1. Introduction

Interference effects in spontaneous parametric down-conversion (SPDC) have been extensively studied over the last few decades. In addition to the fundamental interest, revealing counterintuitive features of quantum mechanics, they also find practical applications in quantum communication [1], computation [2] and metrology [3–7]. In this work we consider the effect of the nonlinear interference of down-converted photons, also known as “induced coherence” [8,9]. When signal and idler photons generated in two nonlinear crystals are superimposed, the interference is observed in the intensity and coincidence counts [9]. In contrast to the classical case, the interference pattern for signal/idler photons depends on phases and amplitudes of all the three interacting photons: the signal, the idler, and the pump [10,11]. This effect is particularly useful in metrology applications when the sample properties are to be measured in the detection challenging spectral range, for instance in the infrared (IR). When the unknown sample is inserted in the path of idler photons, one can infer its properties in the IR range from the interference pattern for signal photons in the visible range. This concept was recently implemented in several practical applications, including IR imaging [12], IR spectroscopy [13–17] and IR tunable optical coherence tomography (OCT) [18].

Earlier works studied influence of various experimental factors on the nonlinear interference, including the effects of spatial and temporal overlap of SPDC modes [19–21], linear losses and dispersion [8,9,22] introduced in the interferometer. Analysis of relevant polarization effects, however, is less extensive [23–25]. Grayson et al. demonstrated that signal photons acquire the non-local Pancharatnam phase, which was introduced into the path of idler photons by a set of retardation elements [24]. Recently, Lahiri et al. used the nonlinear Mach-Zehnder interferometer and introduced a polarizer into the path of signal photons and an attenuator into the path of idler photons to study the degree of polarization in such a system [25].

In this work, we perform a systematic analysis of polarization effects in the nonlinear interference of down-converted photons. We derive explicit relations between polarization transformations of idler photons and the interference pattern of signal photons. Based on this principle, we propose and experimentally realize two new methods for characterization of retardation properties of a sample in the IR range using well-developed components for visible light.

2. Theory

We consider the nonlinear Michelson interferometer, as shown in Fig. 1(a) [17,18]. SPDC photons (signal and idler) generated at the first pass of the pump through a nonlinear crystal enter the interferometer. The pump, signal and idler photons travel in different paths after being separated by corresponding dichroic mirrors (DM₁, DM₂). The photons are reflected by the mirrors and recombined at the crystal. The reflected pump generates another pair of down-converted photons, which interferes with the pair traveled in the interferometer.

 

Fig. 1 (a) The nonlinear Michelson interferometer. The pump beam (green) generates SPDC photons (yellow and red), which are separated into different arms by the dichroic mirror DM₁. The dichroic mirror DM₂ separates signal and pump photons. Mirrors M_s, M_p and M_i reflect all the photons into the crystal, where the pump generates another pair of photons. Then, the interference of signal photons is detected. (b) The beam splitter model which accounts for the double pass through the sample and reflection from the mirror M_i; τ_i is the amplitude transmission of the beam splitter. A mode a_i1 transforms into a mode a_i2 by injecting vacuum modes a₀ and a_0” from open ports of the beam splitter. The mirror M_i inverts the Cartesian coordinate system from the right-handed to the left-handed (x-y).

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The state vector of SPDC photons generated in the single crystal can be written as [26]:

Let us consider type-0 phase matching when pump, signal and idler photons have the same linear polarization σ_s = σ_i = σ (the theory also applies to type-I and type-II phase matching). According to Eq. (1), for the single spatial and temporal mode, the state vector of SPDC photons created at the first ${|\psi 〉}_1$ and at the second ${|\psi 〉}_2$ passes of the pump through the crystal is given by the superposition state [8,9,17,18]:

When the sample is inserted into the path of idler photons, the state vector in Eq. (2) changes. According to the beam splitter model [31], and assuming that idler modes i1 and i2 are matched, the photon annihilation operator for the idler mode i2 is given by:

