1. Introduction

Temporal characterization of the electric field of an ultrashort pulse can now be performed with a great variety of diagnostic techniques. The well-known techniques, such as Frequency-Resolved Optical Gating (FROG) [1] and Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER) [2], have been used to characterize laser beams with conventional polarization (linear, circular, and elliptical). Laser beams with axially symmetric (radial and azimuthal) polarization [3] have unique properties which make them attractive for several applications. For example, focused radially polarized laser beams can produce intense axial electric fields, which could be used for direct laser electron acceleration [4]. Few methods for characterization of laser beams with axially symmetric polarization have been reported to date [5], particularly for short laser pulses.

A technique known as POLarization Labeled Interference versus Wavelength of Only a Glint (POLLIWOG) [6] has been used for measuring polarization shaped pulses in the past. It uses a polarization beamsplitter to split the signal pulse into orthogonal linearly polarized components, which are subsequently measured using Spectral Interferometry (SI). POLLIWOG has the advantage of being fast and sensitive, but it suffers from the loss of spectral resolution and temporal range due to a large time delay between interfering pulses required to retrieve the phase from its interferogram, especially for longer pulses [7]. A similar approach for measuring polarization shaped pulses called Tomographic Ultrafast Retrieval of Transverse Light E-fields (TURTLE) [8] uses several FROG measurements to determine the complete polarization state of the pulse. This technique has the disadvantage of being slow. A significant progress in the Spatial Spectral Interferometry (SSI) technique [9, 10] has been reported over the past 15 years [11, 12]. A modified version of SSI called Spatially Encoded Arrangement for Temporal Analysis by Dispersing a Pair Of Light E-fields (SEA TADPOLE) [13], can be used to extract the spectral phase difference between the signal pulse and the reference pulse. It lacks the capability to simultaneously measure the polarization state of a pulse.

We present a novel technique that extends SEA TADPOLE to an angularly multiplexed geometry, in which the relative spectral phase and the polarization state of the signal pulse can be simultaneously measured. We experimentally demonstrate that this approach can be used to characterize conventional polarization states, such as linear and elliptical polarization, but also more complex, spatially variable polarization states, such as radial polarization.

2. Description of the method

A signal beam with an arbitrary polarization is sent to the arrangement illustrated in Fig. 1(a), in which two orthogonally polarized reference beams are used. The signal pulse is temporally overlapped with the two reference pulses incident at two small angles (θ₁ > θ₂), producing interleaved fringes in the spatial (vertical) axis of the spectrometer image plane, as shown in Fig. 2(a). The spectral phase-retrieval algorithm is similar to the SEA TADPOLE Fourier-filtering algorithm [13]. Two reference beams are used instead of one and the polarization state of the signal pulse is subsequently measured, together with its spectral phase. The retrieval process is illustrated in Fig. 2 and summarized as follows. A one-dimensional Fourier transform of the interleaved spectral interferogram is performed along the spatial (vertical) axis, yielding four sidebands. These sidebands can be filtered and inverse-Fourier transformed back to the spatial frequency domain. The resulting product of the interfering fields is divided by the spectral field of the reference pulse to obtain the spectral field of the signal and therefore the spectral phase difference between the signal and the corresponding reference pulses. Those two spectral phase differences are subsequently subtracted and used in conjunction with the separately measured phase difference between the two reference pulses. The relative spectral phase between the two polarization components of the signal pulse is calculated by using these phase differences. The spectral fields corresponding to two polarization components of the signal beam and their relative spectral phase difference can be used to determine the complete polarization state of the measured signal pulse.

