Frequency domain interferometry

First proposed by Froehly [8], this technique (sometimes referred to as Fourier transform spectral interferometry or FTSI) determines the spectral dependence of phase through Fourier transforming a spectral interferogram, rather than scanning delay. Because of this the experimental setup is the simplest of any of the linear phase measurement techniques, as shown in figure 3.2.

The interferometer is not balanced in this case; the (nominally static) delay, τ, between arms must be chosen carefully. To yield valid results the measured

3. Comparison of existing techniques 40 Retroreflector on translation stage Beamsplitter Spectrometer E₀(ω) E₀(ω) E(ω) Dispersive sample

Fig. 3.2: Frequency domain interferometer, as conceived by Froehly et al

τ

−τ 0

Delay

Intesnity

Filter

Fig. 3.3: Delay vs intensity trace resulting from Fourier transform of spectral

interferogram

interference pattern must both contain sufficient fringes to be able to determine phase and have the fringes spaced sufficiently far apart to be well resolved. The upper limits of delay corresponding to these requisites are wholly dependent on the scanned spectral range and the resolution of the spectral filter respectively. By defining the largest value of delay for which a valid interferogram can still be measured, these two factors determine the maximum spectral resolution of phase for this technique.

3. Comparison of existing techniques 41 as a function of time, as shown in figure 3.3. Two peaks are symmetrically placed around t = 0, centred at t = τ and t = −τ. These peaks correspond to the terms in equation 3.4. A large peak centred at t = 0 represents the sum of the intensity terms (i.e the envelope of the interferogram) and carries no phase information. Filtering out all of the transform other than one of the satellite peaks leaves data corresponding only to intensity and the complex transfer function, R(t − τ). The central peak can be eliminated from the transformed trace by chopping the beams in both arms of the interferometer at different frequencies and using lock-in detection to measure the signal at the difference frequency. The resultant interferogram then corresponds only to the interference between beams and not to their intensities. Removing the central peak allows a broader filter to be applied to the transformed trace. Having filtered one of the satellite peaks, it is necessary to compensate for the finite delay, τ (remembering that a constant delay corresponds to a slope of constant spectral gradient). This is done by centering the filtered peak att= 0, before the inverse transform is applied to it. The argument of the resultant complex frequency plot gives the phase transfer function of the device. The resolution of the complex frequency plot is proportional to the width of the filter applied before performing the inverse transform. The maximum filter width is proportional to the chosen delay, τ.

The interferogram can also be treated without using Fourier transforms; the position of fringe peaks can be noted and the separation of adjacent peaks equated to a phase of 2π radians[9]. If this is done, phase resolution is again proportional to delay (assuming spectral resolution is sufficient to adequately resolve the interferogram), but is independent of the scanned spectral range. Increasing delay decreases the ratio of signal to noise however and for this reason, Fourier transforms offer greater accuracy at high spectral phase reso- lution.

This technique relies on the accuracy of the spectral filter and to ensure a smooth trace, measurements are taken at intervals smaller than the resolution of the spectral filter. This ’running average’ removes sharp features from the delay trace and so eliminates the corresponding high frequency components that would otherwise be seen in the Fourier transform.

3. Comparison of existing techniques 42 Other than the simplicity of the experiment, there are several appealing features to this method. The calculated phase is given as a continuous function of frequency, as opposed to the discrete points given by methods which do not use Fourier transforms. This is ideal when consideration is given to the fact that the phase transfer function will be derivated once to give group delay and again to yield GDD. No fitting functions have to be applied, which avoids the associated problems of smoothing. When applying a fitting function to discrete data it is common to smooth and so eliminate spurious sharp features caused by noise, but this also lessens the accuracy of the fit.

A significant advantage of FTSI is that all spectral components can be measured concurrently. The spectral components can be measured simultane- ously by an optical spectrum analyser or spatially separated and detected by a CCD array. For any significant acquisition time (in comparison to the rate of drift in delay), variations in delay will cause the measured interferogram to lose contrast (as with the compensated Michelson). Despite the unavoidable decrease in signal to noise ratio this causes, data taken this way is still valid. Such measurements are also complicated by the frequency step between pixels changing over the width of a CCD, although compensation can be made for this[10].