Threshold gain and threshold current analysis of circular grating DFB and DBR lasers

2596 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 29, NO. IO, OCTOBER 1993

Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers

Chunmeng Wu, Toshihiko Makino, Member, IEEE, S . Iraj Najafi, Romain Maciejko, Member, IEEE, Mikelis Svilans, Member, IEEE, Jan Glinski, Member, IEEE, and Mahmoud Fallahi

Abstracf-A TE-cylindrical wave threshold gain and threshold current analysis is presented for circular grating distributed feedback (CG-DFB) and circular grating distributed- Bragg-reflector (CG-DBR) lasers. Mode properties of CG-DFBI DBR lasers are discussed. It is found that the threshold gain, threshold current density and threshold current of CG-DBR lasers can be much lower than those of CG-DFB lasers.

I. INTRODUCTION HE IDEA OF using circular gratings as distributed T feedback resonators for surface-emitting lasers was

first introduced in 1988 [ l ] . since then, many theoretical papers have been published on this subject [2]-[4]. Un- like conventional distributed feedback (DFB)/distributed- Bragg-reflector (DBR) lasers which use linear gratings, these lasers are generally called circular grating (CG) lasers and are fabricated by defining a set of concentric circular gratings to form the resonator. There are two cate- gories of CG lasers: the circular grating distributed feedback (CG-DFB) laser and the circular grating distributed- Bragg-reflector (CG-DBR) laser. Second order gratings are used to obtain surface emission.

The first optically pumped surface-emitting CG-DFB laser was demonstrated in 1991 at X = 1.3 pm on an InGaAsP/InP double heterostructure [5], and more re- cently, at X = 0.81 pm on a GaAlAs/GaAs heterostructure [6], [7]. Early this year, we reported the first electrically pumped surface-emitting CG-DBR laser operating at room temperature under pulsed conditions [8], [9]. This latter device was fabricated on an InGaAsP/InP double heterostructure and required two steps of epitaxial growth. It has been shown experimentally that it is possible to obtain low-divergence circularly-symmetric surface-emission from CG-DFB lasers [7] and CG-DBR lasers [9]. These novel surface-emitting lasers are of great interest for many new applications such as optical interconnec- tions and two-dimensional laser arrays.

Manuscript received March 25, 1992; revised November 18, 1992. C. Wu, T. Makino, and M. Svilans are with the Advanced Technology

Laboratory, Bell-Northem Research (BNR) Ltd., Ottawa, ON, Canada KlY 4H7.

S . I. Najafi and R. Maciejko are with the Department de Genie Physique, Ecole Polytechnique de Montreal, Montreal, QE, Canada H3C 3A7.

J . Glinski is with Szkola, Francusko-Polska, ul., 60854 Poznan, Poland. M. Fallahi is with the National Research Council (NRC) and Solid-state

IEEE Log Number 92 11546. Opto-Electronics Consortium (SSOC), Ottawa, ON, Canada KIA OK6.

This paper presents an analysis of the threshold gain and the threshold current of circular grating (both CG- DFB and CG-DBR) lasers. First, in Section 11, a brief introduction is given to the coupled-wave equations for TE-cylindrical waves developed earlier [lo], [ll]. Then, in Section 111, the eigenvalue equation for the threshold condition of circular grating lasers is derived. In Section IV, mode behaviors of CG-DFB and CG-DBR lasers are studied and the threshold gain analyzed. In Section V, the threshold current density and the threshold current are estimated for CG-DFB and CG-DBR lasers on In- GaAsP /InP double heterostructure.

11. COUPLED-WAVE EQUATIONS FOR TE-CYLINDRICAL WAVES

In circular grating devices, the propagating waves are inward- and outward-propagating cylindrical waves described by Hankel functions. As a result, the conventional coupled-mode theory [ 121-[ 151 is not applicable here, new coupled-wave theories have been formulated to treat this novel structure, for example: the coupled-wave theory of Zheng and Lacroix [ 161; the scalar-wave formulation of Erdogan and Hall [4] ; the vector-wave coupled-wave theory of Wu er al. [lo]; the scalar-wave equations of Gong et al. [ 171; and the vector-wave derivation of Erdogan and Hall [18].

Assuming that the circular grating is fabricated on a single-mode planar waveguide, the coupled-wave equations for TE-cylindrical waves are [ l l ] :

where the a’s are the amplitudes of the TE-cylindrical waves (with the superscript “ +” denoting the outward- propagating waves and the superscript “ - ” the inward- propagating waves, respectively). The subscript ‘ ‘n” is an integer and denotes the order of the cylindrical wave.