 

Fig. 1 (a) Principle of the angle-multiplexed spatial-spectral interferometric technique. After being dispersed by a grating, two reference pulses and the signal pulse are vertically multiplexed on a cylindrical mirror and reflected at small angles θ₁ and θ₂. A CCD camera is used to record the 2D interference pattern produced at the focus of the cylindrical mirror. (b) The orientation angle ψ(ω) and the ellipticity angle χ(ω) of the polarization ellipse

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Fig. 2 The spectral phase and polarization ellipse retrieval algorithm using simulated data. (a) Simulated interferogram corresponding to two orthogonal reference pulses and a signal pulse. The reference pulses have a group delay dispersion (GDD) of −100 fs² and −30 fs² and a third-order dispersion (TOD) of −250 fs³ and −50 fs³, respectively. The signal pulse has a GDD of 2000 fs² and a TOD of 1200 fs³. The phase shift δ between the two polarization components of the signal pulse varies between −π to π ( Media 1). (b) The 1D Fourier transform of this interferogram along x-dimension ( Media 2). (c) In the k_x-domain, either the top or bottom sidebands were filtered out. (d) and (e) The extracted sidebands were inverse-Fourier transformed back to the x-domain. (f) and (g) The resulting product of the interfering fields is divided by the spectral field of the reference pulse to obtain the spectral field and phase difference ${\varphi }_i^s(\omega )-{\varphi }_i^r(\omega )$ of the corresponding polarization components of the signal and reference pulses. (h) The spectral phase difference δ(ω) between the two polarization components of the signal pulse was calculated by taking the sum of the extracted value of Δϕ(ω) and Δϕ^r(ω) ( Media 3). (i) The polarization ellipse parameters at central wavelength (λ_⚬ = 800 nm) was retrieved by using the extracted spectral fields $E_1^s(\omega )$, $E_2^s(\omega )$ and spectral phase difference δ(ω) ( Media 4).

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Consider the geometry of Fig. 1(a), where the electric field of an arbitrary polarized signal pulse E⃗^s at any coordinate r⃗ is the sum of two orthogonal polarization components with magnitudes $E_1^s(\omega )\text{exp}\left[i({\overrightarrow{k}}_1^s\cdot \overrightarrow{r}+{\varphi }_1^s)\right]$ and $E_2^s(\omega )\text{exp}\left[i({\overrightarrow{k}}_2^s\cdot \overrightarrow{r}+{\varphi }_2^s)\right]$, where $E_1^s(\omega )$ and $E_2^s(\omega )$ are their amplitudes, k⃗ is the wave number, x is the vertical component of r⃗, and ϕ is the spectral phase. The reference pulses are defined as $E_1^r(\omega )\text{exp}\left[i({\overrightarrow{k}}_1^r\cdot \overrightarrow{r}+{\varphi }_1^r)\right]$ and $E_2^r(\omega )\text{exp}\left[i({\overrightarrow{k}}_2^r\cdot \overrightarrow{r}+{\varphi }_2^r)\right]$. The interferogram I(ω, x) is calculated by taking the sum of ${\left|{\overrightarrow{E}}_i^r+E_i^s\right|}^2$ and has the form

The complete polarization state of the signal pulse can be represented by an ellipse shown in Fig. 1(b), with amplitudes $E_1^s(\omega )$ and $E_2^s(\omega )$, ellipticity angle χ(ω), and orientation angle ψ(ω). The ellipticity angle and the orientation angle are determined by

3. Experimental setup

Fig. 3 shows the experimental test setup with a total of three arms used for propagating the signal beam and two reference beams to achieve angle multiplexing in the vertical (x–z) plane. The setup includes an integrated spatially resolved spectrometer consisting of a diffraction grating (GR), a cylindrical mirror (CM), and a CCD camera. The spectrometer was calibrated using a commercial calibrated spectrometer with a spectral resolution of 0.5 nm. The beams interfere at the focal plane of the cylindrical mirror, where the CCD camera is located. The half-crossing angles (θ₁ and θ₂) between the signal and two reference spectra are ∼0.07° and ∼0.21°, respectively. A diffraction grating with a 600 mm⁻¹ groove density and a cylindrical mirror with a focal length of 50 mm were used to accommodate a bandwidth of ∼70 nm over the CCD width. The CCD has a resolution of 1280×960 pixels with a pixel size of 3.75 μm and 12-bit digital output. The laser source for the experiment was a mode-locked Ti:sapphire oscillator with a center wavelength of 800 nm and FWHM bandwidth of ∼30 nm. The laser beam was split into three by two beam splitters; the first reflected beam served as the signal and the other two as references. The signal beam is rotated to an arbitrary polarization state by a zero-order wave plate (λ/4 or λ/2) and subsequently interferes with two orthogonal reference beams at zero delay. The spectral intensities of all three beams are set to be approximately equal to each other during the experiment in order to achieve a high fringe contrast in their corresponding interferograms and to reduce errors in the reconstruction process. The interferogram is recorded with a LabView software controlled data acquisition system and subsequently analyzed using the described reconstruction method.