Let us now redefine the factor |τ_i|², taking into account the polarization properties of the sample. Without the loss of generality, we consider the sample to be a generic retardation waveplate. We use the Jones matrices formalism [33,34] and introduce the corresponding transformation matrix $T_{wp}=\left(\begin{array}{cc}{\tau }_m{e}^{i\delta /2}& 0\\ 0& {\tau }_m{e}^{-i\delta /2}\end{array}\right)$, where δ is the retardation between extraordinary and ordinary waves, and τ_m is the transmission coefficient of the sample (it accounts for reflection, absorption, and scattering in the sample). Then, the rotation of the Cartesian coordinate system before and after the sample is given by: $J=R(\theta )T_{wp}R(-\theta )$, where $R(\theta )=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$ is the coordinate rotation matrix, and θ is the orientation of the optical axis of the sample. Figure 1(b) shows the detailed description of the propagation of the idler mode i1. It accounts for a double pass of the photons through the sample and reflection by the mirror M_i. The Jones matrix for this system is given by$J^{\prime }=R(\theta )T_{wp}M{T}_{wp}R(\theta )=R(\theta )T_{wp}{T}_{wp}R(-\theta )$, where M is the transformation matrix of the mirror M_i. We can then re-write the resulting Jones matrix in the following form:

We assume that horizontally polarized (along the x-axis) idler photons are created in the first and the second pass of the pump through the nonlinear crystal. The idler photon in mode i1 has an initial polarization vector $\overrightarrow{e}=\left(\begin{array}{c}1\\ 0\end{array}\right)$ [33,34]. After propagation through the sample, the polarization vector is modified as follows: $J^{\prime }\left(\begin{array}{c}1\\ 0\end{array}\right)={\tau }_m^2\left(\begin{array}{c}t\\ -r^{*}\end{array}\right)$. The modulus of the amplitude transmission function for the horizontally polarized component is given by:

Since the interference can only be observed for the horizontally polarized component of the idler photon, we can substitute the transmission function |τ_i|² in Eq. (4a) by Eq. (7). Thus we obtain the following expression for the count rate of signal photons:

Let us analyze this expression in more details and show how it can be used for measurement of the sample retardation.

Method 1. Relative phase shift of interference fringes. We set the fast optical axis of the sample parallel to the initial horizontal polarization of idler photons (θ = 0°). Next, we rotate the sample at 90 degrees, so that the slow axis becomes parallel to the polarization of idler photons (θ = 90°). In these two cases, the idler photon experiences refractive indices n_o, and n_e, respectively (we assume n_o<n_e). According to Eq. (8), the relative phase shift between interference fringes of signal photons is proportional to the sample retardation δ:

Method 2. Visibility of interference fringes. From Eq. (8) we obtain the following expression for the visibility (contrast) of the nonlinear interference:

The visibility is proportional to the modulus of the transmission function of the waveplate at the wavelength of idler photons. Note, that the similar dependence is valid for signal photons. From Eq. (10) it follows that the maximum and minimum values of the visibility are observed at θ = 0°/90° and θ = 45°, respectively. Then, the retardation can be directly found from the ratio of minimum and maximum values of the visibility:

The two methods allow measuring the optical retardation of the sample at the wavelength of idler photons from the measurements of the interference of signal photons. Note that Eqs. (9) and (11) account for the double pass of idler photons through the sample, and δ/2 gives sample retardation in the single pass. Next, we describe the experimental realization of the two methods and discuss the obtained results.

3. Experiment

The experimental setup is shown in Fig. 2. A continuous wave (cw) laser with 532 nm wavelength is used as a pump. The laser is focused by a lens F₁ (f = 200 mm) into a periodically poled Lithium Niobate (PPLN) crystal, where SPDC occurs. The phase matching conditions are set to generate the signal photons at λ_s = 809.2 nm and the idler photons at λ_i = 1553 nm (Λ_PPLN = 7.4 μm poling period; temperature T_PPLN = 403 K). Signal and idler beams are separated by a dichroic mirror DM₂ into visible and IR arms, respectively, and collimated by lenses F’ (both f = 75 mm). Then pump and signal beams are split by the dichroic mirror DM₃ into separate channels. All the three beams are reflected into the crystal by mirrors M_s,p (silver coated) and M_i (gold coated). The reflected pump beam generates another pair of SPDC photons. In the detection part, the pump and idler photons are filtered out by the dichroic mirror (DM₁), the notch filter (NF) and the bandpass filter (BP, 809.2 ± 0.6 nm). The signal beam is collimated by the lens F₂ (f = 200 mm) and detected by an avalanche photodiode (APD) or a CCD camera preceded by lenses F₃ (f = 50 mm) and F₄ (f = 100 mm), respectively. The CCD camera is used to facilitate the setup alignment.