K ( r ) in (1) is the coupling coefficient defined by

0018-9197/93$03.00 0 1993 IEEE

WU er al . : ANALYSIS OF CIRCULAR GRATING DFB AND DBR LASERS 2591

in which ko is the wave number of light in the vacuum, AE ( r , z ) is the permittivity perturbation due to the circular grating. Z is the field distribution in the z direction of the fundamental TE slab-mode which satisfies the following scalar wave equation:

- d 2 Z + [/& - p 2 1 z = 0 dz (3)

with Z denoting the permittivity of the unperturbed planar waveguide and /3 the propagation constant of the fundamental TE slab-mode.

Equation (1) is derived from the large radius approximation [ 1 1 J and agrees with the corresponding equations obtained Erdogan and Hall [ 181.

111. EIGENVALUE EQUATIONS FOR CIRCULAR GRATING LASERS

Fig. l(a) shows schematically the structure of a circular grating (CG) laser. The device is composed of a uniform central region of radius RI and a surrounding grating region RI < r < R2, where R2 is the outer radius of the circular grating. In the case of CG-DFB laser, the active material extends from the central region into the grating region. In the case of CG-DBR laser, the active material is limited within the central region of radius RI and the grating region is passive.

The cross-section of the CG-laser is depicted in Fig. l(b), where without loss of generality, the circular grating is assumed to be an etched grating and have a rectangular profile. The line spacing of the grating is denoted by W,, the linewidth by W,, and the grating period by A (A = WI + W2). Fig. l(b) shows also the distribution of permittivity of the CG laser.

The eigenvalue equation for CG lasers can be derived by solving ( 1 ) with specific boundary conditions. The first step is to find the coupling coefficient K ( r ) . As shown in (2), K ( r ) is determined by the perturbation A E ( ~ , z ) and the field distribution of the fundamental slab-mode in the unperturbed waveguide. Since a CG laser is composed of a central region and a grating region, the unperturbed waveguide for these two regions should be different. Fol- lowing the suggestion of Handa et al. [ 191, we choose the unperturbed waveguides for the central region and the grating region as two different lossless planar waveguides. This is illustrated in Fig. 2.

The derivation of the coupling coefficient K ( r ) for the structure shown in Figs. 1 and 2 is presented in the Ap- pendix. It can be easily shown that:

K ( r ) = ja, (for 0 < r < RI) (4 .4

m and

(for RI < r < R2). (4.b) ai and a2 are the gain (or loss) in the central region (0 < r < R,) and the grating region (R, < r < R?), respec-

1'

RI 0 RI r - R,

E,

(b) Fig. 1. Diagram of circular grating (CG) laser: (a) a schematic view; (b)

cross-section.

central region I grating region t ' : a d z IYt*ZdZ

d , = O =d,=O

0 E$ RI G r

Fig. 2. Structure and permittivity of unperturbed planar waveguides.

tively . K, is the mth order coupling coefficient of the grating and Q , is a phase constant. The definitions of these quantities are given in (A8.a)-(A8.d). The prime (') in the summation symbol of (4.b) means m# 0.

A . Solutions for 0 < r < RI

equation becomes : By substitution of (4.a) into ( l ) , the coupled-wave

for 0 < r < R,. In ( 5 ) , the a ' s denote the amplitudes of the cylindrical waves in the central region and is the propagation constant of the unperturbed waveguide for the central region. Neglecting the fast oscillating terms in (5)


and solve the resulting equations, we get

H,+(r) = a,'(0)eair (6.a)

an-(r) = 6 ; ( 0 ) e - a i r (6. b)

which can be expressed in the following matrix form

(0 < r < R I ) . (7)

B. Solutions for R I < r < R2

into ( l ) , we have For the grating region, R I < r < R2, substituting (4.b)

[-j(-l>"e-j2@'a,f(r) + a , ( r ) ] (8.b)

where is the propagation constant of the unperturbed waveguide for the grating region.

