled DFB lasers. When el deviates from these values, the resonant modes become asymmetrical with respect to co B and with respect to the lowest-threshold mode. We calculated the mode separation as a function of the reflector length, assuming the Bragg-mode operation or the oscillation frequency is at the Bragg-frequency. The Bragg condition (9) reduces to 324
, 2
:5
4
(LI2)12 Fig. 2. Mode separation as a function of the reflector length. I, II, and llI correspond to the cases: I) Kl = 0.5, al = 1, II) Kl = 1, cd = 0.5, and III) Kl = 0.5, el = 0.25, respectively. 1 was assumed to be 500 ,*m. The broken lines show the approximation when (L/2)/l is small.
1 = (m -- ~) nA/(n a - n).
and e is a parameter associated with the refractive index difference between the active region and the reflector region defined as: e = (n a -
I
(6a,b)
where R B and 0 B are defined as follows:
RB eJ0B ~ 3,coth(TL/2) + a + j S '
"0
(10)
The adjustment of the active region length to satisfy (10) is possible by knowing beforehand, possibly experimentally, the refractive index difference between the active region and the reflector region. For ( n - n a ) / n ~ 10 -4 and A = 0.12/am, I becomes 600/am. The calculated mode separation between the Bragg mode and the next mode as a function of the reflector length L / 2 is shown in fig. 2. When L / 2 is small, it can be shown from eq. (7) that the mode separation AX can be approximated as follows: AX = X 2/2neffleff ,
(11)
where kdn] neff = n 1 -- n--dX-]' /eft :- l + L/2. This means that the mode separation of the Bragg-reflector laser is approximately the same as that of a Fabry-P6rot laser of length l + L/2 in this limit; or, as far as the mode separation is concerned, the total laser length is extended by half the length of the Bragg-reflectors. As the L / 2 is increased, AX approaches a constant value. Therefore, further increase of L/2 does not change much the mode separation. The threshold in the Bragg mode operation (aa/)0 and the corresponding mode selectivity were also calculated as a function of the reflector length, and are shown in figs. 3(a) and (b). The mode selectivity was evaluated by calculating the ratio r of the t h r e s h o l d (Otal)l of the next mode to that of the Bragg-frequency mode (aal)o. These figures give
Volume 23, number 3
OPTICS COMMUNICATIONS
December 1977 1.8
°~ o A
g
/ .,<
,0 1.4
2
,.oI 0
I
J
J
2
5
\\N
06 o
I
2
3
4
fL/2)/£
( L/2I/L
(a)
Fig. 4. Mode separation of the Bragg reflector laser with one external reflector (cleaved facet of GaAs) and one Braggreflector. The broken line is an approximation for small values of (L/2)/l. The cases 1, II, and III are the same as in fig. 2.
5
,,.9
"=3 I=1
1.5 o u i:1
I
.5 O
0
~
~
I
2
3
4
0
i
0
I
2
_r
3
4
(L/2)/~ (L/2)/2
(b)
(a)
Fig. 3. a) Threshold of Bragg-reflector lasers operating in the Bragg-mode as a function of the reflector length, b) The ratio of the threshold of the mode next to the Bragg mode to that of the Bragg mode as a function of the reflector length. The cases I, II, and III are the same as in fig. 2.
a g o o d i n d i c a t i o n o f h o w t o c h o o s e t h e Bragg-reflector l e n g t h . F o r case II o f fig. 3, for e x a m p l e , L / 2 ~ 2l is t h e o p t i m u m l e n g t h f r o m the s t a n d p o i n t o f t h e m o d e selectivity. As t o t h e case III, t h e ratio s h o w s a r a t h e r comp l e x b e h a v i o r . H o w e v e r , it will be a good choice also to use t h e r e f l e c t o r l e n g t h L/2 ~ 2l. F r o m the standp o i n t o f t h e low t h r e s h o l d , it is also a g o o d choice as seen f r o m fig. 3(a). A b s o r p t i o n in t h e r e f l e c t o r regions greatly degrades t h e m o d e selectivity. W h e n al = 5 ( a = 100 cm -1 for l = 5 0 0 / a m ) a n d tel = 1, for e x a m p l e , we o b t a i n a m a x i m u m r o f 1.05 w i t h 'C~al ~ 2.3. This m e a n s a p o o r m o d e selectivity. N e x t we discuss t h e case o f a G a A s laser w i t h one
o
g
g v
0
t
2
3
4
(L/2)/1
(b) Fig. 5. a) Threshold of the Bragg-mode as a function of the reflector length. The Bragg-reflector has one external reflector (cleaved facet of GaAs) and one Bragg reflector, b) The threshold ratio r of the same laser. I, 1I and III correspond to the same cases as in fig. 2. 325
Volume 23, number 3
OPTICS COMMUNICATIONS
cleaved facet and one Bragg-reflector without any external reflector. Putting 01 -- 0, L 2 = 0, and/)2 = 0.565 e - j a e -j&l, we obtain from (2) O~al= - 0 . 5 In 0.565 - 0.5 In RBI,
(12a)
0131/2 + rnTr = 61 + (-3,
(12b)
where ® = el + (~0/2)1 + rr/2 + g2/2, and RB1 and 0B1 are obtained from eq. ( 7 ) b y replacingL/2 by L 1 / 2 . The procedure to find the resonant frequencies and the corresponding thresholds is basically the same as before. However, tile difference lies in that O is a completely undetermined value. A change o f l by a fraction of one micron will cause a phase shift o f 2rr in ®, which will change the lasing characteristics o f Bragg reflector lasers. The wavelength separation is shown in fig. 4 for the case of Bragg operation. When the reflector length is small, lef t. can be approximated as leff = l + L / 4 .
(13)
Therefore, again the length of the laser is extended by half the reflector length. This is generally true as long as the reflector length is small compared with the active region. The mode selectivity is shown in fig. 5. Because the exlernal reflector is not spectrally selec-
326
December 1977
rive, the threshold ratio is smaller than that of the twoBragg-reflector case. In conclusion, the mode characteristics of Bragg reflector lasers have been studied. The results indicate that low threshold and the high mode selectivity can be achieved by a proper choice o f the active region length and the Bragg reflector length.
References [ 1] H. Kogelnik and C.V. Shank, J. Appl. Phys. 43 (1972) 2327. [21 C.L. Tang, Laser source considerations in integrated optics, Short Course in Integrated Optics, U.C. Santa Barbara, California (1973); and in: Introduction to integrated opiics, ed. M.K. Barnoski (Plenum Press, New York, 1974) p. 490. [31 S. Wang, IEEE J. Quantum Electron. QE-10 (1974) 413. [4] F.K. Reinhart, R.A. Logan and C.V. Shank, Appl. Phys. Lett. 27 (1975) 45. [5 ] W.-T. Tsang and S. Wang, Appl. Phys. Lett. 28 (1976) 596. [6] W. Streifer, R.D. Burnham and D.R. Scifres, IEEE J. Quantum Electron QE-11 (1975) 154. [7 ] T. Tsukada and C.L. Tang, IEEE J. Quantum Electron., to be published.