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  • Abstract

SECTION 1

INTRODUCTION

Recent years have witnessed the development [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] and application [12], [13], [14], [15], [16] of programmable vector pulse shapers that create desired temporal profiles of amplitude, phase, and polarization of an ultrashort optical pulses. This development builds on numerous applications of single-polarization pulse shaping [17], [18] and necessitates advances in the polarization state measurements of ultrashort laser pulses. Many measurements have made use of dual-polarization spectral interferometry with a well-characterized reference pulse [1], [3], [11], [19]. This technique works well, provided that precise phase stability is maintained between the reference and uncharacterized ultrashort vector-shaped pulses. We have introduced a self-referenced measurement technique called tomographic ultrafast retrieval of transverse light Formula$E$-fields (TURTLE) [20]. TURTLE determines the vector polarization state of the pulse from three or more projected pulse measurements using established single-polarization measurement techniques. The purpose of this paper is to analyze the performance of TURTLE under noisy conditions. We find that, with the correct implementation, TURTLE is a highly reliable vector pulse measurement technique, even under noisy conditions.

SECTION 2

THEORY

Vector ultrafast pulse shapers commonly operate by independently shaping orthogonal polarization components. This motivates us to write the field as the vector sum of two orthogonally polarized components, along the Formula$\mathhat{\bf x}$ and Formula$\mathhat{\bf y}$ Cartesian axes, with the beam propagating along Formula$\mathhat{\bf z}$. Assuming longitudinal field components are negligible, we determine the time-dependent electric field polarization, Formula$E(t)$ by measuring three or more orthogonal polarization projections. With dual-channel spectral interferometry, the reference field is able to relate the phase of the two orthogonal Formula$\mathhat{\bf x}$ and Formula$\mathhat{\bf y}$ polarizations. However, since self-referenced measurements are not sensitive to the absolute phase, this relative phase is not determined by two measurements. The third measurement depends on the sum of the projections of the fields from the two orthogonal polarizations. This interference leads to an additional measurement that is dependent on the relative phase between the two orthogonal polarizations. Using existing single-polarization pulse measurement techniques, the electric field for the three polarization projections can be determined within an unknown absolute time Formula$t_{0}$ and absolute phase Formula$\theta_{0}$. Additionally, such pulse measurements usually do not provide the absolute magnitude of measured electric field profiles. As a result, careful normalization is employed in the measurements, ensuring the correct relative amplitudes of the field polarization components.

The time-domain evolution of the polarization state of the field is described in the spectral domain, related by a Fourier transform. These vector field components are given by Formula$\mathtilde{E}_{x}(\Omega)$ and Formula$\mathtilde{E}_{y}(\Omega)$, where the optical angular frequency Formula$\Omega \equiv \omega - \omega_{0}$ is centered on the carrier frequency Formula$\omega_{0}$. This leads us to write the polarization-shaped pulse field as Formula TeX Source $$\mathtilde{\bf E}(\Omega) = \mathtilde{E}_{x}(\Omega)\mathhat{\bf x} + r\mathtilde{E}_{y}(\Omega)e^{-i(\Omega\tau + \theta)}\mathhat{\bf y}\eqno{\hbox{(1)}}$$ where we have defined the relative delay Formula$\tau \equiv t_{0, y} - t_{0, x}$ and relative phase Formula$\theta \equiv \theta_{0, y} - \theta_{0, x}$. The relative amplitude ratio Formula$r$ can be determined experimentally by measuring the pulse energy Formula$U$, or equivalently the average power Formula$P$, for each linear projection measurement. For example, if Formula$r_{x}$ is an arbitrary scaling factor for the Formula$\mathtilde{E}_{x}(\Omega)$ component and we measure an average power of Formula$P_{x}$, then Formula TeX Source $$P_{x} \propto \int\limits_{\infty}^{\infty}\left\vert r_{x}\mathtilde{E}_{x}(\Omega)\right\vert^{2}d\Omega = r_{x}^{2}\int\limits_{\infty}^{\infty}\left\vert\mathtilde{E}_{x}(\Omega)\right\vert^{2}d\Omega\eqno{\hbox{(2)}}$$ and so, Formula$r_{x}^{2} \propto P_{x}$, provided that we normalize the retrieved fields according to Formula$\int\vert\mathtilde{E}(\Omega) \vert^{2}d\Omega = 1$. Thus, we can find Formula$r$ in Eq. (1) from Formula$r = \sqrt{P_{y}/P_{x}}$. Using the fitting algorithm described in Section 5 below, we find, for complex pulse shapes where Formula$\mathtilde{E}_{x}(\Omega)$ and Formula$\mathtilde{E}_{y}(\Omega)$ are sufficiently dissimilar, Formula$r$ may be treated as an additional fit parameter and the pulse energy measurements are not necessary. By contrast, power measurements are indispensable for the trivial case of pure elliptical polarization [20], where the reconstructed fields are identical except for the amplitude factor.