Let 6 denote the deviation from the Bragg frequency, 6 = P - m?r/A. If 161 << 1, then only the resonant terms in (8) are important. By neglecting the fast oscillating terms in (8), we obtain

The solution of (9) can be obtained in a matrix form with

(for RI < r < R2) (10)

where

- (v) sinh [ y ( r - R I ) ] ] (1l.d)

y = J K ~ + (a2 - j 6 l2 . (11.e)

C. Threshold Condition To find the threshold condition, we must relate the

waves in the central region to those in the grating region. Note that these two regions form two different waveguides. Similar to the case of one-dimensional waveguide discontinuities, by matching the electrical and magnetic fields of the central region with those of the grating region at r = R I , it is not difficult to show that in the present case, the discontinuity is described by

where CO is the power coupling efficiency between the central region and the grating region

In (13), 2 and 2 are the slab mode distributions in the two region, respectively.

Then, from (7) and (12) , we have

= I 0

The amplitudes of the cylindrical waves at r = R2 can be found from (10) to be

2599 WU et al.: ANALYSIS OF CIRCULAR GRATING DFB AND DBR LASERS

Substitution of (14) into (15) , we obtain

Suppose that there is no reflection at r = R2, then to' create self-sustained oscillations, the ratio of the ampli- tude of the incoming-wave to that of the outgoing-wave must be zero at the boundary, i.e.,

~ = 0. an- (R2) a : (R2)

Equation (23) is similar to the eigenvalue equation for conventional DBR lasers with one perfect mirror [20] . In fact, because the field must be finite at r = 0, we have to require

anf(0) = a i ( 0 ) (25) (17)

This condition is identical to that of one-dimensional DFB lasers [14] . Thus, from (16) and (17) , we obtain the eigenvalue equation of circular grating lasers as

. i.e., po = 1. The center thus behaves like a mirror of 100% reflectivity.

Another noteworthv feature is the reflection coefficient d,'(0)T2, ( R 2 ) G e n l R 1 - j ( p - p ) R 1 p R I of the circular grating depends on the order of the cy-

lindrical waves. This can be evidenced by substituting T21 and T22 from ( 1 1 ) into (23) to obtain explicitly the grating (18) +

(0) T22 ('2) e -airi + j t B - @ ) R I = 0 JG reflection coefficient as

or in another form

Note that from (15) and (17), we also have

Now defining the reflection coefficient at r = 0 as

and the reflection coefficient of the grating seen from r = RI outward as

Then, the threshold condition in (19) can be expressed by

~ ~ p ~ p ~ ~ e - j ' e ~ ~ ~ ~ ~ = 1 (23)

with Q denoting the total phase shift

Q = Q, + 2 ( p - @RI ?r

= - m ( W l + 2R1) + 2 ( p - @ ) R I . (24) A

The dependence on the order of cylindrical waves is rep- resented by the factor (- 1)" in (26) . For n = even, we have ( - 1 ) " = 1 > 0; and f o r n = odd, ( - l ) n = - 1 < 0. This sign change is equivalent to a phase difference of ?r between the even and the odd cylindrical waves. The threshold of the even and odd waves is thus ?r-phase apart as will be shown below. However, all the even (or odd) waves are degenerate, i.e., they have the same threshold. The degeneracy among the even (odd) order cylindrical waves is a direct result of the large radius approximation for the Hankel functions [ 4 ] , [ 1 1 3 , [ 181. From physical argument, lower order waves should have a lower threshold since the optical fields of lower order waves have a better interaction with the active medium than do higher order ones. The numerical calculations of Erdogan and Hall [ 4 ] , [18] and those of Gong et al. [ 1 7 ] , indicate that the degeneracy would break down, though the threshold gain difference between even (or odd) waves are small. For surface-emitting CG lasers, the coupling to radiation would also affect the mode selection process.

IV. THRESHOLD GAIN OF CIRCULAR GRATING LASERS In this section, we study the threshold gain of circular

grating lasers. For simplicity, we will not consider the radiation effect in the present paper. That is, we only study CG-lasers with first-order (m = 1 ) circular grating. The results presented here would provide a lower bound for second-order grating devices, because radiation from the surface causes additional losses, usually leading to a higher threshold.

For convenience, we denote the length of the grating

2600

J

0 -

e .

0 .

e

B

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 29, NO. 10, OCTOBER 1993

region by L = R2 - RI and define the normalized inner radius by S = R I / L .

A. Mode Spectrum and Threshold Gain of CG-DFB Lasers

In the case of DFB lasers, because the active material extends over the whole grating region, we essentially have aI = a2 = a, fi = fi and CO = 1.

