Source spectrum, apodization, and temporal resolution

Frequency-domain C2 _{imaging uses a narrow linewidth, swept-wavelength laser to}

facilitate measurements. This laser is swept over a given bandwidth ∆Ω as camera images are acquired, resulting in three dimensional (x, y, ω) intensity data (Eq. (4.6)). We will assume that over the sweeping bandwidth, the laser power is constant. The validity of this assumption is ensured in practice by measuring the laser power during these sweeps and normalizing the data for any variation. There is no laser power outside of the bandwidth ∆Ω, therefore the source spectrum can be written in terms of a continuous frequency variable ∆ω as a rect function: ˜A(ω) = A0rect _∆Ω∆ω
. If

the dispersion-times-length difference between the interferometer arms is zero, then ϕmr (Eq. (4.10c)) is also zero, and the cross-correlation function (Eq. (4.9)) is simply

a delayed Fourier transform of the input spectrum – a sinc function in this case. Ideally, the cross-correlation function is as narrow in time as possible, and quickly falls to the noise floor so as not to interfere with other signals in the modal weight

trace. The sinc function fulfills part of this criteria as it is fairly narrow in the vicinity of the peak. Unfortunately, it also exhibits side-lobes which could potentially be prob- lematic for discerning other nearby signals. In some cases, it is beneficial to apodize the data the in order to remove these side lobes. For these purposes, apodization is defined as multiplying the intensity data (Eq. (4.6)) by an arbitrary function in the frequency domain. This is equivalent to modifying the input spectrum of the light source by some apodization function. Careful selection of the apodization function can smooth the otherwise sharp corners of the input spectrum and suppress side- lobes in the resulting cross-correlation function. This practice has long been in use for side-lobe suppression in filters, both in the electrical and optical domain, includ- ing fiber Bragg gratings and LPGs (Madsen and Zhao, 1999; Kashyap, 1999). Here, super-Gaussian apodization functions are used in order to modify the signals. It is worth mentioning, however, that since apodization happens in the post-processing do- main, any function of arbitrary complexity could be used. The set of super-Gaussian functions considered are defined as

AN(∆ω) = e−  ∆ω ∆ΩN N (4.20)

where N is an even integer describing the order of the super-Gaussian, and ∆ΩN

is an adjusted bandwidth defined for the apodization function. The apodized input spectrum is thus ˜ A′(∆ω) = ˜_{A(∆ω) · A}N(∆ω) = A0rect  ∆ω ∆Ω  e−  ∆ω ∆ΩN N (4.21)

In order to eliminate the steep gradients in the source spectrum, the adjusted band- width of the apodization function ∆ΩN must be chosen such that it falls to zero

within the bandwidth of the source, ∆Ω, otherwise side-lobes will still be present in the cross-correlation function. The following definition for the apodization bandwidth

requires that the function has fallen to 1% of maximum within the source bandwidth: ∆ΩN = 1₂∆Ω [−ln(0.01)]−1/N. Using this definition, the apodized source spectrum is

effectively the apodization function

˜ A′_{(ω) ≈ A} 0e−  ∆ω ∆ΩN N (4.22)

The bandwidth of the apodized light source is given by ∆ΩN, which means that lower-

order super-Gaussian apodization functions will effectively decrease the bandwidth of the light source relative to the un-apodized case. Decrease in bandwidth has ramifications for the temporal resolution of the measurement which is discussed later on in this section. To counteract this effect, the laser can be swept over a larger bandwidth, but doing so will increase the measurement time.

In Fig. 4·1(a), several apodization functions are shown including 12th_{(orange), 6}th

(yellow), and 4th _{(green) order super-Gaussians, a normal Gaussian (blue, N=2), and}

the un-apodized case (red, rect function). As the super-Gaussian order is decreased the sharpness of the corners also decreases. If GVD is ignored, then the cross- correlation function, |C(t)|2_{, is a simple Fourier transform. The cross-correlation}

function for each apodized spectra is shown on a log scale in Fig. 4·1(b). In the un-apodized case, the side-lobes are very strong. With increasing apodization the side-lobes are further suppressed, and for Gaussian apodization, the side-lobes are entirely removed.

Typically, the resolution for a system is defined in terms of the full-width at half maximum (FWHM) of the peak. For the work presented here, the goal is to resolve the signals from different modes in order to determine their relative powers. If the power in a parasitic mode is roughly half that of the desired mode, then the FWHM definition of resolution suffices. Ideally, however, parasitic modes are typically suppressed at least 10 dB relative to the dominant mode, in which case, FWHM

Figure 4·1: (a) Various apodized input spectra including Gaus- sian (blue), 4th _{(green), 6}th _{(yellow), and 12}th _{(orange) order super-}

Gaussians, and the un-apodized case (red); (b) |C(t)|2 _{for all cases}

(offset for clarity); (c) the envelopes of the functions in b) shown for t>0; the −15 dB point is marked with a dashed black line; (d) Group delay resolution as a function of MPI between a dominant and para- sitic C2 _{signal for each apodization function. Reproduced from Fig. 1}

in (Demas and Ramachandran, 2014).

is not necessarily a good definition for the temporal resolution of the system. A more in depth metric for GD resolution, which takes into account the multi-path interference (MPI) between the desired and parasitic modes, must be employed.

The envelopes for the cross-correlation functions for each of the apodized input spectra are shown in Fig. 4·1(c). Conservatively, a parasitic signal cannot be resolved unless it is above the envelope. As a result, the envelope gives the minimum MPI for which a parasitic signal can still be resolved as a function of the relative GD between

the signals. The envelopes can thus be inverted to give GD resolution as a function of MPI, as shown for each apodized spectrum in Fig. 4·1(d).

In Fig. 4·1(c), there is an evident crossing point at −15 dB between all of the envelopes of the cross-correlation functions for the various apodization functions. Above this point, the un-apodized case has the narrowest cross-correlation function, and correspondingly, the GD resolution is better than all of the other apodized spec- tra. Below −15 dB, the side-lobes of the sinc function make the un-apodized case wider than the others. In this regime, the Gaussian-apodized spectrum has the nar- rowest cross-correlation envelope, and thus the best GD delay. The crossing point defines two specific regimes: for MPI > −15 dB, the un-apodized case exhibits the best GD resolution; for MPI < −15 dB, Gaussian-apodization improves resolution. Super-Gaussian apodization of any order is a compromise between the two, with higher orders tending towards the un-apodized case. For the experiments presented in this chapter, the lowest MPI values measured were −14 dB, thus no apodization was used. However, since apodization is a post-processing technique, for the same measurement, the data can be reprocessed with different apodization functions to resolve parasitic features with any MPI with respect to the desired signal.