The essence of TURTLE is then the determination of the relative delay Formula$\tau$ and phase Formula$\theta$ in Eq. (1). To determine these parameters, we measure an additional polarization projection at angle Formula$\eta$, relative to the Formula$\mathhat{\bf x}$ axis, to obtain the field Formula$\mathtilde{E}_{\eta}(\Omega)$ sketched in Fig. 1. The TURTLE algorithm is then implemented as two steps: First, we use existing single-polarization measurement techniques to establish Formula$\mathtilde{E}_{x, y}(\Omega)$, and subsequently using either an analytical or parameter-fitting approach to determine Formula$\tau$, Formula$\theta$ from the Formula$\eta$ projection.

Figure 1
Fig. 1. Schematic visualization of the TURTLE principle. Three linear polarization projection measurements of Formula$E_{x, y, \eta}$ determine the parameters Formula$\tau$, Formula$\theta$ that fully characterize the vector field Formula${\bf E}(t)$.

This projected field can be written as Formula TeX Source $$r_{\eta}\mathtilde{E}_{\eta}(\Omega)e^{-i(\Omega\tau_{\eta} + \theta_{\eta})} = \cos\eta\mathtilde{E}_{x}(\Omega) + r\sin\eta \mathtilde{E}_{y}(\Omega)e^{-i(\Omega\tau + \theta)}\eqno{\hbox{(3)}}$$ where the self-referenced measurement of Formula$\mathtilde{E}_{\eta}(\Omega)$ introduces additional unknowns in the relative delay Formula$\tau_{\eta}$ and phase Formula$\theta_{\eta}$ of the Formula$\eta$-projected component relative to Formula$\mathtilde{E}_{x}(\Omega)$. A power measurement Formula$P_{\eta}$ yields the relative amplitude coefficient Formula$r_{\eta}$, analogously to Eq. (2).