 

Fig. 3 A schematic illustrates the experimental geometry used for angle-multiplexed spatial-spectral interferometry technique. M, mirror; SM, spherical mirror. Polarization of the signal beam is rotated to an arbitrary polarization state by a zero-order wave plate (λ/4 or λ/2) after reflection from the beamsplitter BS₁. Two zero-order half-wave plates (λ/2) after two Glan-Taylor polarizers (LP₁ and LP₂) with an extinction ratio of ∼ 1 × 10⁻⁶ keep the polarization of two reference pulses highly orthogonal. Two delay lines (Delay1, Delay2) equipped with two high precision translation stages provide correct timing between signal pulse and two reference pulses. Three beams are vertically multiplexed at zero delay and collimated before being spectrally dispersed on the CCD using the grating (GR) and the cylindrical mirror (CM).

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4. System calibration

As pointed out in previous studies of other interferometric techniques [2, 13], accurate calibration of the system is paramount. The setup should further exhibit low jitter and low calibration drift on the time scale of the measurement. Since this approach involves two reference beams and a signal beam, any non-zero temporal overlap of the three pulses could introduce error in the measurement of δ(ω). A non-zero delay τ between pulses results in the tilt of interference fringes along the horizontal axis of the CCD, as shown in Fig. 4. In our experiment, we used a signal beam linearly polarized at 45° and subsequently interfering with two orthogonal reference beams. We deliberately moved one of the delay stages in a reference beamline by equal distances in opposite directions from its reference, zero-delay (τ = 0) position, while the other delay stage was kept at a zero-delay position with respect to the signal pulse. We subsequently recorded the corresponding interferograms for these cases and compared them, as shown in Figs. 4(a)–4(c). The spectral phase shifts δ(ω) that correspond to those cases are extracted using Eq. (4) and are shown in Figs. 4(d)–4(f). Figure 4(e) shows a small parabolic phase which is due to small chirp caused by a beam splitter in the system. Figures 4(d) and 4(f) show curves that are similar to linear spectral phase which is introduced by a temporal delay in one of the reference arms.

 

Fig. 4 Experimental interferograms with three different temporal delays τ between the two reference pulses: (a) τ < 0, (b) τ = 0, (c) τ > 0. The extracted spectral phase shift δ(ω) corresponding to respective temporal delays of: (d) τ < 0, (e) τ = 0, (f) τ > 0.

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The calibration first involves setting the zero delay position so that the fringes are horizontal. Second, the signal arm is blocked and the polarization of one reference arm is rotated to match the other, so that the system is reduced to a standard two-arm interferometer. Measuring the spectral phase difference between the two reference arms and subtracting out this phase from all subsequent measurements eliminates the spectral phase introduced by the setup itself.

In practice, a small change of the optical path lengths in the three arms of the interferometer can be manifested both as a fast jitter and a slow phase drift. The main causes of the jitter and drift are mechanical vibrations, air current-induced and temperature-dependent refractive index changes and mechanical expansion, and the finite beam pointing stability. We built an acrylic box around our setup and a tight isolation of the system against the air currents in the lab has been established. The relative spectral phase Δϕ^r(ω) between the two reference arms is needed for obtaining the polarization measurement of the signal pulse. In our experiment, both polarizations are rotated to be the same and the signal beam is blocked; we subsequently obtain the interferograms corresponding to two reference pulses. The resulting reconstruction of the interferogram yields Δϕ^r(ω) and is used in Eq. (4) to extract the relative spectral phase shift δ(ω) of the two polarization components of the signal pulse. The drift of Δϕ^r(ω) is monitored on short and longer time scales by continuous recording of the interferograms corresponding to two reference arms. Figure 5(a) shows the mean-value subtracted zeroth-order spectral phase ${\delta }_{⚬}^r=\mathrm{\Delta }{\varphi }^r({\lambda }_{⚬})$ that was monitored over one-minute period. In our measurement, the RMS variation of ${\delta }_{⚬}^r$ was 0.06 rad over one minute, while the RMS variations in corresponding ellipse parameters $\mathrm{\Delta }{\chi }_{⚬}^r$ and $\mathrm{\Delta }{\psi }_{⚬}^r$ were 0.03 rad and 3.95 mrad, respectively (Figs. 5(b) and 5(c)). The corresponding RMS delay fluctuations ( ${\delta }_{⚬}^r/{\omega }_{⚬}$) for this case are calculated to be 0.02 fs. Figure 5(d) shows the mean value subtracted zeroth-order spectral phase ${\delta }_{⚬}^r=\mathrm{\Delta }{\varphi }^r({\lambda }_{⚬})$ that was monitored over a longer, 20-minute period. In our measurement, the variation of ${\delta }_{⚬}^r$ was ∼0.6 rad over 20 minutes, (Figs. 5(e) and 5(f)). The corresponding delay drift for this case is calculated to be ∼0.25 fs. This drift does not significantly affect the higher orders of the spectral phase (such as group delay dispersion and third-order dispersion), as previously pointed out in related spectral interferometry measurements [13, 14], and discussed further in Section 6.

