Integrated Optics - Applied Optics Fundamentals and Device Applications - page 232

1 dB cm −1 for wavelengths shorter than 0.92 μm in waveguides with an alu-
minum concentration of 40%. More recent studies have measured losses of
3-5 dB cm −1 at wavelengths of 0.86 μm. The high quality material now avail-
able will further reduce losses to an acceptable level. It is actually inad-
visable to reduce losses by going to aluminum concentration greater than
20% because the appearance of deep level defects becomes the limiting fac-
tor. At wavelength greater than 0.92 μm, waveguides with total propaga-
tion loss less than 1 dB cm −1 have been made. For example, Tracey et al.
[18] reported MBE-grown waveguides with 1 dB cm −1 loss in either 1.06 or
1.15 μm waveguides.
Surface scattering from the walls of the waveguide can also be a significant
loss mechanism if the roughness of the walls is not kept within certain lim-
its. Tien [19] derived an expression for scattering loss due to surface rough-
ness based on the Rayleigh criterion. The exponential loss coefficient α s is
given by

2

⎛
⎜

3

⎞
⎟

A cos

sin

θ
θʹ

1

m

α = (5.15)

3

2 t + ( 1 / p ) + ( 1 / q )

m g

where
t g is the thickness of the waveguiding layer
θ m is the angle between the ray of the waveguide light and the normal to
the waveguide surface
p and q are the extinction coefficients in the confining layers

The coefficient A is given by

4

1 2 2 2

) /

1 2

A = ( σ + σ (5.16)

λ

2

where
λ 2 is the wavelength in the guiding layer
σ 12 and σ 23 are the statistical variances of the surface roughness

Although Tien's expression was derived for the case of a three layer planar
waveguide, it can be used to estimate the order of magnitude of surface
scattering loss in a rectangular guide as well. Note that α s is basically pro-
portional to the square of the ratio of the roughness to the wavelength in the
material (represented by A 2 ), weighted by secondary factors that take into
account the shape of the optical mode. For the case of AlGaAs waveguides
and wavelengths in the range of 0.82-1.3 μm, the wavelengths in the material
are a few tenths of a micron; hence, if surface variations of the waveguide
are limited to approximately 0.01 μm, α will be approximately 0.01 cm −1 and