 

Fig. 2 The experimental setup. The cw-laser pumps the PPLN crystal, where SPDC occurs. The PPLN is set to generate signal and idler photons in the visible and IR range, respectively. The photons are split by the dichroic mirror DM₂ into different arms. Pump and signal photons are separated by the dichroic mirror DM₃. All the photons are reflected by the mirrors M_s, M_p, and M_i. Filters DM₁, NF, and BP filter the detected signal photons. The interference is detected either by the avalanche photodiode (APD) or by the CCD camera. Mirror M_s is mounted on the translation stage for adjustment of the optical path Δz_s. The sample is inserted into the path of idler/ signal photons. Mirrors M_i and M_p are placed on piezo-translators for fine scans of interference fringes.

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The interference of the signal photons is observed once the interferometer arms are equalized within the coherence length of the SPDC, refer Appendix for the details. The mirror M_s, mounted on a translation stages (step size ~1 μm), is used to equalize the interferometer arms (Δz_s). Test samples are mounted into a rotation stage and inserted into the interferometer arms. Fine scans of the interference of signal photons are performed by translating the mirror M_i or M_p (Δz_i,p), mounted onto the piezo stages (step size ~2 nm). The interference fringes are measured at different orientations of the sample θ. The interferometer was not actively stabilized and was observed to be stable during the measurements.

4. Results and discussion

4.1 Sample in the path of idler photons

First, we test our method with samples of known retardation, namely quarter- (QWP) and half- (HWP) waveplates designed for 1550 nm. For the initial calibration, we measure the interference fringes without a sample in the IR arm, refer to details in the Appendix. Then, the sample is inserted into the IR arm with its optical axis set parallel to the polarization of the idler photons (θ = 0°). The change of the optical path length is compensated by translation of the mirror M_s. Once the optimal position of the mirror is found, we perform fine scans by translating the mirror M_i in the idler channel using a piezo-stage. Measurements are taken at different orientations of the optical axis of the sample θ.

Figures 3(a) and 3(b) show interference fringes of signal photons measured for the orientation of the sample at θ = 0°, 45°, and 90°. The relative phase shift between patterns at θ = 0° and 90° is equal to δ. As expected from Eq. (9), the shift of the interference pattern is λ_i/2 and λ_i for the QWP and HWP, respectively (idler photons travel twice through the sample). This experiment demonstrates the first method for measurement of the sample retardation. Summary of the obtained results is given in Table 1.

 

Fig. 3 The count rate of signal photons at λ_s = 809.2 nm versus translation of the mirror M_i in the idler channel for (a) QWP at 1550 nm and (b) HWP at 1550 nm. The orientation of the optical axis at 0° (black squares), 45° (red dots) and 90° (blue triangles). Points are experimental data, and solid lines are fits by Eq. (8) (R²>0.99). The relative phase shift between interference patterns at θ = 0° and 90° is equal to retardation δ. (c) The dependence of the visibility on the sample orientation θ for QWP (blue triangles) and HWP (red circles) at 1550 nm. Points are experimental data, and solid lines are fits by Eq. (10) (R²>0.99). The inset shows zoomed results for QWP at 1550 nm at visibility values close to zero.

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Table 1. Results of retardation measurements at 1553 nm by the two methods (idler mirror scan).

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Figure 3(c) shows the dependence of the visibility of the interference on the orientation of the sample θ. In accordance with Eq. (10) the interference fringes have maximum visibility at θ = 0° and 90°, while at θ = 45° the visibility reaches its minimum. For the QWP at θ = 45° the visibility is nearly equal to zero (V = 0.005 ± 0.005). Solid lines in Fig. 3(c) correspond to theoretical curves for waveplates designed for 1550 nm. The sample’s absorption data was obtained in independent measurements and is summarized in the Appendix. Thus we realize the second method for measurement of the sample retardation. The measurement results are summarized in Table 1.

Next, we perform measurements of samples with arbitrary retardations. We used HWP and QWP designed for operation at 532 nm. These waveplates operate as unconventional retarders for the probing beam at λ_i = 1553 nm. Figures 4(a) and 4(b) show interference fringes of signal photons measured for the two samples at orientations θ = 0°, 45°, and 90°. Corresponding retardation values are calculated by Eq. (9), and the results are shown in Table 1. Figure 4(c) shows the dependence of the visibility on the orientation of the two samples. Solid lines are fits with Eq. (10). Sample absorption data was obtained in independent measurements, and it is summarized in the Appendix. The retardation is inferred from the ratio of the minimum and maximum values of the visibility, see Eq. (11). The measurement results are summarized in Table 1.

