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Circular grating DFB and DBR semiconductor lasers: threshold current analysis
OPTICS COMMUNICATIONS
Optics Communications 90 ( i 992) 297-300 North-Holland
Circular grating DFB and DBR semiconductor lasers: threshold current analysis Toshihiko M a k i n o a n d C h u n m e n g W u Bell-Northern Research Ltd., P.O. Box 3511, Station C, Ottawa, Ontario, KI Y 4H 7 Canada
Received 15 October 1991; revised manuscript received 17 January 1992
The threshold current is analyzedfor distributed feedback (DFB) and distributed Braggreflector (DBR) semiconductorlasers with circular gratings. It is shownthat in circular grating DFB lasers, the threshold current becomesminimum at a certain cavity radius, while in circular grating DBR lasers it increases monotonicallyas the active region radius increases.
1. Introduction In recent years, there have been several works on circular grating resonators and DFB lasers [ 1-7 ]. The circular grating DFB laser is expected to have potential advantages as a surface-emitting light source, such as a circular-symmetric, low-divergence output beam and the ability to phase-lock a two-dimensional array [ 1,3,4 ]. Very recently, the authors and their colleagues have demonstrated the first lasing action in surface-emitting circular grating DFB lasers by optical pumping [ 7 ]. For many applications, electrical pumping of semiconductor lasers is preferred over optical pumping, since the former retains the major advantages of semiconductor laser technology, namely, compactness, ruggedness, and high electrical-to-optical power conversion efficiency. In order to realize circular grating DFB lasers pumped directly by electrical injection, it is important to assess the threshold current (density). However, although a threshold-gain analysis has been reported [ 4 ], the threshold current has not yet been analyzed. It is well known that there are two basic types of conventional semiconductor lasers with gratings: DFB lasers in which the grating covers the entire active (or pumped) region of the laser, and DBR laser in which the grating is located in the passive (unpumped) region of the laser [ 8 ]. These two types can be realized with circular grating laser structures
as well. It is expected that these two types of circular grating lasers will show different structural dependences of the threshold current. In this communication, we present the first analysis of the threshold current for circular grating DFB and DBR lasers.
2. Model A schematic diagram of the circular grating laser is shown in fig. 1. The structure consists of a cladding layer on the bottom, an active layer (the hatched region in fig. 1 ), and a grating layer on the top. The grating provides a laser feedback, and also emits a radiation power vertically from the surface if the grating is second-order for Bragg reflection. In this communication, we assume a first-order grating, and hence ignore the radiation field because the theory has not been developed enough to predict correctly the normal radiation component in cylindrical waves. The analysis of the case of the first-order grating can give an estimate for the minimum bound for the threshold gain (and hence the threshold current) because the normal radiation always increases the threshold gain [ 9,10 ]. In the DFB configuration, the whole active region is pumped, and the region without a grating (O~r<~Ra in fig. 1 ) is chosen small, and used as a phase-adjust region. In the DBR configuration, only the region 0 ~
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved.
297
Volume 90, number 4,5,6
OPTICS COMMUNICATIONS
15 June 1992
Denoting the field reflection coefficients at r = 0 and r=Ra by Fo and F~, respectively, we can derive from eq. ( 1 ) the following threshold condition, similarly to one-dimensional DFB lasers [ 11 ],
FoF8 exP[½~(g-Ot,os~)2Ra] e x p ( - i 2 f l R ~ ) = 1,
(2)
with
Ijj J
O
Ro
r
h. V
Fig. 1. Cutaway view of the geometry for the circular grating DFB or DBR semiconductor laser. Fo and F s denote the field reflection coefficients looking inward at r = 0 , and outward at r=R., respectively.