In Fig. 3(a) and (b), the normalized threshold gain athR2 is plotted against the normalized frequency deviation 6R2 for Q = 7r/2 and 27r. The open and the solid circles represent the even-order and the odd-order cylindrical waves, respectively. The calculation is done for KR2 = 1.0 and S = 0.0. It is seen in Fig. 3(a) that for Q = 7r/2, the even cylindrical waves and the odd cylindrical waves have the same threshold gain, but the even waves oscillate at a lower frequency (6R2 < 0) than the odd waves (6R2 > 0). The even waves and the odd waves are located sym- metrically with respect to the Bragg frequency (6R2 = 0). The same can be said for Q = 3 ~ / 2 except that the lasing frequency of the even waves and that of the odd waves are now interchanged. For Q = 27r, as shown in Fig. 3(b), the even waves have a much lower threshold gain than the odd waves. The reverse is true for Q = U. This clearly shows that by properly choosing the phase Q , we can select either the even or the odd order cylindrical waves. This selectivity may be very important in designing CG- DFB lasers.

As shown by (25), the total phase-shift Cl is a function of the inner radius R I . However, because RI is usually very small in comparison to the outer radius R2 (RI << R2) in CG-DFB lasers, once the total phase Q is fixed, the influence of RI on threshold gain is negligible. This fact is shown in Fig. 4, where we compare the case of S = 0.0 and S = 0.01 for even-order cylindrical waves. For a grating length of L = R2 - RI = 140 pm, S = 0.01 means an inner radius of RI = 1.4 pm. The threshold gain for S = 0.01 is only slightly lower than for the case where S = 0. Therefore, for a given phase-shift Q , our calculation for S = 0.0 offers a good estimation of the threshold gain of CG-DFB lasers.

In Fig. 5(a), the normalized threshold gain of the lowest mode is shown as a function of Q (0 < Q < 27r). The corresponding lasing frequency is presented in Fig. 5(b). We conclude from these two figures that if we want even (or odd) waves to lase, it can be achieved by choosing Q = 27r (or 0 = 7r) and the lasing frequency is right at the Bragg frequency (6R2 = 0).

The normalized threshold gain is shown as a function of the grating radius R2 in Fig. 6 . It is seen that the normalized threshold gain athR2 decreases as the grating radius R2 increases.

I

B. Mode Spectrum and Threshold Gain of CG-DBR Lasers

For CG-DBR lasers, the gain region is limited within a circle of radius RI and the grating is a passive region with

Circular Grating DFB Laser ....1....1....1....,....,....

Q=nl2, KR,=l.O

1.4 I

9 l l e o e o

0 n=odd

e 0

0 e

e o

0 .

0.4 - 1 5 - 1 0 - 5 0 5 1 0 1 5

6R2

(a)

Circular Grating DFB Laser

0 0

e 0

0 0

e .

0

J

0 .2 U - 1 5 - 1 0 - 5 0 5 1 0 1 5

6R2

(b) Fig. 3 . Mode spectrum of CG-DFB laser with KR2 = 1.0 and S = 0.0:

(a) Q = 1r/2; (b) 0 = 21r.

a length of L = R2 - RI (see Fig. l(b)). Note that the inner radius RI of CG-DBR lasers is large compared to that of CG-DFB lasers. Since the grating region is inde- pendent of the active region, we have a = a1 and the eigenvalue equation (23) can be rewritten in another form:

1 aR, = In ( ) (27.a) I POI 1 PRII

arg ( p R I ) - Q = 2q7r (q = integer) (27.b)

WU el al.: ANALYSIS OF CIRCULAR GRATING DFB A N D DBR LASERS 260 I

Circular Grating DFB Laser 1.9

n=even R = 2 R

C - ' 1.3 E c

s e 0.9

0.3 O a 5 - 1 5 U - 1 0 - 5 0 5 1 0 1 5

SR2

Fig. 4. Effect of normalized inner radius S on the mode spectrum of CG-DFB laser: (a) Q = ~ / 2 ; (b) Q = 2 ~ .

In our numerical calculations, we ignore the losses in the grating region, i.e., with a2 = 0. We also assume that CO = 1. The imperfect coupling between the gain region and the grating region (CO < 1) leads inevitably to additional losses, and thus increase the threshold gain [25].

