Phase Reflectance at Narrow Gap

The condition for maximal transmission of light with wavelength $\lambda$through a Fabry-Perot interferometer is

$$m \lambda = 2 \mu L \cos \theta ,$$ (9)

where m is the order of interference, L is the spacing between the plates, $\theta$ is the interior angle of incidence and $\mu$ is the refractive index of the gap medium. For TTF, this medium is air with $\mu = 1.00$. Equation (9) is an approximation suitable for high orders and therefore large plate spacings. However, it fails to take into account the wavelength-dependent phase-change inherent in reflections between the optical coatings on the inner plate surfaces. Such coatings are optimised to reflect the design wavelength (819 nm for TTF) with zero phase change but incur a lead or lag phase elsewhere. The phase change becomes increasingly important as the plate spacing becomes comparable to the thickness of the optical coatings. Equation (9) can be modified to

$$\biggl( m_{\rm T} + \frac{\epsilon_\lambda}{\pi} \biggr) \lambda = 2 \mu L \cos \theta ,$$ (10)

accounting for phase change through the introduction of an order correction term, $\epsilon _\lambda$ ([Atherton et al. 1981]; [Knudtson, Levy & Herr 1996]). Here, $m_{\rm T}$ is the true order number associated with phase correction. The subscript $\lambda$ denotes the wavelength dependence of $\epsilon$. At large orders $m_{\rm T}$, the effect of the phase change term becomes negligible. The inner coatings of the red TTF cause $\epsilon _\lambda$ to vary by $\pm \pi$ over 630 - 950 nm.

The ratio $m_{\rm T}/m$ characterises the relative influence of phase change at a given wavelength. Combining Eqns. (9) and (10) for a common wavelength and plate spacing L, gives

$$\biggl( \frac{m_{\rm T}}{m} \biggr)_L = 1 - \biggl( \frac{\lambda}{2 \mu} \frac{\epsilon_\lambda}{\pi} \biggr) \frac{1}{L} .$$ (11)

By this we see that the relative size of the phase correction will become larger towards smaller gap at a given wavelength. In practice, the phase correction will also alter the free spectral range and bandpass of the instrument, in addition to the shape and location of the transmission profile ([Atherton et al. 1981]). Table 1 contains a list of $\epsilon_\lambda/\pi$ values at selected wavelengths, as measured by Queensgate Instruments. Also included are values for the coefficient in Eqn. (11).

Figure 5(a) shows four TTF scans at the lowest plate spacings reached by our instrument. The scans show blended lines of Ne (659.9, 667.8 and 671.7 nm). The lines are unresolved at all plate spacings except the largest (L = 4.7 $\mu$m), where they are labelled in (a). The scans were made at various values of (Z_c, Z_s) and transformed to physical units of spacing by Eqn. (2). Changes in the software scan increment ( $\Delta Z_s$) are evident in the different sampling densities of each scan. Observe that the transmission peaks are evenly separated by $\lambda/2 \mu = 0.33$ $\mu$m, confirming that the calibration in Eqn. (2) is robust over all settings of Z_c and Z_s used. Also note the broadening of the transmitted profile as plate spacing and resolution decreases. The flat background levels are from CCD regions that were not used in the charge shuffle.

In Fig. 5(b) we plot the change in $m_{\rm T}/m$for the same orders of the Ne blend shown in (a). The ratio $m_{\rm T}/m$ was calculated at each plate spacing measured in (a) by Eqn. (11). Queensgate Instruments have measured $\epsilon_\lambda = -0.79 \pi$ for TTF at 666 nm. Observe in Fig. 5(b) that phase correction is a $\sim 10$ % effect at 670 nm. The dotted lines show the effect to be significantly less at wavelengths near the centre of the TTF coverage. The narrowest spacing (2.5 $\mu$m) is a self-imposed limit to which we are prepared to drive the plates. Any closer and we run the risk of damaging the inner coatings by pressing dust particles between the two coating surfaces. We conclude that at the narrowest spacings of TTF we are in a regime where phase effects are non-negligible, particularly for wavelengths at the extremities of the optical coating curve.