 

Fig. 4 The count rate of signal photons at λ_s = 809.2 nm versus translation of the mirror M_i in the idler channel for (a) QWP and (b) HWP at 532 nm inserted in the path of idler photons. The orientation of the optical axis of the sample is at 0° (black squares), 45° (red dots) and 90° (blue triangles). Points are experimental data, and solid lines are fits by Eq. (8) (R²>0.99). The relative phase shift between interference patterns at θ = 0° and 90° is equal to δ. (c) The dependence of the visibility on the orientation of the sample θ for QWP (blue triangles) and HWP (red circles) designed for 532 nm. The solid curves show fits by Eq. (10) (R² = 0.98).

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Then, we investigate the dependence of interference fringes on the translation of the mirror in the pump arm. Figure 5 shows interference fringes obtained for QWP and HWP designed for operation at 1550 nm and 532 nm for different orientation of the optical axis (at θ = 0°, 45°, and 90°). We perform fine scans of the mirror M_p, placed on the piezo-stage. The pump wavelength λ_p now defines the periodicity of the pattern. Similar to the above procedure, we infer the sample retardation from the phase shift and the visibility ratio. The results are summarized in Table 2. They agree with the ones reported in Table 2.

 

Fig. 5 The count rate measured by translating pump mirror M_p for (a) QWP at 1550 nm, (b) HWP at 1550 nm, (c) QWP at 532 nm, (d) HWP at 532 nm inserted in the idler arm with orientations of the optical axis at 0° (black), 45° (red) and 90° (blue). Points are experimental data, and solid lines are fits by Eq. (8) (R²>0.99).

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Table 2. Results of retardation measurements at 1553 nm by the two methods (pump mirror scan).

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4.2 Sample in the path of signal photons

Next, we study polarization transformations by the sample, when it is placed in the signal channel. We use the zero-order QWP designed for 800 nm and perform scans of the mirror in the idler channel. Similar to previous measurements, we observe the phase shift of the interference fringes at different orientations of the sample θ, see Fig. 6(a). Also, we measure the visibility dependence at each orientation of the sample (Method 2), see Fig. 6(b). The retardation is then inferred from the shift of the interference fringes (Method 1) and the ratio of minimum/ maximum visibilities (Method 2). The results are summarized in Table 3.

 

Fig. 6 (a) The count rate measured by translating the mirror M_i in the idler channel when the QWP at 800 nm is inserted in the signal channel. Orientations of the optical axis are at 0° (black squares), 45° (red dots) and 90° (blue triangles). (b) The dependence of the visibility of the interference on the orientation of the sample. Points are experimental data, and solid lines are fits with Eq. (8) in (a) and Eq. (11) in (b) (both R² > 0.99). The inset shows zoomed visibility values near zero for QWP at 800 nm (yellow rhombuses and line), and for QWP at 1550 nm (blue line) for reference.

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Table 3. Results of retardation measurements at 800 nm by the two methods (idler mirror scan).

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4.3 Discussion

Based on the obtained experimental data we are now able to compare the two methods for characterization of samples retardation. The “phase shift” method (Method 1) is fast, as we infer the retardation from two sets of fringes at different orientations of the sample. However, the interferometer should be well stabilized during the measurements to minimize drift of the fringes due to thermal and mechanical fluctuations, see Appendix A3 for the analysis of introduced uncertainties. The best experimental accuracy in determination of the retardation δ is about ± 0.006π.

In contrast, the “visibility method” (Method 2) is more tolerable to the phase drifts, as they do not strongly affect the visibility. However, the measurement takes longer time than the “phase shift” method. The accuracy of the method is defined by the contrast between minimum and maximum visibility values attained in the experiment. In this method, the best achieved experimental accuracy is at the order of ± 0.002π, which also accounts for the accuracy in setting up the sample orientation. The accuracy of the two methods reaches specifications reported for industrial-grade optical elements

As our results show, the period of the interference fringes is defined by the scanning configuration: it is given either by λ_p or by λ_i, depending on which mirror is scanned. The configuration with the scan of the mirror in the pump beam may represent a certain advantage when the translation range of the piezo-stage is limited. We also note that when the sample is inserted in the path of the signal photon, the retardation can be measured at the visible wavelength with the same configuration. Moreover, our method allows us to distinguish fast and slow axes of the sample, see Appendix A4.