The mode spectrum of CG-DBR lasers is presented in Fig. 7(a) and (b) with S = 0.2 and KL = 1.0. The solid line is obtained from (27.a). The open and the solid circles represent the even-order and the odd-order cylindrical waves, respectively. For a grating length L = R2 - R I = 140 pm, S = 0.2 is equivalent to an inner radius of R I = 28 pm. Similarly to the case of CG-DFB lasers, for Q = ~ / 2 , the even and the odd waves have the same threshold gain, but the even wave oscillates below, and the odd wave above the Bragg frequency, this is shown in Fig. 7(a); for Q = 37r/2, the roles of the even waves and the odd waves are reversed. For Q = 27r, the even wave has the lowest threshold and oscillates right at the Bragg frequency, this is shown in Fig. 7(b) for Q = T, the reverse is true.

In Fig. 8(a) and (b), the normalized threshold gain and the lasing frequency is plotted against the total phase-shift Q , respectively. As in the case of CG-DFB laser, the even order waves have the lowest threshold gain and oscillate at the Bragg frequency when Q = 27r; The same can be said about the odd order waves when Q = 7r.

In Fig. 9, the normalized threshold gain is shown as a function of the gain region radius R I . The phase Q is set to 27r and the coupling coefficient K is used as a parame- ter. Note that for Q = 27r and n = even, the lasing frequency is right at the Bragg frequency. By substituting 6 = 0 into (26) and (27), the normalized threshold gain q h R I is found to be a function of the coupling strength

0.8

K: 0 . 7

a-

3 0.6 0 c

t! c c U N 0

0.5

- - E

0.4


I " ' " " ' I ' " '

KR,=1 .O . n=even


- 3 :

l . . . . l . . . . l . . . . - 4

1 .5 2 0 0.5 1

R i a

(b) Fig. 5 . Effect of the total phase-shift Q on CG-DFB lasers: (a) normalized

threshold gain versus Q; (b) normalized frequency as a function of Q.

KL only. For n = odd and Q = 27r, because the lasing frequency is not located at the Bragg frequency and is dependent of the ratio S = R l / ( R 2 - R I ) , the normalized threshold gain decreases monotonously as RI increases. This difference between the even waves and the odd waves makes the corresponding threshold current behave differently as to be seen in the next section.


Circular Grating DFE Laser

2 .5

N

= j2 c d E 1.5 c 2

0 0 4 0 0 0 1 2 0 1 6 0 2 0 0

R,("

Fig. 6. Normalized threshold gain of CG-DFB laser as a function of grating radius R2.

V. THRESHOLD CURRENT OF CIRCULAR GRATING LASERS

A. Model In the last section, we have just calculated the mode

field gain for CG-DFB and CG-DBR lasers. In a practical design, it is necessary to know the threshold current density and the threshold current. For such an analysis, it is more appropriate to use the numerical gain g.

As in [22], the relationship between the material gain and the field gain can be obtained as

gthra = 2ath + %t (28)

where Fa is the optical confinement factor of the active layer, gth is the material threshold gain, (;Yth is the threshold field gain calculated from (23) or (27), and cyint is the total internal power loss.

Assuming that the gain satisfies the following linear re- lation [22]:

g(N) = A O W - No) (29) where g is the peak gain, A. is the gain coefficient, N is the carrier density and No is the carrier density to achieve transparency, the threshold camer density Nth is obtained from (28) and (29) as

Nth = NO + (2ath -k '%nt)/(ACJra). (30)

The threshold current density Jth can be calculated from [23]:

Jth = e(da/qi) Beff(Nth)2* (31)

In (31), e is the charge of an electron, d, is the thickness of the active layer, qi is the internal quantum efficiency,

Circular Gratlng DER Laser 5 1 . * . . 1 . . . . 1 . . . . 1 . . . . 1

E

2 c

C I-

n=n/2, s 0 . 2 KLd.0, a,L=O.O

- 1 0 - 5 0 5 1 0

SL (a)

Circular Grating DER Laser 5

4

3

2

1

- 1 0 - 5 0 5 1 0

SL (b)

Fig. 7 . Mode spectrum of CG-DBR laser: (a) Q = u / 2 ; (b) Q = 2u.

BeR is the effective recombination constant. The threshold current Ith is obtained from:

Ith = 7r(R2)2Jth, for circular grating DFB lasers

(32.a)

= 7r (R1)*Jth, for circular grating DBR lasers.