To simplify the notation in Eq. (3), we introduce the scaled fields, Formula$E_{x}(\Omega) \equiv \cos\eta\mathtilde{E}_{x}(\Omega)$; Formula$E_{y}(\Omega) \equiv r\sin\eta\mathtilde{E}y(\Omega)$; and Formula$E_{\eta}(\Omega) \equiv r_{\eta}\mathtilde{E}_{\eta}(\Omega)$. Rearranging, we obtain Formula TeX Source $$-\!i(\Omega\tau + \theta) = \ln\left[{E_{\eta}(\Omega)e^{-i(\Omega\tau_{\eta} + \theta_{\eta})} - E_{x}(\Omega) \over E_{y}(\Omega)}\right].\eqno{\hbox{(4)}}$$ Since Formula$\tau$ and Formula$\theta$ are defined as real parameters, the left side is purely imaginary, so that we may set the real part of Eq. (4) to zero. Writing the complex fields in terms of (real) amplitudes Formula$A$ and (real) spectral phases Formula$\varphi$ as Formula$E(\Omega) = A(\Omega)e^{-i\varphi(\Omega)}$ and noting that Formula$\Re \{\ln[E(\Omega)]\} = \Re\{\ln[A(\Omega)]\} - \Re\{i\varphi(\Omega)\} =\!\!$ Formula$(1/2)\ln[A^{2}(\Omega)] \!=\! (1/2)\ln\vert E(\Omega)\vert^{2}$, we find Formula TeX Source $$0 = {1 \over 2} \ln \underbrace{\left\vert E_{\eta}(\Omega)e^{-i(\Omega\tau_{\eta} + \theta_{\eta})} - E_{x}(\Omega)\right\vert^{2}}_{\equiv\Gamma} - {1 \over 2} \ln \left\vert E_{y}(\Omega)\right\vert^{2}.\eqno{\hbox{(5)}}$$ Let us define the argument of the first logarithm as Formula$\Gamma$ and expand it using the identity Formula$\vert X + iY\vert^{2} = X^{2} + Y^{2}$. This results in an expression of the form Formula$\Gamma = A_{\eta}^{2}(\Omega) + A_{x}^{2}(\Omega) - 2A_{\eta}A_{x}\{\cos[-\varphi_{\eta}(\Omega) - (\Omega\tau_{\eta} + \theta_{\eta})]\cos[-\varphi_{x}(\Omega)] \!+ \sin[- \varphi_{\eta}(\Omega) \!-\! (\Omega\tau_{\eta} \!+ \theta_{\eta})]\sin[-\varphi_{x}(\Omega)]\}$ Using the difference angle formula, this becomes Formula$\Gamma = A_{\eta}^{2}(\Omega) + A_{x}^{2}(\Omega) - 2A_{\eta}A_{x}\cos[\varphi_{x}(\Omega) - \varphi_{\eta}(\Omega) - (\Omega\tau_{\eta} + \theta_{\eta})]$. Equation (5) thus becomes Formula TeX Source $$\cos\left[\varphi_{x}(\Omega) - \varphi_{\eta}(\Omega) - (\Omega \tau_{\eta} + \theta_{\eta})\right] = {A_{\eta}^{2}(\Omega) + A_{x}^{2}(\Omega) - A_{y}^{2}(\Omega) \over 2A_{\eta}(\Omega)A_{x}(\Omega)}\eqno{\hbox{(6)}}$$ which may also be expressed as Formula TeX Source $$\Omega\tau_{\eta} + \theta_{\eta} = \varphi_{x}(\Omega) - \varphi_{\eta}(\Omega) - \cos^{-1}\left[{A_{\eta}^{2}(\Omega) + A_{x}^{2}(\Omega) - A_{y}^{2}(\Omega) \over 2A_{\eta}(\Omega)A_{x}(\Omega)}\right].\eqno{\hbox{(7)}}$$

SECTION 3

SAMPLE PULSES FOR NOISE PERFORMANCE SIMULATIONS

The key question with TURTLE is to what accuracy the algorithm is capable of retrieving the correct relative delay Formula$\tau$ and phase Formula$\theta$ values. In order to probe the quality of TURTLE reconstructions, we performed a number of simulations using idealized pulses, summarized in Table 1. In the first case, we generated 50-fs transform-limited Gaussian pulses and introduced a relative delay and phase of Formula$\tau = 140\ \hbox{fs}$ and Formula$\theta = \pi/6\ \hbox{rad}$ between the Formula$\mathtilde{E}_{x, y}(\Omega)$ polarizations. The next four pulses were obtained by chirping initially 30-fs Gaussian pulses by Formula$+300\ \hbox{fs}^{2}/\hbox{rad}$ for Formula$\mathtilde{E}_{x}$ and Formula$-500\ \hbox{fs}^{2}/\hbox{rad}$ for Formula$\mathtilde{E}_{y}$. The relative phase was kept at Formula$\theta = \pi/3\ \hbox{rad}$, and the relative delay was varied as Formula$\tau = +10, +50, 100, -75\ \hbox{fs}$. The 30-fs pulses were also chirped more strongly with −1600, Formula$+900\ \hbox{fs}^{2}/\hbox{rad}$ for Formula$\mathtilde{E}_{x, y}$ and delayed by Formula$\Delta\tau = -150\ \hbox{fs}$. Finally, we generated two arbitrary random pulses by filtering a randomly generated field with a Formula$\Delta t = 150\hbox{-}\hbox{fs}$ wide Gaussian window in the time domain and a Formula$\Delta\omega = 0.2\ \hbox{rad/fs}$ Gaussian window in the frequency domain. We used identical pulses in both the noise analysis in the analytic solution (see Section 4) and the 2-D fitting algorithm (see Section 5).