grating region R~ ~ r ~ R is unpumped, where R is the total cavity radius. The coupled-wave equations have been used in analyzing conventional one-dimensional (linear) DFB and DBR lasers [8, l l ]. For the analysis of circular grating DFB and DBR lasers, a different type of coupled wave equations are required because outward- and inward-propagating cylindrical waves are generated [ 2,4 ]. Recently, the authors and their colleagues have developed a self-consistent coupledwave theory for circular gratings, and derived the most general coupled-wave equations [ 5 ]. It can be shown that the coupled-wave equations for the lowest order TE cylindrical waves are, to a very good approximation, expressed as [ 12 ]
da+/dr= ( o q - i O ) a + +xexp(i£2) a - , - d a - / d r = (c~g- it~)a - - x e x p ( - i £ 2 ) a + ,
( 1)
where a + and a - are the amplitudes of the outwardand inward-propagating cylindrical waves, respectively, ot8 is the net field gain for R~
Fg= - x e x p ( - i I 2 ) t a n h [ y ( R - R a ) ] , ~ - (ag -ic~) tanh[~,(R-Ra) ]
(3)
72=1¢2+ (a s --i~) 2 ,
(4)
where ~, g, ot~.... and fl are the optical confinement factor, the power gain, the loss coefficient, and the propagation constant of the waveguide in the region without corrugations, respectively The center (r = 0) can be regarded as a perfect reflector, so we have Fo= 1 [4]. In the DFB configuration, R~ is chosen to be short so that exp[ ½~(g-ot~oss)2R] can be approximated by 1, and exp(-i2flRa) is adjusted to give a proper phase-shift. In the DBR configuration, F~ is a passive reflection coefficient in which the gain is zero. The threshold current density can be related to the threshold gain as follows. For the sake of simplicity, we assume that the lasing wavelength is equal to the gain-peak wavelength. This can be realized by choosing a proper phase-shift so as to give a lasing at the Bragg wavelength [ 11 ], which can be set equal to the gain-peak wavelength by using a proper grating period. Then, the threshold current density Jth is given by [13]
Jth = eBeff[g,h/((~'1o) + no ] 2da/r/i ,
(5)
where e is the electron charge, Bar is the effective recombination constant, gth is the threshold power gain, Ao and no are the parameters expressing the peak gain gp as a function of carrier density n by gp +Ao ( n - n o ) , d, is the active layer thickness, and ~]i is the internal quantum efficiency. The threshold current Ith is given by Ith = nR 2Jth, for the DFB laser,
=TrR2Jth, forthe DBRlaser.
(6)
Volume 90, number 4,5,6
OPTICS C O M M U N I C A T I O N S
3. Results
40
We take, as numerical examples, 1.3 ttm InGaAsP-InP circular grating DFB and DBR lasers. The following parameters were used in the calculation: A o = 2 . 5 × 1 0 -~6 cm 2, n o = l . 5 × 1 0 ~s cm -3, Bar= 1.0X 10 -1° cm 3 s - I [ 10], ~=0.5, da=0.2 ~tm, and ?~i" ~ " I. The grating phase D was assumed to be 0 for simplicity. In fig. 2, the threshold current density Jth and the threshold current Ith for the circular grating DFB laser are plotted by the solid and dashed curves, respectively, as a function of the cavity radius R with the coupling coefficient x as a parameter. 2#R, was chosen to be ~ in order to have Bragg wavelength oscillation. As R increases, Ith decreases first, and then increases. This can be explained as follows: for small R, the decrease of Ith caused by the decrease of Jth due to the decrease of gth is faster than the increase of lth due to the R 2 factor in eq. (6). For larger R, the increase of Ith due to the R 2 factor becomes dominant, and therefore Ith increases as R increases. Figure 3 shows similar plots for the circular grating DBR laser as a function of the active region radius Ra with the normalized coupling coefficient x ( R - Ra) of the DBR mirror as a parameter. In the DBR laser, Jth
90 80
,
r
,
,
,
,
1100
,
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70
I000
\\
" 900 .............
~" 'E,, 50
\~\
~ .
~
~Y~
- 800
~
/( = 50 cm "1 _4 75cm-1
/
L\
40
-700
lOOcm-1
"'.><;--~_~
12s cm-'
<
..
......
-
30
600 E
I
,
,
200
'
//
30
//
150
/
'E
K[R- R,)
.~
: 1.0,.
20
=_
\
//
/
//2.o\"~
0
0
--t10
i//// //
t 20
E
.-
/
/ y...-
~o
oo<
..
v
/.,%...
t 30 Ra (wm)
/
t 40
5o
0
50
Fig. 3. Threshold current density J,h (solid curves) and threshold current lth (dashed curves) for the circular grating D B R laser as a function o f the active region radius R, with normalized coupling coefficient x(R-R,) as a parameter.
decreases and Ith increases monotonically as Ra increases. In this case, the R~ factor in eq. (6) is always dominant even though Jth decreases due to the decrease of gth as Ra increases.