 

Fig. 5 Effect of measured fast phase jitter: (a) phase drift corresponding to the relative phase difference between the two reference arms; drifts in corresponding polarization ellipse parameters, such as (b) ellipticity angle $\mathrm{\Delta }{\chi }_{⚬}^r$ and (c) orientation angle $\mathrm{\Delta }{\psi }_{⚬}^r$. The corresponding drifts for those quantities is also measured over longer time scales and shown in (d)–(f).

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In our experiments, a measurement is performed over a relatively short period (less than a minute) following the calibration of the phase between the two reference beams. As a result, the effect of slow drift is not significant and only the fast jitter affects the accuracy of the measurement. If a more conservative, peak-to-peak criterion is used to quantify the fast phase jitter shown in Fig. 5(a) and we slightly overestimate it to be π/16, this would be consistent with a fast time jitter of 0.085 fs. Thus with this level of short-time jitter one would expect to be able to perform measurements with ∼10% accuracy over bandwidths > 300 nm, provided that the measurements are performed within a time period of ∼5 minutes following the calibration.

It is important to take into account the differences in transmission of the two orthogonal polarizations through the system, arising primarily from the dependence of the diffraction grating efficiency on polarization. Our measurement shows that the ratio of the grating efficiencies for S and P polarization components is 0.69, which is accounted for in the reconstruction. The grating we use is coated for high efficiency at the central wavelength of 800 nm and its spectral response over the bandwidth of the beam used in this measurement is nearly constant. Therefore, no consideration of the spectrally dependent efficiency was needed in our experiment.

5. Elliptically polarized pulse measurement

The ability to perform simultaneous measurement of the relative spectral phase shift and polarization is first validated by characterizing an elliptically polarized pulse. A signal beam linearly polarized at 45° is generated by a Glan-Taylor polarizer and sent to a zero-order quarter-wave (λ/4) plate in the signal arm of the interferometer. The fast axis of the quarter-wave plate is referenced to the transmission axis of the Glan-Taylor polarizer and subsequently rotated to a known angle. When an axis of the quarter-wave plate and the polarizer are parallel, the interferogram produced by the signal beam and the two orthogonal reference beams is recorded and the extracted phase shift δ(ω) is used as a calibration phase Δϕ^r(ω) for this measurement. Successive rotations of the quarter-wave plate (taken over a time interval of less than a minute) create a set of signal pulses with various δ(ω). The extracted phase differences Δϕ(ω) from multiple measurements are summed with the calibration phase Δϕ^r(ω), resulting in the relative phase shift δ(ω) for each of the quarter-wave plate positions. Figure 6(a) shows the measured relative spectral phase shifts δ(ω) produced in this way. The measured spectral phase shifts δ(ω) show a good agreement with their expected values.

 

Fig. 6 (a) Extracted relative phase difference between the two components of the polarization state created by the quarter-wave plate; (b) polarization ellipse parameters δ_⚬, χ_⚬ and ψ_⚬ at the central wavelength (λ_⚬), corresponding to various polarization states set by the quarter-wave plate. Solid lines represent the calculated values for these parameters.