Table 1
TABLE 1 Pulses used to examine the retrieval fidelity of the analytic algorithm, with 1% Gaussian noise added to Formula$\mathtilde{E}_{x, y, \eta}(\Omega)$. We tabulate the deviations Formula$\Delta\tau$ between applied and retrieved delays Formula$\tau$, initially using only Eq. (4) and the correct values of Formula$\tau_{\eta}$, Formula$\theta_{\eta}$ and then deviations in Formula$\tau_{\eta}$ and Formula$\tau$ determined from the full analytic retrieval

Under realistic measurement conditions, noise and fluctuations in the experimentally measured data will lead to imperfections in the fields Formula$\mathtilde{E}_{x, y}(\Omega)$ determined in the first step of TURTLE. Further, since the arbitrary offsetting of local time and phase shifts within Formula$\mathtilde{E}_{x, y}(\Omega)$ directly impact the numeric values of Formula$\tau$ and Formula$\theta$, the success of TURTLE is difficult to assess in this way. In the simulations below, we supply the generating fields Formula$\mathtilde{E}_{x, y}$ to the second retrieval step so that we can directly compare target and retrieved values of Formula$\tau$ and Formula$\theta$.

SECTION 4

NOISE PROPAGATION IN THE ANALYTIC SOLUTION

In the above derivation, we require only the amplitudes and spectral phases of each of the fields Formula$\mathtilde{E}_{x, y, \eta}(\Omega)$ to be known. These quantities can be determined for a single-polarization component using existing techniques, including SPIDER [22], FROG [23], or MIIPS [24]. Equation (7) and the imaginary part of Eq. (4) are in the form of a straight line over optical frequency Formula$\Omega$, with offsets Formula$\theta$, Formula$\theta_{\eta}$, and slopes Formula$\tau$, Formula$\tau_{\eta}$. Commonly, the complex fields Formula$E(\Omega)$ will be numerically sampled at discrete frequencies Formula$\Omega_{i}$, so that each equation naturally lends itself to solution by linear regression [25]. The parameters Formula$\tau$, Formula$\tau_{\eta}$, Formula$\theta$, Formula$\theta_{\eta}$ can be calculated from the measured data samples analytically.

We plot the straight-line data obtained from Eq. (7) in the lower half-panels in Fig. 2. From the figure, we see the right-hand side (RHS, red dashed line) follows the linear regression fit (solid blue) only over certain regions (heavy black). In other regions, the ambiguity Formula$\cos(\theta) = \cos(\pi - \theta)$ causes the RHS to deviate so these regions must be omitted from the fit. The RHS of Eq. (6) is shown in the upper half-panels and is found to closely follow the left-hand side so that Eq. (6) is satisfied over the whole spectral range Formula$\Omega$. This is true even for noisy fields, as shown in Fig. 2(b), where the fields Formula$\mathtilde{E}_{x, y, \eta}(\Omega)$ have been perturbed by 1% Gaussian noise.

Figure 2
Fig. 2. Examples of (red dashed) (left-hand side) and (blue solid) (right-hand side) of (upper half-panels) Eq. (6) and (lower half-panels) Eq. (7). (a) Chirped pulse #6. (b) Pulse #4 with 1% Gaussian noise added.