4. Concluding remarks We have presented the results of threshold current analysis for circular grating DFB and DBR semiconductor lasers. The threshold current of the circular grating DFB laser has been shown to become minimum at a certain cavity radius because of the radius dependence of threshold gain and active area. In the case of circular DBR lasers, the threshold current increases monotonically as the active region radius increases.
500
20
~
10 020
15 June 1992
400 300
i
30
,
40
i
50
i
60
,
70
, 80
0 9_
200
100
R (;Jm) Fig. 2. Threshold current density Jth ( solid curves) and threshold current lm (dashed curves) for the circular grating DFB laser as a function of the cavity radius R with coupling coefficient ~: as a parameter.
Acknowledgments The authors wish to thank J. Glinski for useful discussions.
299
Volume 90, number 4,5,6
OPTICS COMMUNICATIONS
References [ 1 ] R.M. Shimpe, U.S. patent 4743083 (1988). [2] X. Zheng, Electron. Lett. 25 (1989) 1311. [3] M. Toda, IEEE J. Quantum Electron. QE-26 (1990) 473. [4] T. Erdogan and D.G. Hall, J. Appl. Phys. 68 (1990) 1435. [ 5 ] C. Wu, T. Makino, J. Glinski, R. Maciejko and S.L. Najafi, J. Lightwave Technol. 9 ( 1991 ) 1264. [ 6 ] M. Fallahi, C. Wu, C. Maritan, K. Fox and C. Blaauw, LEOS 1991 Summer topical meetings: Microfabrication for photonics and optoelectronics (Newport Beach, California, 1991).
300
15 June 1992
[7 ] C. Wu, M. Svillans, M. Failahi, T. Makino, J. Glinski, C. Maritan and C. Blaauw, Electron. Lett. 27 ( 1991 ) 1819. [ 8 ] S. Wang, IEEE, J. Quantum Electron. QE- l 0 (1974) 413. [ 9 ] R.E Kazarinov and C.H. Henry, IEEE J. Quantum Electron. QE-21 (1985) 144. [10] T. Makino and J. Glinski, J. Quantum Electron. QE-24 (1988) 73. [11] N. Eda, K. Furuya, F. Koyama and Y. Suematsu, J. Lightwave Technol. LT-3 (1985) 400. [ 12 ] C. Wu, to be submitted. [ 13 ] K. Iga and S. Uchiyama, Opt. Quantum Electron. 18 (1986) 403.
Optics Communications 90 ( i 992) 297-300 North-Holland
Circular grating DFB and DBR semiconductor lasers: threshold current analysis Toshihiko M a k i n o a n d C h u n m e n g W u Bell-Northern Research Ltd., P.O. Box 3511, Station C, Ottawa, Ontario, KI Y 4H 7 Canada
Received 15 October 1991; revised manuscript received 17 January 1992
The threshold current is analyzedfor distributed feedback (DFB) and distributed Braggreflector (DBR) semiconductorlasers with circular gratings. It is shownthat in circular grating DFB lasers, the threshold current becomesminimum at a certain cavity radius, while in circular grating DBR lasers it increases monotonicallyas the active region radius increases.
1. Introduction In recent years, there have been several works on circular grating resonators and DFB lasers [ 1-7 ]. The circular grating DFB laser is expected to have potential advantages as a surface-emitting light source, such as a circular-symmetric, low-divergence output beam and the ability to phase-lock a two-dimensional array [ 1,3,4 ]. Very recently, the authors and their colleagues have demonstrated the first lasing action in surface-emitting circular grating DFB lasers by optical pumping [ 7 ]. For many applications, electrical pumping of semiconductor lasers is preferred over optical pumping, since the former retains the major advantages of semiconductor laser technology, namely, compactness, ruggedness, and high electrical-to-optical power conversion efficiency. In order to realize circular grating DFB lasers pumped directly by electrical injection, it is important to assess the threshold current (density). However, although a threshold-gain analysis has been reported [ 4 ], the threshold current has not yet been analyzed. It is well known that there are two basic types of conventional semiconductor lasers with gratings: DFB lasers in which the grating covers the entire active (or pumped) region of the laser, and DBR laser in which the grating is located in the passive (unpumped) region of the laser [ 8 ]. These two types can be realized with circular grating laser structures
as well. It is expected that these two types of circular grating lasers will show different structural dependences of the threshold current. In this communication, we present the first analysis of the threshold current for circular grating DFB and DBR lasers.
