To devise an efficient freezing phase scheme, we first conduct a series of model calculations. The coherent optical pulse input into a 4-f pulse shaper is assumed to have a Gaussian profile with pulse duration of 60 fs (FWHM) and a central wavelength of 1.252 µm. The coherent pulse is assumed to be phase distorted with a phase profile of15[(i-64)/64] +7[(i-64)/64] -7[(i-40)/64] -8[(i-60)/64]2 3 4 5, where i denotes the position index of pixel.
Figure 3.3 describes the flow chart of our freezing-phase procedure. The simulation starts with grouping the SLM pixels into two classes: The first group plays the role of phase modulation. The other group is used as the reference. We vary the phase of the modulation group from 0 to 2π to generate SH modulation pattern. The spectral phase of the modulation group can be determined from the modulation pattern and is fixed to this value. The procedure is repeated by regrouping the SLM pixels until the phase retardations of all pixels are adjusted.
Fig. 3.3: Flow chart depicts the procedure used to implement the freezing-phase scheme into the theoretical and experimental studies
A.1 Freezing procedure with a left-to-right scan scheme
The intensity profile of the input pulse is shown in Fig. 3.4(b) with filled circles.
In Fig. 3.4(c) the curve with open circles denotes the corresponding distorted spectral phase profile. A phase-freezing procedure with a left-to-right scan of SLM pixels is applied to compensate for the phase distortion. Fig. 3.4(a) depicts the time course of the maximum SH intensity during the phase-freezing procedure. Note that the SH signal is normalized to that with a transform-limited pulse (TLP). We found that significant SH signal change always occurs at those pixels locating within the spectral range (i.e., 1.20 µm→1.29 µm) of the input pulse. SH intensity reaches 75% of the transform-limited value after first scan and 98% after second scan.
Fig. 3.4: (a) Time course of the maximum SH intensity during the phase-freezing procedure with a left-to-right scan scheme; (b) the intensity profiles of
transform-limited pulse (TLP), phase-distorted input pulse (DP), and
phase-compensated pulse after first, second, and third freezing phase scans; (c) the spectral phase profile of the input pulse (open circles), compensating phase profile (crosses), and error phase profiles after first (thin solid curve), second (short dashed), and third (thick solid curve) freezing phase scans.
The corresponding pulse profiles after the first (open squares), second scan (open triangles) and third scan (thick solid curve) are presented in Fig. 3.4(b). The pulse spectrum (dashed curve), the distorted phase (DP, open circles), the compensating phase (CMP, cross symbols), and the error phase obtained with a summation of DP and CMP are shown in Fig. 3.4(c). After the first scan, the root-mean-squared (rms) deviation of the error phase from a linear-phase decreases to 1.995. It can be further reduced to 0.640 by the second scan and to 0.095 by the third scan. In the best result of phase compensation only linear error phase term was left.
Note that linear phase term in the frequency domain only results in a temporal shift (see Fig. 3.3(b)) of the entire pulse. Thus with the time invariance principle, our phase retrieval process does not lose any information about pulse characteristics.
A.2 Freezing procedure with a center-to-two sides scan scheme
As shown in the previous section, SH signal was found to change significantly with phase retardances of those pixels locating within the spectral range of the input pulse. To use this finding efficiently, we design a new scan scheme which starts from the central pixel and proceeds helically toward both sides of SLM. Fig. 3.5 presents the simulation results. We found that this scan scheme is indeed more efficient than the first scheme. The SH intensity increases to 92% of the transform-limited value after the first scan and the rms deviation of the error phase decreases to 1.356. With
just two scans the rms deviation is reduced to as small as 0.374.
Fig.3.5: Time course of the maximum SH intensity during the freezing phase procedure with a center-to-two sides scan scheme; (b) the intensity profiles of transform-limited pulse (TLP), phase-distorted input pulse (DP), and
phase-compensated pulse after first, second, and third freezing phase scans; (c) the spectral phase profile of the input pulse (open circles), compensating phase profile (crosses), and error phase profiles after first (thin solid curve), and second (short dashed) freezing phase scans.
A.3 Freezing procedure with a cascading thinning-out scheme
To devise an efficient procedure with higher noise immunity, we implement a special thinning-out scheme in our phase freezing process. We used a pixel-grouping method similar to that reported by Mizoguchi, et al [11]. The pixel numbers of the reference and the modulation groups vary from 64:64 (2s), 96:32 (4s), 112:16 (8s),
120:8 (16s), 124:4 (32s), 126:2 (64s), and 127:1 (128s) with ns denoting the number of pixel segments.
The freezing process starts with 2s thinning-out scheme and ends with 128s.
Except the scheme 2s, which has only one configuration, there are n different arrangements for the scheme ns. The pixel segments cycle through SLM during each freezing stage. From the simulation result shown in Fig. 3.6(a), we found that the phase compensation almost completes at the stage 64s. The resulting pulse profiles and phase patterns after each phase compensation stage are presented in Fig. 3.6(b) and 3.6(c). Although the rms deviation of the error phase appears to be larger than that with the center-to-two sides scan scheme, this method is superior in noise immunity, which will be detailed in the next section.
Fig. 3.6: Time course of the maximum SH intensity during the freezing phase procedure with a cascading thinning-out scheme; (b) the intensity profiles of
transform-limited pulse (TLP), phase-distorted input pulse (DP), and
phase-compensated pulse after first, second, and third freezing phase scans; (c) the spectral phase profile of the input pulse (open circles), compensating phase profile (crosses), and error phase profiles after 8s (short dashed), 32s (dotted), and 128s (thick solid curve) freezing phase scans.
A.4 Noise influences on various freezing phase schemes
To properly assess the noise influences on the freezing phase schemes, we divide noise source into two terms: the first one is a multiplicative noise which could originate from the intensity fluctuation of coherent pulses. The second term is an additive noise which mainly comes from thermal noise in detection and feedback electronics. We assume each of the multiplicative noise and additive noise to be 5% of the maximum optical signal received.
By averaging ten phase compensation runs with each freezing scheme, we can obtain high-quality time course of maximum SH signal as a function of the number of freezing steps used. The results are shown in Fig. 3.7. It can be found that noise can degrade the quality of SH signal detection and therefore leads to an incorrect phase determination. We can combine N-neighboring pixels into one group to improve the accuracy of phase determination. Among the three above-mentioned methods, phase freezing with cascading thinning-out scheme appears to be the fastest method to reach the optimum solution. The slight decrease in SH intensity after 90 freezing steps is mainly caused by erroneous phase determination. This is supported by the observation that the signal level after the thinning-out scheme of 128s is close to the noise level (10%). Further phase freezing step beyond this point can not improve the result.
Therefore in real application, we shall stop the freezing procedure in time based on the noise level encountered.
Fig. 3.7: Time courses of SH signal as a function of the number of freezing steps with a variety of freezing procedures: cascading thinning-out scheme (solid curve), a center-to-two sides scan scheme (long-dashed curve), and left-to-right scan scheme (short-dashed line).