The cosine term in Eq. (6) arises due to the spectral beating of the two fields Formula$\mathtilde{E}_{x, y}(\Omega)$ projected to be co-polarized, analogous to interference fringes observed in spectral interferometry [21]. If Formula$\mathtilde{E}_{x}(\Omega)$ and Formula$\mathtilde{E}_{y}(\Omega)$ have matching higher order spectral phases, then the periodicity of these fringes immediately yields Formula$\tau$, and the phase at Formula$t = 0$ is determined by Formula$\theta$. This can be seen comparing Fig. 2(a) and (b), where the delays of Formula$\vert \tau\vert = 150$, 100 fs, respectively, are inversely proportional to the observed fringe spacings in both the spectral amplitude and the cosine term.

The parameters Formula$\tau_{\eta}$, Formula$\theta_{\eta}$ are calculated using contiguous regions where Eq. (7) holds. The region is determined by finding a range of Formula$\Omega$ over which the argument of the inverse cosine is within the range 0.1–0.95. Once these parameters have been found, we calculate the parameters Formula$\tau$, Formula$\theta$ from Eq. (4), again using linear regression. Results from these regression calculations are presented in Table 1 for each of the pulses with 1% Gaussian amplitude noise added to Formula$\mathtilde{E}_{x, y, \eta}$. The table lists deviations between the target and retrieved values of each of the parameters. The first two columns of Retrieved Values show deviations for Formula$\tau$ and Formula$\theta$ measured when the algorithm is seeded with the correct values of Formula$\tau_{\eta}$, Formula$\theta_{\eta}$ used in the simulations. In a real measurement, the values for Formula$\tau_{\eta}$, Formula$\theta_{\eta}$ must first be determined via Eq. (7). The Full Analytic Retrieval section of Table 1 lists deviations in the fitted Formula$\tau_{\eta}$, Formula$\theta_{\eta}$, and finally the deviations in Formula$\tau$, Formula$\theta$ obtained using these values. Inspection of the table indicates that even small amounts of noise in the field measurements deteriorates the retrieved values and renders the analytic approach unreliable in practice.

SECTION 5

NOISE PROPAGATION IN 2-D TURTLE PARAMETER FITTING

Several single-polarization ultrashort pulse characterization techniques measure overdetermined, 2-D data traces. We are therefore motivated to propose instead a multiparameter fit that uses the whole trace. We demonstrated in our initial TURTLE measurements [8] that the parameter-fitting approach on 2-D pulse measurement data provides substantially more reliable vector pulse measurements than the analytic approach described above. These measurement techniques include second-harmonic generation frequency-resolved optical gating (SHG FROG) [23], [26], [27], [28] and SHG multiphoton intrapulse interference phase scan (SHG MIIPS) [24], [29]. In the first step of TURTLE, the pulse fields Formula$\mathtilde{E}_{x, y}(\Omega)$ are retrieved with high accuracy from traces of Formula$\mathhat{\bf x}$ and Formula$\mathhat{\bf y}$ polarization components using the established reconstruction technique. We demonstrate below that additional redundancy in the trace measured for the Formula$\eta$ polarization allows accurate values for Formula$\tau$ and Formula$\theta$ to be obtained by fitting to the data trace. There is no need to determine the values of Formula$\tau_{\eta}$ and Formula$\theta_{\eta}$ explicitly.

The technique calculates 2-D data traces of the Formula$\eta$-projection for trial values of Formula$\tau$, Formula$\theta$, and compares these to the measured traces. We quantify differences between the measured and calculated trial Formula$I_{\eta}^{\rm calc}$ traces by the root-mean-square (RMS) deviation Formula TeX Source $$e = {\sqrt{\sum_{i, j = 1}^{N}\left[I_{\eta}^{\rm meas}(i, j) - I_{\eta}^{\rm meas}(i, j)\right]^{2}} \over \sum_{i, j = 1}^{N}I_{\eta}^{\rm calc}(i, j)\sum_{i. j = 1}^{N}I_{\eta}^{\rm meas}(i, j)}\eqno{\hbox{(8)}}$$ where the denominator ensures that the fitness function Formula$e(\tau, \theta)$ is appropriately scaled. The correct relative delay and phase between Formula$\mathtilde{E}_{x}(\Omega)$ and Formula$\mathtilde{E}_{y}(\Omega)$ are determined by finding those values of Formula$\tau$, Formula$\theta$ that minimize Formula$e$.