5. Conclusions

In conclusion, we performed a detailed study of polarization effects in the nonlinear interferometer. We showed that the change of the polarization of idler photons affects both the visibility and phase of the interference fringes of signal photons. This effect allows us inferring the sample retardation at the frequency of idler photons from (a) shift of the interference fringes and (b) from the visibility ratio. The suggested “phase shift” and the “visibility” methods allow characterization of the sample retardation with the accuracy up to Δδ = ± 0.002π, which meets the requirements of the optical industry.

In comparison with conventional polarimetry systems, which require an IR light source and a detector, our technique has a benefit for wavelength tunability and the use of visible range source and detector. An extension of the operating wavelengths of the presented methods can be easily achieved by choice of the periodic poling of the crystals and/or by tuning crystal temperature. Operation within the range of 1.5–4.3 um was shown in our earlier works with Lithium Niobate crystals [15–18]. The retardation measurement for a given sample takes about 4 minutes using the “phase shift” method and about 20 minutes using the “visibility method”. These measurement times are comparable with corresponding values for the conventional techniques [35].

Our technique can be readily extended to the IR polarimetry of arbitrary waveplates and samples with optical activity, such as Faraday rotators and chiral media. Furthermore, it is possible to measure the spatial uniformity of the retardation properties of a sample using an additional lens in the path of probing photons [18].

Also, active control over polarization of probing photons allows enhancing the signal-to-noise ratio by compensating polarization changes in the realistic samples. This idea forms the basis for polarization-sensitive optical coherence tomography (PS-OCT) [36] and polarization-sensitive quantum optical coherence tomography (PS-QOCT) [37], which have already been set forth as high-contrast methods for birefringence measurements of layered samples. Consequently, our technique can be used for the PS-OCT development, extending the conventional methods to the mid-and far-IR range.

Appendix

A.1 Visibility of interference

The state vector of type-0 SPDC the in Eq. (2) is given by:

(12)$$|\Psi 〉={|\psi 〉}_1+{|\psi 〉}_2=|vac〉+e^{i\left({\phi }_s+{\phi }_i\right)}{|H〉}_{s1}{|H〉}_{i1}+e^{i{\phi }_p}{|H〉}_{s2}{|H〉}_{i2},$$

where $|H〉$ is the single-photon Fock state with horizontal polarization. When the waveplate is introduced into the path of signal photons the state vector in Eq. (12) is given by:

(13)$$|\Psi 〉=|vac〉+e^{i\left({\phi }_s+{\phi }_i\right)}{J}^{\prime }{|H〉}_{s1}{|H〉}_{i1}+e^{i{\phi }_p}{|H〉}_{s2}{|H〉}_{i2},$$

Then, the polarization state vector of the signal photons ${|H〉}_{s1}$ is transformed as $J^{\prime }{\left(\begin{array}{c}1\\ 0\end{array}\right)}_{s1}=\left(\begin{array}{c}{t}_s\\ -r_s^{*}\end{array}\right)$, where index s denote signal photons. Then, the state vector of the SPDC photons in Eq. (13) is given by:

(14)$$|\Psi 〉\text{=}{e}^{i\left({\phi }_s+{\phi }_i\right)}\left(t_s{|H〉}_{s1}-r_s^{*}{|V〉}_{s1}\right){|H〉}_{i1}+e^{i{\phi }_p}{|H〉}_{s2}{|H〉}_{i2}$$

Assuming that idler photon modes are aligned i1=i2=i, the count rate of the signal photons is given by:

(15)$$\begin{array}{l}{P}_s\propto 〈\Psi |a_s^{+}{a}_s|\Psi 〉={\left|e^{i\left({\phi }_s+{\phi }_i\right)}{t}_s+e^{i{\phi }_p}\right|}^2+{\left|e^{i\left({\phi }_s+{\phi }_i\right)}{r}_s^{*}\right|}^2=\\ =2+2\left|t_s\right|\cos \left({\phi }_s+{\phi }_i-{\phi }_p+\arg t_s\right)\end{array}$$

The visibility of the interference fringes obtained from Eq. (15) is proportional to the modulus of the transmission function of the waveplate at the wavelength of signal photons V=|t_s|.