(32.b)

B. Numerical Results We choose 1.3-pm InGaAsP/InP double heterostruc-

ture CG-DFB and CG-DBR lasers as our numerical ex-

WU et al . : ANALYSIS OF CIRCULAR GRATING DFB A N D DBR LASERS

Circular Grating DER Laser

0.6 : - - ' I - - - I - - - - I - - - - . : K k 1 . 0

I . . . . I . . . . I . . . . .

0 0.5 1 1.5 2 0.1

Q l n (a)

2603

Fig. 8.

Circular Grating DER Laser

- 3 :

I . . . . I . . . . I . . . . .

1.5 2 - 4

0 0.5 1

Rllr

Effect of the total phase-shift 62 on CG-DBR lasers: (a) normalized (b)

threshold gain versus 62; (b) lasing frequency against 62.

amples. The following parameters are used in the calculation: A. = 2.5 X 10-l6 cm2, No = 1.5 X 10l8 ~ m - ~ , Ber = 1.0 x lo-'' cm3 s - ' [23], r = 0.5, da = 0.2 pm, v i = 1 and cyint = 50 cm-I.

Consider CG-DFB lasers first. Fig. 10(a) shows the threshold current density as a function of the grating radius R2 for different values of the coupling coefficient K. For both even and odd cylindrical waves, the threshold current density decreases as the grating radius increases, and then levels off with further increase of the grating ra-

1 0

a: a- 1 I

(II 0

E

g 0.1 c

c c U t - - 0

0.01

2

0.001 ............ 0 2 0 4 0 6 0 8 0 1 0 0

R , ( P m )

Fig. 9. Normalized threshold gain of CG-DBR laser as a function of gain region radius R , .

dius. This is expected because the threshold gain (Yth is a decreasing function of R2 (see Fig. 6) and because a certain amount of current is needed to achieve transparency and compensate the total internal loss.

The corresponding threshold current is shown in Fig. 10(b). For each value of K, there is an optimum grating radius that gives the lowest threshold current. This is similar to the case of conventional DFB lasers [24]. Such a dependence of the threshold current of CG-DFB lasers on the grating radius R2 can be explained as follows: from (32.a), the threshold current is expressed by the product of (R2)2 and the threshold current density J t h . Fig. 6 shows that as R2 increases, cyth decreases much faster than 1 /R2. Since the threshold current density is proportional to the square of the threshold carrier density which is a linear function of q,, this means that J t h decreases much faster than (1/R2)2, thus overcomes the increase of zth due to (R2)2. For larger R2, the factor (R2)2 in (32.a) becomes dominant, therefore, I th goes up as R2 increases.

The threshold current of the odd waves following the same trend as the even waves but has a higher value as a result of Q = 27r. At the optimum radius, the threshold current of the odd waves is about twice that of the even waves.

The results for CG-DBR lasers are presented in Fig. l l (a) and (b) for Q = 27r, where the threshold current density and the threshold current are shown respectively as a function of the gain region radius RI . The threshold current of even waves increases monotonously as a function of RI , while that of odd waves has a minimum at a certain radius. For a reasonable gain region radius, both the threshold current density and the threshold current of even waves are much less than those of odd waves. The

2604

5 0 0 “ ‘ ‘ ‘ ‘ . I “ ‘ I ” ’ ” ”

R.2x L=l4Opm a -0.0 L *

i o 0 0

- “E . a x Y

-- 1 0 0


c C !?

i o I!

?! C

C +

1

2000

2 i 5 0 0 E v - c

!? ~ 1 0 0 0

E

!? C

E 5 0 0


0 0 4 0 8 0 i 2 0 1 6 0 2 0 0

Rz(Pm)

(b) Fig. 10. Threshold current density and threshold current of CG-DFB laser

as a function of grating radius R,: (a) Jth versus Rz; (b) I , , versus R,.

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 29, NO. IO. OCTOBER 1993

Circular Grating DBR Laser

reason for the even waves to behave differently from the odd waves is that for even waves, the threshold gain (Yth

decreases as l / R 1 (see Fig. 9). When substituted into (32.b), Zth becomes proportional to thus gives the monotonously increasing behavior. The explanation for odd waves is similar to the case of CG-DFB lasers. Be- cause (Yth decreases much faster than l /RI (see Fig. 9), Jth decreases much faster than (1 thus overcomes the increase of Zth due to in (32.b). For larger R I ,

the factor (R,)2 in (32.b) becomes dominant, therefore, Zth

goes up as R, increases. Comparing the results for CG-DFB lasers with those

for CG-DBR lasers, we observe that the latter has a much lower threshold current density and a much lower threshold current than the former. For example, for K = 100 cm-I, the threshold current density is 9.82 kA/cm2 and the threshold current is 494 mA for a CG-DFB laser (n = even, = 27r, R2 = 490 pm, see Fig. 10(b)), while the

WU et al.: ANALYSIS OF CIRCULAR GRATING DFB AND DBR LASERS 2605

- corresponding values of a CG-DBR laser (n = even, Q = 2?r, RI = 40 pm) is only 1.47 kA/cm2 and 74 mA (Fig. 1 l(b)). The values of CG-DBR lasers are about seven times less than those of CG-DFB lasers.

