66

Chapter 3 The Semiconductor Laser

3.4 Laser Beamwidth and the Fraunhofer Transform

In a laser such as the buried heterostructure type, the optical wave

inside the device is guided by the difference in indices of refraction be-

tw^een the active and surrounding regions. Even in the absence of an in-

dex step, the high gain tends to confine the wavefront to the vicinity of

the active region. Whether "index-guided" or "gain-guided," the beam

profile can be reasonably well described by a Gaussian, exp( -ax^-

by^),

the coefficients dependent on the height and width of the active region.

As the beam emerges from the laser, it propagates through free space,

obeying the electromagnetic wave equations, and we derive here what is

called

ihQ far-field distribution

produced by such a coherent source.

As shown in Figure 3.15, we ask for the electric field, Ep, at a long

distance from a plane containing a

near-field

distribution,

E

^. If we

define E(t) =

RelEeJ^^l,

then, from radiation theory

(Rama

et a/., 1984, p.

611)

we may write the contribution to

Ep

due to E^as

dEp(x',y')

=

-^E^(x,y)e-J^'dA

A,r

(3.12)

where fc, the wave vector, equals 2n/X. This expression is called the

"paraxial" approximation which assumes that the angle between r and

Figure 3.15 Far-field pattern

E

and near-field pattern E .

3.4

Laser Beamwidth and the Fraunhofer Transform

67

the z-axis is small. For clarity, we first treat only the behavior in the

y-z

plane as shown in Figure 3.16. We again assume that the distance

r is much greater than the extent of the near-field distribution in the y-

dimension and that dy is small compared to unity. In this case, we can

replace r in Eq. (3.12) by the expression (R - 0 v), since the distance

from the x-y plane to the far-field point decreases by

Oyy

if we move

fromy

=

0 toy

=

y. The final value of the far-field then becomes, from

Eq. 3.14,

dEp(e:,,ey)-=

1

MR-e^x-Oyy)

-N

(x.y)e-^'^^-'^''-'y'^dA

Setting r ^

K

in the denominator of the coefficient and ignoring the /,

which is an optical quarter-wave phase shift, we obtain

.-m

Ep(k^,ky)

=

^\\E^(x.y)e^^''^>^y^dxdy

XR

(3.15)

where the angles in the x- and y-directions have been replaced by the

X- and y-components of the fc-vector, k^ = kO^ and k - kO . This

expression is the Fraunhofer transform and is seen to oe a two-

dimensional Fourier transform, k replacing o), and ;c or y replacing t

in the more familiar usage. Thus, for example, the far-field angular pat-

= rey

Figure 3.16 Construction for calculating far-field pattern. The two vertical dashed lines

represent a large distance between the y and y' coordinate frame.

68

Chapter 3 The Semiconductor Laser

tern from a slit, which corresponds to a square pulse in the time domain,

is a

sin(a9)/(a0),

the same as the frequency spectrum of the pulse.

We use two important near-field/far-field relationships in our an-

alyses. The first is the far-field pattern of a uniform circular near-field

excitation such as that produced by a plane wave radiated from a

circular aperture. The second is the pattern produced by a Gaussian

field distribution, which turns out to be also a Gaussian in the far field.

First consider a uniform distribution of constant amplitude E^,

bounded by a circle of radius, a. The transform can be calculated using

Eq. (3.15) after conversion to circular coordinates. The result includes a

Bessel function of first order and is given by

2/i

ImO

ImO

(3.16)

where the near and far fields are given as scalars since they are each the

same component of polarization. This distribution is similar to the

sinxlx solution for a one-dimensional square pulse except that the var-

iation is now in the radial direction and the zeros occur at the zeros of the

Bessel function

j-^(x).

The field amplitude is shown in normalized form

Figure 3.17 The far-field distribution of a circular aperture of radius,

a.

The normalized

amplitude is given as a function of the variable,

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