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Figure 6(b) shows the values of the reconstructed δ_⚬ and the polarization ellipse parameters χ_⚬ and ψ_⚬ at the central wavelength (λ_⚬ = 800 nm), corresponding to each angle set by the quarter-wave plate. Our measurement indicates that χ_⚬ varies linearly between ±π/4, while ψ_⚬ varies between 0 and π/2.

6. Measurement of a linearly polarized pulse with dispersion

The spectral phase of the signal beam is measured by a two-step procedure. First, the interferogram produced by the setup with no additional dispersion introduced is recorded and the corresponding spectral phase shift is retrieved. Second, the interferogram with additional dispersion introduced into the signal beam is recorded. The two measured spectral phases are subtracted to obtain the dispersion introduced into the signal beam. A typical experimental interferogram of a linearly polarized signal pulse with dispersion introduced is shown in Fig. 7(a).

 

Fig. 7 (a) Experimental spectral interferogram of a signal pulse linearly polarized at 45° passing through SF11 glass rod and interfering with two orthogonally polarized reference pulses. (b) The retrieved spectrum (red) is compared to the measured spectrum (blue). The extracted net spectral phase (green) introduced by the SF11 glass rod is also compared to the calculated phase (magenta) due to material dispersion. (c) Extracted polarization angles for several tested polarization states are plotted against the set linear polarization angle of the signal.

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To test the accuracy of this technique, group delay dispersion (GDD) of a 2-cm thick SF11 glass rod was measured. Figure 7(b) shows the retrieved spectral intensity and net spectral phase corresponding to insertion of the SF11 rod. We have experimentally determined that the spectra of two polarization components are identical. The retrieved spectrum is also in close agreement with the spectrum separately measured using the same spectrometer. The GDD was calculated by fitting the extracted spectral phase to a quadratic function in frequency, resulting in an extracted value of 1952 fs²/cm which is in reasonable agreement (3% error) with the GDD calculated by the known Sellmeier equations for SF11 (1897 fs²/cm). A half-wave plate in the signal arm was used to set the linear polarization to a known polarization angle; this angle has been retrieved by analyzing the corresponding interferogram. The extracted and set linear polarization angles are shown in Fig. 7(c) and on average they agree at a level of 98%. The extracted polarization angles in Fig. 7(c) are the statistical average of 10 successive measurements for each corresponding polarization angle and the error bars represent one statistical standard deviation.

An important feature of this technique is that the spectral phase difference between the signal pulse and the reference pulses can be independently extracted for both horizontal and vertical polarizations. This feature could be used as a systematic check when the dispersion is polarization independent. In our example, the GDD resulting from the insertion of the SF11 glass rod for each polarization state is retrieved, resulting in agreement at a level of 99.5%. This cross check validates the stability and accuracy of the reconstruction algorithm.

7. Radially polarized pulse measurement

The ability to perform characterization of beams with spatially varying polarization is demonstrated on an example of a radially polarized pulse. The radially polarized state was generated by sending a linearly polarized laser beam into a liquid-crystal polarization converter that operates on the basis of a twisted nematic effect [3]. When the orientation of liquid-crystal molecules of the converter with respect to the incident beam is set properly by the external voltage, the beam experiences a rotation of its polarization direction by the twist angle. This rotation is a function of the angular position with respect to the crystal axis. Figure 8(a) shows the measured far-field image of the radially polarized beam produced in this way, while the Figs. 8(b)–8(e) show the same far field image after the radially polarized beam is passed through a linear polarizer oriented at various angles.

 

Fig. 8 (a) Measured far-field image of a radially polarized beam. Also shown is the far-field image of the same beam after passing through a rotating linear analyzer set at (b) 0°, (c) 45°, (d) 90°, and (e) 135°.