A2. Alignment of interferometer

Initially, the interferometer is balanced, and the interference pattern is observed around the zero position of the translation stage of M_s. The reference interference pattern is shown by black squares in Fig. 7. Once the sample is inserted in the path of idler photons, the mirror M_s has to be moved to compensate for the introduced optical delay. Figure 7 also shows interference pattern after introducing samples into the path of idler photons.

 

Fig. 7 The shift of the interference pattern after the introduction of the sample in the path of idler photons. Data for HWP at 1550 nm, QWP at 1550 nm, HWP at 532 nm and QWP at 532 nm with the orientation of the optical axis at θ = 0°.

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For the HWP at 1550 nm, QWP at 1550 nm, HWP at 532 nm phase shifts are approximately the same, as they are all purchased from the same supplier (Thorlabs) and have close values of the optical thicknesses. The QWP at 532 nm is purchased from another vendor (DayOptics) and the optical thickness is somewhat smaller.

Figure 8 shows the shift of the reference interference pattern due to the introduction of the QWP at 800 nm into the path of signal and idler photons. As the waveplate is designed for 800 nm, its absorption for signal photons is smaller than for idler photons.

 

Fig. 8 The shift of the interference pattern due to the change of the optical path for the signal and idler photons after insertion of the HWP at 800 nm with the orientation of the optical axis at θ = 0°.

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By measuring the interference at the position of maximum visibility of the interference pattern, it is possible to define the transmission function |τ_m|², see Table 4. This data is used to plot theoretical curves for the visibility function in Figs. 3, 4, and 6. The experimental data in Table 4 is in a good agreement with reference measurements, obtained with a conventional IR spectrophotometer (Shimadzu UV3600). The accuracy of the transmission coefficient measurement is ±0.012, which is higher than shown in [17] due to a nearly 5-times increase in the visibility of the interference pattern.

Table 4. Transmission coefficient |τ_m|² for different samples.

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A3. Uncertainties of measurements

A3.1 “Phase shift” method

The accuracy of measurements in the “phase shift” method is determined by temperature fluctuations of the PPLN crystal, mechanical perturbations in the interferometer, and by the uncertainty of the position of the scanning piezo stage.

The temperature of the PPLN crystal is controlled with an accuracy of ±0.01K, which results in the uncertainty of generated wavelengths of 0.22 nm for the idler photons at 1550 nm and 0.1 nm for the signal photons at 810 nm. Therefore, the uncertainty of the phase estimation due to temperature fluctuation is about 3.2×10⁻⁴π.

To study the effect of mechanical perturbations, we perform measurements of the phase fluctuations at a fixed temperature of the crystal, see Fig. 9. Considering that the signal, shown in Fig. 9, is at the slope of the interference fringe, the drift of the phase is proportional to the standard deviation of the signal detected by the APD $\text{Δδ}=\frac{{\text{σ}}_{\text{A}}}{\text{A}}\approx 0.0056\text{π}$, where A is the amplitude of the interference fringe. The standard deviation is taken during one measurement cycle of 50 seconds.

 

Fig. 9 Measurement of the phase drift in the interferometer.

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During the experiment, we first measure the interference fringe with the orientation of the waveplate at θ=0°. We take the first period of the interference fringe (see region “1” in Fig. 10) as a reference point for future measurements at orientations θ=45° and 90°. Next, we start to scan the interference fringe again at θ=0°, and change the orientation of the waveplate to θ=45°/90° (region “2” in Fig. 10). By matching the initial point of the interference fringe at region “1”, we are able to compensate for the long term phase drift in the interferometer.

 

Fig. 10 The experimental procedure of the “phase shift” measurements with the QWP designed for 532 nm wavelength. Region “1” corresponds to the reference point, where each scan starts from the θ = 0° orientation of the waveplate; “2” corresponds to the region, where the orientation of the waveplate is changed; “3” is the region where the retardation of the waveplate is measured.

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The uncertainty of the position of the scanning piezo stage is determined by the fitting of the obtained interference fringes by a cosine function (see Eq. (8)). Fitting of the interference fringes is done using Origin software with a built-in least-square method. The phase of the interference fringes is determined as:

(16)$$\delta =\frac{2\pi }{\lambda }2x,$$

where x is the position of the piezo stage, additional factor 2 takes into account the double pass of the photons in the interferometer. The phase uncertainty is determined as:

(17)$$\Delta \delta \approx \frac{2\pi }{\lambda }2\Delta x=\frac{\Delta x}{\lambda /4}\pi$$

For idler photons, the accuracy is $\Delta \delta \approx \frac{1.3}{389}\pi =0.003\pi$, see Figs. 3, 4, and 6. And for pump photons, the accuracy is $\Delta \delta \approx \frac{0.45}{133}\pi =0.003\pi$, see Fig. 5.