2. Model A schematic diagram of the circular grating laser is shown in fig. 1. The structure consists of a cladding layer on the bottom, an active layer (the hatched region in fig. 1 ), and a grating layer on the top. The grating provides a laser feedback, and also emits a radiation power vertically from the surface if the grating is second-order for Bragg reflection. In this communication, we assume a first-order grating, and hence ignore the radiation field because the theory has not been developed enough to predict correctly the normal radiation component in cylindrical waves. The analysis of the case of the first-order grating can give an estimate for the minimum bound for the threshold gain (and hence the threshold current) because the normal radiation always increases the threshold gain [ 9,10 ]. In the DFB configuration, the whole active region is pumped, and the region without a grating (O~r<~Ra in fig. 1 ) is chosen small, and used as a phase-adjust region. In the DBR configuration, only the region 0 ~
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved.
297
Volume 90, number 4,5,6
OPTICS COMMUNICATIONS
15 June 1992
Denoting the field reflection coefficients at r = 0 and r=Ra by Fo and F~, respectively, we can derive from eq. ( 1 ) the following threshold condition, similarly to one-dimensional DFB lasers [ 11 ],
FoF8 exP[½~(g-Ot,os~)2Ra] e x p ( - i 2 f l R ~ ) = 1,
(2)
with
Ijj J
O
Ro
r
h. V
Fig. 1. Cutaway view of the geometry for the circular grating DFB or DBR semiconductor laser. Fo and F s denote the field reflection coefficients looking inward at r = 0 , and outward at r=R., respectively.
grating region R~ ~ r ~ R is unpumped, where R is the total cavity radius. The coupled-wave equations have been used in analyzing conventional one-dimensional (linear) DFB and DBR lasers [8, l l ]. For the analysis of circular grating DFB and DBR lasers, a different type of coupled wave equations are required because outward- and inward-propagating cylindrical waves are generated [ 2,4 ]. Recently, the authors and their colleagues have developed a self-consistent coupledwave theory for circular gratings, and derived the most general coupled-wave equations [ 5 ]. It can be shown that the coupled-wave equations for the lowest order TE cylindrical waves are, to a very good approximation, expressed as [ 12 ]
da+/dr= ( o q - i O ) a + +xexp(i£2) a - , - d a - / d r = (c~g- it~)a - - x e x p ( - i £ 2 ) a + ,
( 1)
where a + and a - are the amplitudes of the outwardand inward-propagating cylindrical waves, respectively, ot8 is the net field gain for R~
Fg= - x e x p ( - i I 2 ) t a n h [ y ( R - R a ) ] , ~ - (ag -ic~) tanh[~,(R-Ra) ]
(3)
72=1¢2+ (a s --i~) 2 ,
(4)
where ~, g, ot~.... and fl are the optical confinement factor, the power gain, the loss coefficient, and the propagation constant of the waveguide in the region without corrugations, respectively The center (r = 0) can be regarded as a perfect reflector, so we have Fo= 1 [4]. In the DFB configuration, R~ is chosen to be short so that exp[ ½~(g-ot~oss)2R] can be approximated by 1, and exp(-i2flRa) is adjusted to give a proper phase-shift. In the DBR configuration, F~ is a passive reflection coefficient in which the gain is zero. The threshold current density can be related to the threshold gain as follows. For the sake of simplicity, we assume that the lasing wavelength is equal to the gain-peak wavelength. This can be realized by choosing a proper phase-shift so as to give a lasing at the Bragg wavelength [ 11 ], which can be set equal to the gain-peak wavelength by using a proper grating period. Then, the threshold current density Jth is given by [13]
Jth = eBeff[g,h/((~'1o) + no ] 2da/r/i ,
(5)
where e is the electron charge, Bar is the effective recombination constant, gth is the threshold power gain, Ao and no are the parameters expressing the peak gain gp as a function of carrier density n by gp +Ao ( n - n o ) , d, is the active layer thickness, and ~]i is the internal quantum efficiency. The threshold current Ith is given by Ith = nR 2Jth, for the DFB laser,
=TrR2Jth, forthe DBRlaser.
