The Practical Determination of Waveguide Parameters

The fundamental parameter of the optical waveguide is the effective refractive index N. Equivalently, one may refer to the phase velocity, propagation constant, reflection phase change, etc.—all are equivalent (Ramsden 1993a). N characterizes a guided mode. Modes are discrete, and for the thinnest waveguides, which are the most sensitive to changes in the interfacial region, are well separated. It is important to realize that even though the overall waveguiding structure through which light is propagating comprises several layers, each with a distinct refractive index, the light propagates through all of them with a common phase velocity.

The two main methods for determining the lightmode spectrum are (i) coupling, and (ii) interferometry. In this chapter we shall only deal with the former, for it most readily and directly allows the precise and accurate determination of the absolute values of the effective refractive indices of two or more modes, which are required for the structural analysis that is our ultimate goal. Interferometry is covered in the references cited earlier (Ramsden 1994,1997), and see also the chapter by Freeman in this volume.

Coupling requires a grating (or, less conveniently, a prism). Consider a slab structure consisting of a support layer S (typically made from optical glass about a few tenths of a mm thick), a thin high refractive index transparent film F (typically a metal oxide ~ 100 nm thick), and the cover medium C (typically air or water). An optical grating is incorporated into the waveguide at the F,S or F,C interface. A shallow grating, a few nm deep, is sufficient for coupling the few percent of light needed for measurement purposes. In a typical incoupling configuration, an external beam impinges onto the grating, making an angle a with the grating normal. The wavenumber component in the direction of guided propagation is then nair sin a + 2ni/A, where i = 0, ±1, ±2,... is the diffraction order and A the grating constant. If this matches a guided mode with wavenumber kN, incoupling will occur according to the coupling condition

The procedure is, therefore, to measure the light emerging from the end of the waveguide while varying a. The emerging light will appear as a series of sharp peaks, successively TMm=o,i=1,TEm=o,i=1, etc., called the mode spectrum. At the time of writing, the highest precision in the determination of N is achievable by mechanical goniometry in a temperature-controlled environment, with which a can typically be determined to microradian precision.

The refractive index of most substances varies significantly with temperature. The variation is directly related to the coefficient of thermal

Table 4. Contributions to the uncertainty of N in an input grating coupler. It should also be noted that since dN/dnC ~ 0.1 and dnC/dT ~ 10-4K-1 for water, aqueous solutions should be thermostated to ±0.1 °C to keep within the bounds of uncertainty imposed by the other terms

Parameter

Typical value

Uncertainty

Physical origin

nair

1.0002673

10-7

Temperature fluctuations (T ± 1 °C)

A

416.147 nm

0.001

Grating lateral thermal movement

a

0.09 rad

1.25 X 10-6 rad

Goniometer mechanical instability

A

632.816 nm

0.001 nm

Laser mode jumping

expansion of the substance, as can be seen from the Lorentz-Lorenz equation, which connects the macroscopic index of refraction with various atomic or molecular parameters, i. e. molecular mass Mr, density p and molar refractivity RM:

Of these, only the density varies with temperature. Therefore, it is obviously important to record the temperature of the sample under investigation when making optical measurements. Table 4 lists the various sources of uncertainty in the determination of the mode spectrum peaks. After applying the usual combining laws it can be seen that all factors contribute roughly equally to the overall uncertainty in N, typically ±1 - 2 X 10-6.

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