Interferometry has been proven to be a robust and reliable approach for the characterization of electric fields. Examples include the measurement of the temporal profile of an ultrashort pulse via spectral phase interferometry for direct electric-field reconstruction (SPIDER) [1] or wavefront measurements via lateral shearing interferometry [2]. A key advantage of those techniques is that the phase information is directly encoded in the interference pattern, which means that it can be extracted with fast and analytic reconstruction algorithms, from data with low signal resolution. Depending on the application, different variants can be implemented to measure a test field. In referencing interferometry, a known reference field is required in order to extract information about the unknown probe field [3]. On the other hand, in self-referencing interferometry, the probe field is brought to interference with a sheared copy of itself. If the shear is small enough, the measured phase difference can be interpreted as a finite difference and thus as a gradient of the phase. The importance of self-referencing methods lies in the fact that in many applications, a known reference field is not available. It is also possible to combine interferograms obtained for multiple shears to increase the accuracy and precision in the reconstruction of a wavefront [4] or of an ultrashort pulse [5].

In this Letter, we introduce the concept of mutual interferometric characterization of two electric-fields (MICE). We show that when an electric field is interfered with another field that has been sheared in a chosen dimension, it is possible to retrieve both fields via multiple-shearing interferometry. Importantly, the fields do not have to be identical replicas of each other. The only requirements are that the fields are mutually coherent and that they overlap in the degree of freedom of the fields that is associated with the shear.

Consider the interference pattern formed by two fields, $E_1(\gamma )$ and $E_2(\gamma -\mathrm{\Gamma })$, where the latter has been sheared by the amount $\mathrm{\Gamma }$ (see Fig. 1). Here $\gamma$ represents the spatiotemporal field variables $x$, $y$, $z$, and $t$ or their counterparts in the Fourier domain, $k_x$, $k_y$, $k_z$, and $\omega$. In cases where there is no coupling between these variables (i.e., no spatiotemporal structure), it is sufficient to consider each independently. In the interest of clarity, we describe the MICE procedure for such a case. The resulting interference pattern $I_{j,k}$, sampled over $J$ points and $K$ shears, then reads as the following, where $j$ and $k$ represent the sample indexes:

 

Fig. 1. Two complex fields can be retrieved from their interferogram obtained for multiple shears.

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The interferometric product $E_1({\gamma }_j)E_2^{*}({\gamma }_j-{\mathrm{\Gamma }}_k)$ gives rise to the fringe modulations and contains all the phase information. Therefore, the first step in our phase reconstruction procedure consists of isolating this component from the rest of the interferogram. As in conventional SPIDER, this is done by filtering the sideband or AC term in the Fourier domain [6]. We refer to this measured quantity as ${\mathrm{AC}}_{j,j-k}^{\text{meas}}$. The contributions of $E_1$ and $E_2$ in ${\mathrm{AC}}_{j,j-k}^{\text{meas}}$ can be separated using the method of least squares. We define the error $ϵ$ as

Minimizing $ϵ$ with respect to the two fields, $E_1$ and $E_2$, identifies those that best reproduce the measured interferogram. Setting to zero the derivative of Eq. (2) with respect to $E_1$ or $E_2$ leads to the following set of equations:

In order to solve this system of equations, a field representing $E_2$ is chosen randomly from a uniform distribution and inserted into Eq. (3). This yields a function assigned to the field $E_1$, which can then be inserted into Eq. (4). By iterating this procedure, one rapidly obtains the two fields, $E_1$ and $E_2$, which minimize $ϵ$. Each field is thus successively used as a reference to retrieve the other field. Consequently, the general MICE approach can be considered as mutually referencing interferometry.

The key requirement for this process to succeed is that the interferometric product contains redundant information. Consequently, the number of measurement points must be larger than the number of points at which the field is to be retrieved, and there must be a region of overlap between consecutive interferograms. Moreover, in the case of fields that depend on multiple variables, such as 2D wavefronts, MICE is not restricted to the case of lateral shears. The approach equally can be applied to a set of radial or rotational shears or any other known transformation that increases the redundancy of the data. Finally, we note that this process is similar to that found in generalized projections algorithms, such as used in frequency resolved optical gating (FROG) [7] or in ptychography [8].