The fitness function Formula$e(\tau, \theta)$ in Eq. (8) takes the form of an infinite cylinder: The phase Formula$\theta$ wraps at Formula$\pm\pi$, while the delay Formula$\tau$ is unbounded positive or negative. In practice, an appropriate range of values for Formula$\tau$ are estimated based on the time overlap between the individual durations of Formula$E_{x, y}(t)$. As we show below, the fitness function Formula$e(\tau, \theta)$ can have a complex structure with many local minima, presenting a challenge to many search algorithms. We simplify the search by evaluating Formula$e(\tau, \theta)$ over the bounded range, and seed the minimization routine at the minimum sampled on this grid. With this approach, sophisticated search algorithms are unnecessary; we find the Nelder–Mead simplex routine [30], implemented as Formula${\tt fminsearch}$ in Matlab, is adequate despite the algorithm's known susceptibility to becoming trapped in local minima. We can further improve the robustness of the parameter-fitting algorithm by adding fitness metrics for additional projected measurements into Eq. (8).

1. SHG FROG Trace

A SHG FROG trace is measured experimentally by splitting the pulse to be measured into two replicas and recording the SHG spectrum while the delay Formula$T$ between the replicas is varied. The FROG trace for a single, linearly polarized pulse temporal field Formula$E(t)$ measured by SHG FROG is [23] Formula TeX Source $$I_{\rm FROG}(\omega, T) = \left\vert\int \limits_{-\infty}^{\infty}E(t)E(t - T)e^{-i\omega t}dt\ \right\vert^{2}.\eqno{\hbox{(9)}}$$ Since Formula$I(\omega)$ is the Fourier transformation of the product of both field replicas, it will be centered near Formula$2\omega_{0}$, the second harmonic of the fundamental pulse central frequency. An iterative reconstruction algorithm is used to find the full intensity and spectral phase of the pulse [31], [32] with high accuracy. In practice, the FROG trace is an Formula$N \times N$ matrix Formula$I_{\eta}^{\rm calc}(i, j)$, understood to represent the trace of Eq. (9) sampled at regular intervals over a frequency axis Formula$\omega_{i}$ and delay axis Formula$T_{j}$.

Since TURTLE relies on single-polarization measurements, it inherits any ambiguities in the underlying pulse characterization techniques. The form of the second-order nonlinearity that gives rise to the SHG FROG, i.e., Eq. (9), introduces ambiguities in the reconstructed fields Formula$\mathtilde{E}_{x, y}(\Omega)$. These ambiguities are well documented and can be alleviated with an additional measurement or a modification of the measurement apparatus. The ambiguity most significantly impacting TURTLE is that the direction of time cannot be determined from a single SHG FROG measurement [33]. Therefore, one cannot experimentally distinguish between Formula$\mathtilde{E}(\Omega)$ and its conjugate, Formula$\mathtilde{E}^{\ast}(\Omega)$. TURTLE introduces an additional complication in SHG FROG because we must determine the relative direction of time between the orthogonal polarization components. This relative ambiguity can be resolved by calculating the fitness function surface Formula$e(\tau, \theta)$ twice, once for the combination Formula$\mathtilde{E}_{x}(\Omega)$, Formula$\mathtilde{E}_{y}(\Omega)$, and once for Formula$\mathtilde{E}_{x}(\Omega)$, Formula$\mathtilde{E}^{\ast}_{y}(\Omega)$. The correct relative time leads to a lower fit function value—even in the presence of measurement noise. As a result, SHG FROG TURTLE making use of 2-D parameter fitting correctly distinguishes the relative direction of time between two independent measurements of orthogonal field components.

The fitness function surface Eq. (8) for pulse #1 described in Table 1 is shown in Fig. 3(a). Since the constituent fields are transform limited, the surface is symmetric in the Formula$\tau$-axis; indeed, both values Formula$\pm\tau$ retrieve the same vector field. As shown below, applying quadratic spectral phases to Formula$\mathtilde{E}_{x, y}(\Omega)$ results in asymmetric surfaces, from which the correct sign of Formula$\tau$ can be determined.