E g m ( Z ) = (€cl E , ) e -j(r/A)mwl ?rm

- sin (am 9) (m f 0). (A2.c)

VI. CONCLUSIONS We have presented a detailed threshold analysis for TE-

cylindrical waves in CG-DFB and CG-DBR lasers. Our numerical results show that by proper choice of the phase Q (in other words, the inner radius R I ) , we can select either the even wave or the odd wave. This is an important result for the design of practical CG-DFB and CG-DBR lasers. We have also shown that CG-DFB lasers normally have a higher threshold gain, a higher threshold current density and a higher threshold current than CG-DBR lasers.

APPENDIX DERIVATION OF COUPLING COEFFICIENT

For the CG laser shown in Fig. l(b), the permittivity is described by:

C(z, r) = c s , z < dl = O

dl < z < d2

d2 < z < d3

= E’ + j ~ ” ,

- - Eb,

d3 < z < 03 (A1.a) - - E , ,

for 0 < r < R I , and by

Following Handa et al. [19], we choose the unperturbed waveguide for the central region as a lossless planar waveguide whose permittivity is:

E ( z ) = E ~ ,

= E’,

z < dl = 0

dl < z < d2 d2 < z < d3 - - Eb,

(0 < r < R I ) (A3.a)

and for the grating region,

E(z) = E , , z < d ; = 0

= E ’ ,

- - E,[ ,

d i < z < d ;

d ; < z < d ;

(9) = E u = E,[ - (E , [ - E , )

d j < z < d ; - - E , , z > d ;

(RI < r < R2) (A3. b)

This is illustrated in Fig. 2. The perturbation is then calculated from

- - Ed,

= E&, r) ,

- - Ec, d ; < z < 00 (A1.b)

forRI < r < R2. In (A1.a) and (Al.b), E, is the permittivity of the sub-

strate region, E, the permittivity of the cover region (normally air E, = l ) , Eb the permittivity of the cladding for 0 < r < R I , E,[ the permittivity of the cladding for RI < r < 03. The permittivity of the active laser in the center region (0 < r < R I ) is denoted by E (with the real part E‘

and the imaginary part E”) . Ef is the permittivity of the guiding layer in the grating region for d [ < z < d ; and RI < r < R2.

The permittivity of the grating layer can be expressed by :

d ; < z < d ;

d ; < z < d i

m

C E g m ( Z ) e J ( 2 r / A ) m ( r - R ~ ) (A2.a) m = - m

E&, r) =

where

for each region. Suppose both unperturbed waveguides are single-

moded. Let p and /3 denote the propagation constants, 2 and 2 represent the field distributions along the z-direction of the fundamental TE-modes in the two unperturbed waveguides, respectively. From (A1 .a), (Al.b), (A2), (A4), and (2), the coupling coefficient K ( r ) can be obtained as:

K ( r ) = j a l (for 0 < r < R I ) (A5.a)

and 03

(for RI < r < R2). (A5.b)

The prime (’) in the summation symbol means m # 0.

al = (k ; /2p)E~i ra (A6. a)

a2 = (k;/2p)q r, (A6.b)

In (A5.a) and (A5.b), we have

2606 IEEE

where K,,, is the coupling coefficient of the mth-order circular grating, 0, is the phase-shift due to the center region. al and a2 denote the field gain (or loss) in the central region and the grating region, respectively. The r’s in (A6) are the overlap integrals defined by

P d2 I P m

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R. Maciejko (M’80). photograph and biography not available at the time of publication.

M. Svilans (M’84), photograph and biography not available at the time of publication.

Jan Glinski (M’86), photograph and biography not available at the time of publication.

M. Fallahi, photograph and biography not available at the time of publication.

Threshold gain and threshold current analysis of circular grating DFB and DBR lasers

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