Despite the elegance and speed of the algorithm, there remains one ambiguity in the reconstruction: $E_1(\gamma )E_2^{*}(\gamma -{\mathrm{\Gamma }}_k)=\alpha E_1(\gamma )·(1/\alpha )E_2^{*}(\gamma -{\mathrm{\Gamma }}_k)$, where $\alpha$ stands for a complex constant. This means that the absolute phase and intensity of each field remains unknown. The absolute intensity can, however, be retrieved directly from the interferogram component, ${|\alpha ·E_1|}^2+{|(1/\alpha )E_2|}^2$ (the baseband or DC term in the Fourier domain), since here the symmetry between the fields is broken.

In order to demonstrate the technique, in our first example, we perform a 2D wavefront reconstruction on simulated data [here $\gamma$ belongs to $(x,y)$]. Consider an intense laser beam with a large spatial extent that can be neither duplicated nor spatially sheared, and that we refer to as Field 1. It is interfering on a sensor with a small secondary beam, Field 2, which is aligned at an angle with respect to the first beam, Field 1. The second beam Field 2 is then scanned across Field 1 while leaving the latter untouched, see Fig. 2(a). The resulting interferograms form $256\times 256$ pixel images and are recorded for $16\times 16$ spatial positions of Field 2. After two iterations of Eqs. (3) and (4), one obtains the results shown in Fig. 2(b). The wavefronts and spatial profiles of both fields, including their relative phase, are accurately reconstructed, despite their complex shape. The small deviations from the original fields can be diminished by decreasing the shear step since this increases the redundancy in the data. Conversely, the upper limit on the shear step is set by the size of the smallest beam, Field 2, since consecutive interferograms must overlap. To test the algorithm’s performance in the presence of noise, we added to each image a fixed amount of additive noise of 30% of the maximum intensity of the most intense interferogram, as shown in the right panel of Fig. 2(a). Even in such an extreme scenario, the reconstruction is very good. This is due to the fact that the Fourier filtering and the least-squares minimization strongly suppress noise.

 

Fig. 2. (a) Interferograms for three different shears. The white dots represent the locations of the different spatial shears used for the reconstruction. The interferogram in the right panel represents the high noise case. (b) Simulated and reconstructed wavefronts of both fields. Insets: Spatial intensity profiles of the two fields. The linear term in the wavefront of Field 2 has been removed for clarity.

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In a second example, we consider the case of spectral shearing interferometry applied to the temporal characterization of ultrashort pulses. In this case $\gamma$ now represents the optical angular frequency $\omega$. To transpose MICE into the spectral domain, the ultrashort pulse must be sheared multiple times in the frequency domain. This can be done in a single-shot way using techniques known as chirped arrangement for SPIDER (CAR-SPIDER) [9] and spatially encoded arrangement (SEA) CAR-SPIDER [10]. In those implementations of SPIDER, a test pulse $E_1(\omega )$ is frequency upconverted in a nonlinear crystal with an ancillary pulse exhibiting a large spatial chirp. Thus each spatial slice of the test pulse mixes with a monochromatic portion of the ancillary pulse. The resulting pulse ${\mathcal{E}}_1(\omega -{\mathrm{\Omega }}_1)$ is a copy of the original pulse $E_1$, but its central frequency ${\mathrm{\Omega }}_1(x)$ now varies linearly across the beam. This pulse is then interfered with a second pulse ${\mathcal{E}}_2(\omega -{\mathrm{\Omega }}_2)$, which has the opposite spatial chirp. The fringe pattern is detected with an imaging spectrometer. The spatial axis of the interferogram thus maps the frequency shear between the pulses.