From the above analysis, one concludes that the uncertainty due to temperature fluctuations can be neglected. Finally, taking into account the phase drift in the interferometer and the fitting errors, the total uncertainty of the “phase shift” method is approximately 0.006 π.

A3.2 “Visibility” method

In the “visibility” method the accuracy is determined by the fluctuation of the photocount rate in the APD detector. According to Eq. (11) the retardation δ is determined as:

(18)$$\delta =\mathrm{arc}\cos \left(\frac{V_{\min }}{V_{\min }}\right).$$

Therefore, the uncertainty of the retardation measurement is determined as:

(19)$$\Delta \delta =\sqrt{\frac{\Delta V_{\min }^2}{V_{\max }^2-V_{\min }^2}+\frac{\Delta V_{\max }^2}{V_{\max }^2-V_{\min }^2}\frac{V_{\min }^2}{V_{\max }^2}},$$

where ΔV_min and ΔV_max are uncertainties for the minimum and maximum visibilities. These uncertainties are calculated as fitting errors in the Origin software.

For example, the error for the QWP@1550 nm is given by:

(20)$$\Delta \delta =\sqrt{\frac{{0.00444}^2}{{0.74846}^2-{0.00511}^2}+\frac{{0.00444}^2}{{0.74846}^2-{0.00511}^2}\frac{{0.00511}^2}{{0.74846}^2}}\approx 0.002\pi$$

The error for the HWP@1550 nm is:

(21)$$\Delta \delta =\sqrt{\frac{{0.00584}^2}{{0.76132}^2-{0.75454}^2}+\frac{{0.00586}^2}{{0.76132}^2-{0.75454}^2}\frac{{0.75454}^2}{{0.76132}^2}}\approx 0.026\pi$$

We can see that the accuracy of this method depends on the ratio between the maximum and minimum values of the visibilities. The higher is the visibility ratio, the more accurate are the measurements.

Comparing these two methods of the retardation measurement, we can conclude that the “phase shift” measurements is faster, but it has a slightly worse accuracy. At the same time, the “visibility” method takes a longer time, but it yields a higher accuracy.

A4. Determination of the fast and slow axis of the waveplates

Our experimental configuration allows to distinguish slow and fast axis of a sample by applying the “phase shift” method. Indeed, any change in the optical path length in the interferometer arms results in the shift of interference fringes (see Fig. 7). We can determine the direction of the shift of the fringes which correspond to the increase or decrease of the optical path.

In Fig. 7, when we place waveplate in the path of the idler photons, the optical path for the signal photons should be increased. To compensate for the additional optical path we translate the stage with mirror M_s to the positive direction of z (we can also consider moving mirror M_i to the negative direction –z).

For the arbitrary sample, we need to determine the positions of the highest visibility of the interference fringes. The highest visibility value corresponds to the fast or to the slow axis of the waveplate. Next, the waveplate is rotated by 90 degrees. If the sample is rotated in the direction to its slow axis (n_o→n_e), it results in an increase of the optical path length and a shift of interference fringes to a specific direction. And vice versa, moving from the fast to the slow axis (n_e→n_o) results in a shift of the interference fringes in the opposite direction.

We can test this statement based on the results shown in Figs. 3, 4, 5, 6, and 10. We say that angle at θ=0° corresponds to the fast axis of the waveplate, which corresponds to the ordinary refractive index n_o (n_o <n_e). When we change the orientation of the waveplate (θ=45° or θ=90°), the refractive index starts to increase (n_o→n_e), which leads to the shift of the interference fringes to the negative direction.

Therefore, by observing the direction of the shift of the interference fringes with the rotation of the sample, we can unambiguously determine its fast and slow axis.

Funding

Quantum Technology for Engineering program (QTE) of A*STAR (project # A1685b0005); Singapore International Graduate Award (SINGA) fellowship.

Acknowledgments

We acknowledge fruitful discussions with Sergei Kulik, Rainer Dumke, and Berthold-Georg Englert.

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