(6)
Volume 90, number 4,5,6
OPTICS C O M M U N I C A T I O N S
3. Results
40
We take, as numerical examples, 1.3 ttm InGaAsP-InP circular grating DFB and DBR lasers. The following parameters were used in the calculation: A o = 2 . 5 × 1 0 -~6 cm 2, n o = l . 5 × 1 0 ~s cm -3, Bar= 1.0X 10 -1° cm 3 s - I [ 10], ~=0.5, da=0.2 ~tm, and ?~i" ~ " I. The grating phase D was assumed to be 0 for simplicity. In fig. 2, the threshold current density Jth and the threshold current Ith for the circular grating DFB laser are plotted by the solid and dashed curves, respectively, as a function of the cavity radius R with the coupling coefficient x as a parameter. 2#R, was chosen to be ~ in order to have Bragg wavelength oscillation. As R increases, Ith decreases first, and then increases. This can be explained as follows: for small R, the decrease of Ith caused by the decrease of Jth due to the decrease of gth is faster than the increase of lth due to the R 2 factor in eq. (6). For larger R, the increase of Ith due to the R 2 factor becomes dominant, and therefore Ith increases as R increases. Figure 3 shows similar plots for the circular grating DBR laser as a function of the active region radius Ra with the normalized coupling coefficient x ( R - Ra) of the DBR mirror as a parameter. In the DBR laser, Jth
90 80
,
r
,
,
,
,
1100
,
"\ \
70
I000
\\
" 900 .............
~" 'E,, 50
\~\
~ .
~
~Y~
- 800
~
/( = 50 cm "1 _4 75cm-1
/
L\
40
-700
lOOcm-1
"'.><;--~_~
12s cm-'
<
..
......
-
30
600 E
I
,
,
200
'
//
30
//
150
/
'E
K[R- R,)
.~
: 1.0,.
20
=_
\
//
/
//2.o\"~
0
0
--t10
i//// //
t 20
E
.-
/
/ y...-
~o
oo<
..
v
/.,%...
t 30 Ra (wm)
/
t 40
5o
0
50
Fig. 3. Threshold current density J,h (solid curves) and threshold current lth (dashed curves) for the circular grating D B R laser as a function o f the active region radius R, with normalized coupling coefficient x(R-R,) as a parameter.
decreases and Ith increases monotonically as Ra increases. In this case, the R~ factor in eq. (6) is always dominant even though Jth decreases due to the decrease of gth as Ra increases.
4. Concluding remarks We have presented the results of threshold current analysis for circular grating DFB and DBR semiconductor lasers. The threshold current of the circular grating DFB laser has been shown to become minimum at a certain cavity radius because of the radius dependence of threshold gain and active area. In the case of circular DBR lasers, the threshold current increases monotonically as the active region radius increases.
500
20
~
10 020
15 June 1992
400 300
i
30
,
40
i
50
i
60
,
70
, 80
0 9_
200
100
R (;Jm) Fig. 2. Threshold current density Jth ( solid curves) and threshold current lm (dashed curves) for the circular grating DFB laser as a function of the cavity radius R with coupling coefficient ~: as a parameter.
Acknowledgments The authors wish to thank J. Glinski for useful discussions.
299
Volume 90, number 4,5,6
OPTICS COMMUNICATIONS
References [ 1 ] R.M. Shimpe, U.S. patent 4743083 (1988). [2] X. Zheng, Electron. Lett. 25 (1989) 1311. [3] M. Toda, IEEE J. Quantum Electron. QE-26 (1990) 473. [4] T. Erdogan and D.G. Hall, J. Appl. Phys. 68 (1990) 1435. [ 5 ] C. Wu, T. Makino, J. Glinski, R. Maciejko and S.L. Najafi, J. Lightwave Technol. 9 ( 1991 ) 1264. [ 6 ] M. Fallahi, C. Wu, C. Maritan, K. Fox and C. Blaauw, LEOS 1991 Summer topical meetings: Microfabrication for photonics and optoelectronics (Newport Beach, California, 1991).
300
15 June 1992
[7 ] C. Wu, M. Svillans, M. Failahi, T. Makino, J. Glinski, C. Maritan and C. Blaauw, Electron. Lett. 27 ( 1991 ) 1819. [ 8 ] S. Wang, IEEE, J. Quantum Electron. QE- l 0 (1974) 413. [ 9 ] R.E Kazarinov and C.H. Henry, IEEE J. Quantum Electron. QE-21 (1985) 144. [10] T. Makino and J. Glinski, J. Quantum Electron. QE-24 (1988) 73. [11] N. Eda, K. Furuya, F. Koyama and Y. Suematsu, J. Lightwave Technol. LT-3 (1985) 400. [ 12 ] C. Wu, to be submitted. [ 13 ] K. Iga and S. Uchiyama, Opt. Quantum Electron. 18 (1986) 403.