We can rethink the concept of CAR-SPIDER in the framework of mutually referencing interferometry. The interference patterns being obtained by interfering multiple spectrally sheared copies of ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$, it becomes possible to extract both underlying pulses via MICE. Moreover, since the full interferogram is encoded into a single spatially resolved spectrum, this measurement can be performed in a single shot.

An example of SEA-CAR-SPIDER interferogram, measured in the same experimental conditions as in [11], is shown in Fig. 3(a). MICE, and shearing interferometry in general, requires the shear to be accurately known prior to any phase reconstruction. Thus a calibration of the shear axes ${\mathrm{\Omega }}_{1,2}(x)$ may be performed by blocking one arm of the interferometer and by measuring the variation of the central frequency of each spectrum with $x$, see the red and blue dashed lines in Fig. 3(a).

 

Fig. 3. (a) MICE interferogram of two pulses; the dashed blue line and the red line stand for the axes ${\mathrm{\Omega }}_{1,2}(x)$, along which the pulses are spectrally sheared. The black dotted line indicates the shear used for the SPIDER reconstruction. (b) Moduli of the original (inset) and retrieved (main figure) interferometric products associated to the interferogram in (a) after 10 iterations of Eqs. (5)–(7). (c) Evolution of the error between the original and retrieved interferometric products.

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There is, however, a noticeable difference with the previous example depicted in Fig. 2. In the original concept of MICE, the interfering fields only undergo a shear without any other modifications. However, in the present case, the amplitude and phase properties of the monochromatic ancillary beam are transferred to the frequency sheared pulses ${\mathcal{E}}_{1,2}$ during the nonlinear process. This can be mathematically described by multiplying the upconverted fields by a complex factor depending solely on the shear, ${\mathcal{E}}_{1,2}(\omega -{\mathrm{\Omega }}_{1,2})a_{1,2}({\mathrm{\Omega }}_{1,2})$. Using the transformed variable $\omega -{\mathrm{\Omega }}_1$, the product of the two fields appearing in Eq. (1) takes the form ${\mathcal{E}}_1({\omega }_j){\mathcal{E}}_2^{*}({\omega }_j-{\mathrm{\Omega }}_k)\mathcal{A}({\mathrm{\Omega }}_k)$ where $\mathcal{A}({\mathrm{\Omega }}_k)=a_1{a}_2^{*}({\mathrm{\Omega }}_1-{\mathrm{\Omega }}_2)$.

The MICE approach easily can be adapted to this problem. To do so, we modify Eq. (2) to take into account this new model for the interferometric product, and we differentiate it with respect to the three quantities of interest. We are now left with a set of three equations:

These equations can be solved iteratively. As a starting point, one inserts two randomly chosen fields representing ${\mathcal{E}}_2$ and $\mathcal{A}$ into Eq. (5), leading to a distribution representing ${\mathcal{E}}_1$. These three fields are then updated by applying successively Eqs. (6), (7) and (5).

In addition to the absolute amplitude ambiguity $\alpha$, this extension of MICE to three fields leads to an additional ambiguity: ${\mathcal{E}}_1(\omega ){\mathcal{E}}_2^{*}(\omega -\mathrm{\Omega })\mathcal{A}(\mathrm{\Omega })={\mathcal{E}}_1{e}^{\beta \omega }·{\mathcal{E}}_2^{*}{e}^{-\beta (\omega -\mathrm{\Omega })}·\mathcal{A}{e}^{-\beta \mathrm{\Omega }}$, where $\beta$ is a complex constant. The real part of $\beta$ leads to an ambiguity on the retrieved spectral densities, which hence are known up to an exponential factor. As in the case of the ambiguity $\alpha$, this can be resolved by a separate measurement of the spectra of each arm. The imaginary part of $\beta$ defines the spectral phases up to a linear term; that is, an additional delay. However, since the reconstructed fields are only defined up to an arbitrary delay, the value of $\beta$ does not modify the precision of the estimation of the fields. Moreover, note there is no ambiguity of the relative delay between the fields.