Figure 3
Fig. 3. Fitness function surface Formula$e(\tau, \theta)$ for pulse #1 by fitting (a) SHG FROG and (b) SHG MIIPS trace. Dark blue colors indicate lower Formula$e$.

2. SHG MIIPS Trace

Measuring a MIIPS trace [24] involves adding calibrated, known spectral phases Formula$f(\Omega, p)$ to the pulse to be measured and recording SHG spectra as a function of the reference parameter Formula$p$. When the added quadratic spectral phase chirp Formula$[\partial^{2}/\partial \Omega^{2}]f(\Omega, p)$ cancels that of the pulse to be measured, Formula$[d^{2}/d\Omega^{2}]\varphi(\Omega)$, a stronger SHG signal is obtained at the optical frequency where the cancellation occurs. For a specific optical frequency of interest Formula$\Omega^{\prime}$, the quadratic phase can be found according to Formula TeX Source $$\left.{d^{2}\varphi(\Omega) \over d\Omega^{2}}\right\vert_{\Omega = \Omega^{\prime}} = \left.{\partial^{2}f(\Omega, p_{\rm max}) \over \partial\Omega^{2}}\right\vert_{\Omega = \Omega^{\prime}}\eqno{\hbox{(10)}}$$ where Formula$p_{\max}$ is the parameter value that maximizes the SHG signal at Formula$\Omega^{\prime}$. A double integration with respect to Formula$\Omega$ retrieves the full spectral phase. In our simulations, we used parabolic reference functions Formula$[\partial^{2}/\partial\Omega^{2}]f(\Omega, p) = p$, with Formula$p$ determining the quadratic chirp so that Formula$f(\Omega, p) = (p/2)\Omega^{2}$.

The MIIPS measurement technique is particularly suited to TURTLE when the polarization-shaped pulses are generated by a SLM pulse shaper. For accurate pulse measurements, the pulse shaper must be well calibrated. A MIIPS trace fitness function surface is shown in Fig. 3(b) for pulse #1. Since the sign of the applied phase is known, the TURTLE fitness surface determined with a MIIPS measurement does not exhibit a Formula$\pi$-rad phase ambiguity of SHG FROG.

3. Retrieval Fidelity

We applied the parameter-fitting method of TURTLE for each of the pulse simulations described in Section 3 and listed in Table 1. In each case, we calculated a 256 × 256 point SHG FROG or SHG MIIPS trace for the Formula$\mathtilde{E}_{\eta}(\Omega)$ projection as defined in Eq. (3). As described earlier, we used the generating fields Formula$\mathtilde{E}_{x, y}(\Omega)$ directly in the Nelder–Mead minimization search algorithm. Arbitrarily, we used Formula$r = 1$ and Formula$\eta = 45^{\circ}$.

In order to ensure that the simplex minimization finds the appropriate global minimum of the fitness function surface of Eq. (8), we first evaluated the TURTLE fitness surface over the ranges Formula$-200 \leq \tau \leq 200\ \hbox{fs}$ and Formula$-\pi \leq \theta \leq \pi\ \hbox{rad}$. Normalized surfaces, calculated from SHG FROG traces, are shown in Fig. 4 for a selection of the sample pulses. The surfaces ranged from simple with high symmetry as in the case of transform-limited pulse #1 to highly structured for the randomly generated pulses. Fig. 4(a)(c) show the progression of fitness surfaces for the set of pulses comprised of two identical Gaussian components as their relative delay Formula$\tau$ is varied from 10 to 150 fs. The increasing complexity of the 45°-projected FROG trace introduces more structure into the fitness surface and leads to tighter minima at Formula$\tau$ and Formula$\theta$. The fitness function trace for random pulse #8 is shown in Fig. 4(d). The structure of each surface depends on the components Formula$\mathtilde{E}_{x, y}(\Omega)$ and their Formula$\eta$-projection, but in each case the global minimum is easily distinguished. The increased specificity in Formula$\mathtilde{E}_{x, y}(\Omega)$ of the random pulse makes the minimum more confined than those in Fig. 4(a)(c).