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The measurement of relative spectral phase and polarization state of a radially polarized beam can be performed by sampling the output beam at different spatial positions. The insets in Figs. 9(c) and 9(d) indicate the location of the two sampling positions used for this demonstration: vertical bottom (1) and corner left (2). In this experiment, a calibrated iris with the aperture size of 1.5 mm was mounted on an x–y translation stage and used to sample the beam. The sampled beamlet in the signal arm of the interferometer interferes with two orthogonally polarized reference beams. The sizes of two reference beams are set to be larger than the sampled beamlet, so that the signal beamlets are overlapped with both reference beams, producing interferograms that are always be contained within the CCD field of view. Figures 9(a) and 9(b) show the interferograms corresponding to the sampling positions (1) and (2), respectively.

 

Fig. 9 Two interferograms were taken with two reference beams interfere with the sampled signal beam taken at point 1 (a) and point 2 (b), respectively. (c) Extracted relative spectral phases corresponding to two sampling points (1 and 2) were compared. The inset indicates the sampling position in the beam. (d) The ellipse orientation angles ψ that represent sampled points (1 and 2) were calculated and compared.

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The extracted relative spectral phase corresponding to the sampling positions (1) and (2) is shown in Fig. 9(c). The difference of the relative spectral phase between those two points at the central wavelength (λ_⚬ = 800 nm) is 2.92 rad, close to the expected relative spectral phase of 3.14 rad. Their reconstructed group delay dispersion are also very similar, differing by ∼ 9%. The polarization ellipse orientation angle ψ for the two sampled positions is shown in Fig. 9(d). The difference in absolute values of ψ at central wavelength (λ_⚬ = 800 nm) is 0.71 rad, which is close to the expected value of π/4. The quality of the reconstruction of the spectral phase and ellipse orientation angles is reduced further from the central wavelength, mainly due to reduced fringe contrast. It is important to note that since the polarization continuously varies across the beam, the finite sampling size results in an ensemble of polarizations at the sample location being reconstructed as an average. Making the sampling size smaller could help reconstruct the local polarization with greater fidelity, at the expense of the reconstruction time needed to collect and analyze a greater number of interferograms, and is ultimately limited by the stability of the interferometer.

8. Conclusions

In summary, a simple technique capable of simultaneous reconstruction of the relative spectral phase and the polarization state has been proposed and experimentally demonstrated. The use of angle multiplexing with two orthogonally polarized reference pulses is a convenient approach to achieve this simultaneous reconstruction. The technique has several other advantages over POLLIWOG [6] and TURTLE [8], including the ease of alignment, high spectral resolution, and the speed of reconstruction. When compared to SEA TADPOLE, this technique involves linear processes and inherits all the advantages of SEA TADPOLE, while extending it to an angularly multiplexed geometry. Unlike SEA TADPOLE [13], it has the capability to measure the relative spectral phase and polarization state of two orthogonal polarization components of an arbitrary polarized signal pulse independently and simultaneously. Furthermore, this technique may also have applications in single attosecond pulse generation via the polarization gating technique [15, 16], provided that sufficiently low calibration jitter can be achieved. Another possible application of this approach is measurement of the phase and the polarization state of the polarization-modulated high-energy pulses suitable for efficient high-harmonic generation. As in other similar techniques [6, 13], a full characterization of the two linearly polarized reference pulses by using one of the many available techniques such as FROG [1] and SPIDER [2] is also needed to make the absolute spectral phase and intensity measurement of the signal. Similar to other interferometric techniques, our technique is limited by the interferometric phase drift, necessitating a calibration prior to each data taking. However, it is shown experimentally on the example of a beam with complex spatially varying polarization – a radially polarized beam – that sufficient stability can be practically achieved, resulting in acceptable accuracies of the reconstructed spectral phase and spatially varying polarization. The slow drift in our spectral phase measurement indicates that a calibration is needed approximately every 5 minutes in our setup in standard experimental conditions. To reduce the time between the calibration and measurement even further, a fast shutter could be placed in the signal arm of the setup to block the signal beam while an electro-optic modulator can be placed in one of the reference arms in order to establish a fast flipping of the polarization. With the addition of these two devices, the recording of the calibration data would become hassle-free, and data collection could resume in a rather quick way so that the effects of drift would be reduced further. Shot-to-shot variations could be reduced further by improving the mechanical stability and the beam pointing stability (while the latter could also be simultaneously measured to reject measurements exhibiting too large pointing errors). More complex shaped pulses with spatially varying polarization can be characterized by the use of this technique.