The SEA-CAR-SPIDER interferogram in Fig. 3(a) may be processed using the MICE algorithm. After Fourier filtering, we obtain the term ${\mathrm{AC}}_{j,j-k}^{\text{meas}}$, the modulus of which is shown in the inset of Fig. 3(b). The interferometric product in the main panel of Fig. 3(b) is then retrieved after 10 iterations of Eqs. (5)–(7) and shows good agreement with the measured data. The normalized error $ϵ/{\mathrm{\Sigma }}_{j,k}{|{\mathrm{AC}}^{\text{meas}}|}^2$ between the measured and retrieved products decreases with the number of iterations, as shown in Fig. 3(c). The algorithm converges rapidly, so that the error becomes lower than 2% after only three iterations. The good agreement between the measured and retrieved AC terms, together with the small number of iterations required, highlights the relevance of Eqs. (5)–(7) to describe the measurement.

The fields ${\mathcal{E}}_1$, ${\mathcal{E}}_2$, and $\mathcal{A}$ extracted from the measurements in Figs. 3(a) and 3(b) are shown in Figs. 4(a) and 4(b). The spectra were obtained from the measurement of each arm alone averaged over 50 shots. The spectral phases are shown for 10 consecutive single-shot acquisitions. The phase of $\mathcal{A}$ appears to be purely linear. This is due to the experimental arrangement in which the fringes are spatially encoded, and the shear varies linearly along the spatial axis. As for ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$, the phases are relatively stable from shot to shot and nearly independent of frequency, indicating that the associated pulses have a duration close to the Fourier limit.

 

Fig. 4. (a) Fields ${\mathcal{E}}_1$ (dashed blue line), ${\mathcal{E}}_2$ (red line), and (b) $\mathcal{A}$ retrieved for 10 single-shot measurements. (c) Single-shot pulse profiles of the fields ${\mathcal{E}}_1$ (dashed blue line) and ${\mathcal{E}}_2$ (red line) reconstructed from Figs. 3(a) and 3(b). The black dotted lines stand for the spectral phase (a) and pulse profile (c) retrieved with SPIDER.

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Figure 4(c) shows the pulses that are simultaneously retrieved from the interferogram in Fig. 3(a). As expected, the pulses are almost identical. One can, however, notice slightly different peak intensities and a delay of $\sim 10\mathrm{fs}$ between the pulses. For comparison, we perform a SPIDER reconstruction from the data obtained for a small shear [see Fig. 3(a)]. The resulting spectral phase and pulse profile are very close to the results obtained with MICE.

Finally, the MICE technique has a number of features that make it well suited to a number of applications that are challenging for conventional FROG and SPIDER approaches. For example, MICE should allow the characterization of pulses that have large, complementary, spectral gaps. Moreover, compared to SPIDER, there is no need for a calibration of the linear phase term [1] in MICE, since no concatenation or integration of the phase is performed. The algorithm is rapid enough to allow it to be used as an online diagnostic for any pair of pulses. Mutually referencing interferometry is especially interesting when the beams are difficult to manipulate. For example, it can be challenging to duplicate and shear spatially or spectrally high-intensity laser beams, extreme ultraviolet beams, or ultra-broadband pulses. However, MICE makes it possible to transfer the shearing constraint to the more flexible secondary beam while leaving the first one untouched.

In conclusion, we have demonstrated a technique that can retrieve the electric fields of two beams simultaneously from a single interferogram of the pair. Among its main advantages are its simplicity and general applicability, since it is possible to measure a beam in any single- or multidimensional degree of freedom in which it is possible to shear one of the two fields. Applications of our method include wavefront sensing and single-shot ultrashort pulse measurement. Consequently, mutually referencing interferometry should allow the characterization of fields that were until now inaccessible for measurement through classical self-referencing shearing interferometry. Our technique is therefore well suited for the study of laser–matter interactions and opens the route toward the characterization of new light sources.