Microelectronic Engineering 47 (1999) 329-331

Cavity Size and Detuning Effects on Threshold Gain and Oscillation Frequency in a Planar Microcavity Laser

Ichiro Takahashi and Kikuo Ujihara

The University of Electra-Communications

1-5-l Chojkgaoka, ChoJi, Tokyo 182-8585, Japan

The threshold condition of a one-dimensional planar microcavity laser with output coupling is analyzed semiclassically. The theory assumes homogeneously broadened atoms and a constant population inversion and the noise is neglected. Modified expressions with relative corrections against conventional formulas for the oscillation frequency and the threshold gain for a planar microcavity laser are obtained. The relative corrections depend on the cavity size and the detuning including its sign.

1. Introduction

The conventional semiclassical laser theory [l] is mainly concerned with the evolution of the field of a predetermined field mode, and spatial behavior of the field is not considered. The size effects in a microcavity laser, therefore, cannot be treated in the usual theory which does not consider the spatial behavior of the field and the cavity boundary conditions explicitly. Ujihara [2] has developed a theory of a laser, incorporating the output coupling exactly and treating the field modes of the entire space. In the theory the slowly varying amplitude approximation (S.V.A.A.) both in time and in space was used. However, the S.V.A.A. in space is no longer valid in a microcavity laser. Therefore the threshold gain and the oscillation frequency should be re-examined for a microcavity laser.

In this paper we follow the theory described in Ref. [2] but discard the S.V.A.A. in space, and derive expressions for the oscillation frequency and the threshold gain for a one-dimensional planar microcavity laser with output coupling. The present theory is essentially linear and semiclassical in the sense that the population inversion is kept constant in time and fluctuating forces associated with atomic dampings are neglected.

2. Laser Cavity and Laser Equation of Motion

The model cavity comprises a dielectric slab extending in the region -1 c z < 1 with dielectric constant E, . The space outside is a vacuum with dielectric constant Ed. We put infinitely thin mirrors at z = f 1, and assume that the reflectivity rof the mirror is independent of E~ and eO

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330 I. Takahashi, K. Ujihara I Microelectronic Engineering 47 (1999) 329-331

and can take a positive or a negative value as long as ]r] < 1 .We impose a periodic boundary condition with a large period Z+ L in the z direction. The mode function of the j -th mode describing the spatial distribution of the field is denoted by U,(z). In this case if r > 0 the mode functions, the cavity resonant frequencies, and the decay constant are the same as those in Ref. [3], and if r < 0 the only changes are the cavity resonant frequencies and the decay constant, which can be written as C$ = 2q&, , co: = ALO,( 2q+ 1) and yC = ( ACD, /Ic)ln( 1 / I r I), respectively. Here CD; and 0,” are resonant frequencies for a and b modes , respectively, yC is the cavity decay constant, ACD~ is the mode separation, and 4 is an integer.

Let us now consider our laser consisting of active two-level atoms located at z=z,,zz,.. , Z In,**- in the cavity. The transition frequency of the m-th atom is v, . From the Heisenberg equations of motion for the field modes and the atoms under the dipole approx- imation and the rotating wave approximation, the slowly varying (in time) field amplitude of the negative frequency part of electric field operator can be written as

kz, f)’ i+z,t)

l-4 ~ i$T”, ’

+c 2Aw IX[ ~j(Z)Uj(Z,)ei(“,“)(‘-“) II 1’ ,I~(~-VIII)+YBI l(~B-l’)&Zm, f)dfdi

m O i 0

, (2.1)

where v(z, t) is the initial field, and w is the center frequency of oscillation to be determined. P d, and ym are the dipole matrix element, the unsaturated inversion, and the damping c&rk.nt for the atomic polarization, respectively. This is the basic equation for the laser field valid both inside and outside the cavity. The equation states that we can calculate the field outside the cavity if the field at each atom k(z,,t) is known. To obtain the threshold condition, however, it is enough to study the field inside the cavity, which can also be obtained from eq. (2.1).

3. Field Equation Inside the Cavity

We assume that the atomic transition frequency, the dipole moment, and the damping constant , as well as the population inversion, are the same for all the atoms:

v, =vo, pm=p., ym=y, c&o. (3.1) Using the mode functions inside the cavity with their normalization constants expanded in the Fourier series [3], we calculate eq. (2.1) to obtain the differential equation, which can further be transformed to a system of inhomogeneous linear equations for the electric field amplitudes at the location of the atoms in the Laplace transformed domain:

G E(z,,s)-- C( s+f m' a” z,,z,,,s)E(z,.,s)=V(z,,s), (m, m'=l,2,3 ,... 1, (3.2)

where G is the gain per unit time and unit atom, and y’ = y - i (v, - O) . The solution of eq. (3.2), if found, would yield the general solution for &z, t) .To find the threshold condition we need only the determinant of coeffkients in eq. (3.2) because the vanishing determinant gives the poles for &, ,s) from which the threshold condition can be determined.

I. Takahushi, K. Ujihara I Microelectronic Engineering 47 (1999) 329-331 331

4. Threshold Condition

Let us assume that the atoms are distributed with a uniform density N per unit length in the z-direction (axial direction) and that the density is large enough to have a number of atoms within a distance in which the amplitude of the field changes appreciably. These

assumptions allow us to write C,,+ j Ndz,, .

We write the oscillation frequency 0 as 0 = w, + 60 , with co, =m&,,where o, is the cavity resonant frequency, m is a positive integer (longitudinal mode index), and 60 is the difference. Assuming that G CC y, 6~ CC h, and yC CC b, the determinant of eq. (3.2) can be written, in the first order approximation, by

2GN1 A(s) =I --

s+y’ ~~~l-“[s~~)&z~)l * (4.1)

bC Here the sign + corresponds to r > 0 and r < 0 , respectively. By setting A(s) = 0 in eq. (4.1), we obtain from the imaginary part of the solution an expression for the oscillation frequency at threshold :

(4.2)

The first factor corresponds to the oscillation frequency for a conventional laser showing the frequency pulling [ 1,2], while the second term in the square brackets can be considered as a relative correction which depends on the detuning v, -0, = S, C between the atomic transition

frequency and the cavity resonant frequency. From the real part of the solution the threshold gain is determined by

yyc[l+[-; ~]=GNc,[&(~)] . ( 4.3)

The first term on the r.h.s. gives the conventional threshold gain [1,2]. We define the second term in the square brackets on the r.h.s. to be a relative correction for the threshold gain. It should be noted that the correction term in eq. (4.3) is proportional to the detuning S, C (= v, -0, ) . This means that for a microcavity laser with r > 0 and a large detuning, less

gain per atom is required to lase if S,, > 0 and vice versa. The situation is reversed for

r < 0: More gain per atom is required to lase if S,, > 0 and vice versa.

The relative corrections arise from propagation effects of waves inside the cavity, which cannot be derived from the usual semiclassical theory. It can be shown that for a long , conventional laser, the relative corrections can be neglected.

References

[l] H. Haken, Laser Theory, Springer-Verlag, Berlin, 1984, p. 186. [2] K. Ujihara, Jpn. J. Appl. Phys. 15 (1976) 1529. [3] X. P. Feng and K. Ujihara, Phys. Rev. A 41 (1990) 2668.