Figure 4
Fig. 4. Fitness function surfaces Formula$e(\tau, \theta)$ by fitting to SHG FROG traces. Darker blue colors indicate lower values of deviation Formula$e$. The target Formula$\tau$, Formula$\theta$ is indicated by the crosshairs.

In order to assess the impact of noise in the TURTLE algorithm, we added increasing fractional Gaussian noise to the Formula$\eta$-projected SHG FROG trace, and clamped negative intensity values to zero. We ran the TURTLE parameter-fitting retrieval algorithm, giving it the initial Formula$\mathtilde{E}_{x, y}(\Omega)$ fields, the projected FROG trace Formula$I_{\eta}^{\rm FROG}$ with added noise, as well as Formula$r = 1$ and Formula$\eta = 45^{\circ}$. We then compared the optimized Formula$\tau_{\rm opt}$, Formula$\theta_{\rm opt}$ retrieved by the TURTLE algorithm, to the initial Formula$\tau$, Formula$\theta$ applied to Formula$\mathtilde{E}_{x, y}(\Omega)$. Without noise, the retrieved values were within numerical accuracy of the applied values, confirming that the deviation metric surface Formula$e(\tau, \theta)$ has a minimum at the desired Formula$\tau$, Formula$\theta$ values. We plot in Fig. 5 the RMS deviation calculated for the set of eight trial pulses, as a function of the added noise fraction Formula$\alpha$. The deviations between retrieved and the applied relative delay and phase are listed in Table 2 for Formula$\alpha = 20\%$ noise.

Figure 5
Fig. 5. RMS deviations for the set of pulses shown in Table 1. (a) Deviations in retrieved delay Formula$\tau_{\rm opt}$ as compared with target value Formula$\tau$. (b) Deviations in retrieved phase Formula$\theta_{\rm opt}$. The deviations are plotted as a function of the added noise fraction Formula$\alpha$. Solid lines show linear regression fits to the data up to 20% noise, with slopes Formula$\tau_{\rm RMS} = 0.90$, 0.57 fs/20% for (blue diamonds) SHG FROG and (red circles) SHG MIIPS, respectively.
Table 2
TABLE 2 Deviations between applied delay Formula$\tau$ and phase Formula$\theta$, measured with SHG FROG and SHG MIIPS for the different pulse types listed in Table 1. 20% Gaussian amplitude noise has been added to the projected traces at Formula$\eta = 45^{\circ}$

From the data in Fig. 5, we performed a linear regression fit for the RMS values up to 20% noise and found that slopes Formula$\tau_{\rm RMS} = 0.91\ \hbox{fs}/20\%$ and Formula$\theta_{\rm RMS} = 0.014\pi\ \hbox{rad}/20 \%$. Experimental FROG measurements will usually have substantially less than 20% noise, so that the TURTLE reconstructions are expected to be better than ±1 fs and ±0.04 rad.

SECTION 6

CONCLUSION

In this paper, we have shown that while the analytic inversion algorithm of TURTLE is not robust to noise, 2-D parameter fitting of TURTLE data with SHG FROG and MIIPS enables reliable and robust characterization of ultrafast vector laser pulses. With 2-D TURTLE, we find that for < 5% measurement noise, that the relative delay and phase that TURTLE extracts are found with a accuracy better that 0.2-fs and 10 mrad, respectively, for those pulses tested. While this is not an exhaustive search, it is likely that TURTLE can be used with confidence for the characterization of the temporal polarization evolution of ultrafast laser pulses.

ACKNOWLEDGMENT

The authors are grateful to R. Trebino and L. Xu at Georgia Institute of Technology for useful discussions.

Footnotes

First published Online. This research was supported in part by the Monfort Family Foundation and the National Science Foundation under Grant ECCS-0348068. Corresponding author: R. A. Bartels (e-mail: bartels@engr.colostate.edu).

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