Interference fringes in a nonlinear Michelson interferometer based on spontaneous parametric down-conversion

Quantum nonlinear interferometers (QNIs) can measure the infrared physical quantities of a sample by detecting visible photons. A QNI with Michelson geometry based on the spontaneous parametric down-conversion in a second-order nonlinear crystal is studied systematically. A simplified theoretical model of the QNI is presented. The interference visibility, coherence length, equal-inclination interference, and equal-thickness interference for the QNI are demonstrated theoretically and experimentally. As an application example of the QNI, the refractive index and the angle between two surfaces of a BBO crystal are measured using equal-inclination interference and equal-thickness interference.


Introduction
Interferometers have been the most useful and important metrology tools. A novel type of interferometer known as a nonlinear interferometer [1][2][3][4][5] was recently realized. Different from the traditional interferometers, the interference in the novel interferometers is caused by the indistinguishability between photons from two nonlinear sources. The nonlinear sources can be realized in many physical systems based on four-wave mixing or parametric down-conversion, including atomic ensemble [6], integrated silicon photonic chip [7], nonlinear optical fiber [8,9] and bulk χ(2)-nonlinear medium [10][11][12].
In this work, we focus on the nonlinear interferometers based on spontaneous parametric down-conversion (SPDC) [13,14] of bulk χ(2)-nonlinear crystals. Because SPDC is usually used in a quantum source [15], we refer to these interferometers as quantum nonlinear interferometers (QNIs) to distinguish them from other nonlinear interferometers (using other nonlinear effects, for example, the parametric amplification). In recent years, these interferometers have seen growing interest due to the properties of correlations and nonlinearity. Owing to the correlations of momentum and frequency, information can be extracted by a photon never detected [10,11]. On the other hand, owing to the nonlinearity, two photons in the two arms of a QNI can be frequency highly nondegenerate. These allow that an infrared photon in the interferometer is used to contact samples and extract information, while another visible photon is detected. Because the performances of infrared light detectors are generally inferior to that of visible light detectors, an advantage of QNIs is that one can take advantage of the high quality of visible light detectors to record infrared information. Another advantage is that one can easily tune the photon wavelength in a relatively wide band by changing the phasematching conditions in some nonlinear sources. For example, silicon chips and some biological tissues are high transmittances in infrared bands, so QNIs can be used to extract their image information by infrared photons and recorded by visible photons using silicon intensified CCD (ICCD) cameras [16]. Moreover, these interferometers also have applications in imaging [10,[16][17][18][19][20], metrology [6,12], polarimetry [21,22], optical coherence tomography [23][24][25], spectrum shaping [26], spectroscopy [11,18], testing the complementarity principle [27,28], and measuring bi-photon correlations [29,30].
References [2] and [30] reported equal-inclination interference in their QNIs and proved that the changing of interference pattern is characterized by a combination of wavelengths of signal and idler photons. In this work, we use a Michelson geometry of QNI [12,16,[18][19][20]23] that is different from the configuration in Ref. [2] and [30], to study both equal-inclination and equal-thickness interference theoretically and demonstrate the theoretical model experimentally. In our configuration, a smaller equivalent wavelength for the equal-inclination interference is obtained compared to Ref. [2]. Moreover, we present two applications of metrology for equalinclination interference and equal-thickness interference. Our work provides a method to measure the optical properties of crystals. Especially in the mid-infrared band, traditional Michelson interferometers may not applicable in some laboratories for lack of mid-infrared sources or detectors. In these cases, our scheme could be a good alternative.

The basic phenomenon and principle of QNI
Our experiment is based on the SPDC of a second-order nonlinear crystal. A simplified schematic shown in Fig. 1 (a) is used to introduce the basic phenomenon and principle of QNI. A monochromatic pump beam sequentially passes through two identical nonlinear crystals, and the down-converted idler and signal photons generated by the two crystals are aligned. The interference is caused by the indistinguishability: one cannot say which crystal the photon pairs are generated from. Although the idler photons also have an interference effect in this scheme, we only focus on the detection probability of signal photons and ignore the idler photons, because in our experiment we refer to the idler photons as infrared photons that cannot be detected by a standard silicon CCD.
This QNI has only one output, unlike the traditional interferometers and the nonlinear interferometers reported in [2,30,31], where the energy of light redistributes between two outputs. Here, the energy redistributes nonlinearly between pump photons and the downconverted photons, i.e., the photon number of the down-converted photons is not conserved in the interferometer. The two spatial separated crystals should be treated as a whole system and the total average number of photon pairs generated in the whole system has the form  is the probability of photon pair generated from one crystal; the phase  is determined by the length and refractive index of the medium between the two crystals.
The principle of the QNI in Fig. 1  indicate pump, signal, and idler; ω, c, and n represent angular frequency, speed of light, and refractive index of air; l represents the path length between the two crystals. The phase jump can be inserted in the phase-matching condition [32] because they have similar form when the three photons in SPDC process have equal path length l. Assuming that the phase-matching condition is satisfied, 2 4  photon pairs can be generated when  is equal to zero (the efficiency of SPDC is approximately proportional to the square of the crystal length [14,33]). However, when  is equal to  , it is equivalent to the case of phase mismatch, therefore, no photon pairs are generated. In this sense, this setup can also enable the nonlinear interference of three-wave mixing process [34].
The output bi-photon state from the QNI is given by Here,  is a constant and the modulus square

QNI with a Michelson geometry
The schematic for the nonlinear interference in Fig. 1 (b) has a modified geometry compared with that in Fig. 1 (a), in which the pump beam and the signal and idler photons are reflected back into the same crystal. This schematic resembles traditional Michelson interferometers, which allows better flexibility because the phase factors can be introduced independently for signal and idler photons as well as for the pump light [1, 3]. Moreover, this scheme has been used for testing the complementarity principle [27,28]. Our experiment is performed using this scheme, therefore, the following theoretical models are based on this scheme. In this schematic, a lens (L1) is positioned at the center between the crystal and the mirrors (M1 and M2); the crystal and the mirrors located at the front and rear focal plane of L1; L1 and M1 (or M2) form a 4-f imaging system where the photons pass through L1 twice and the object and image planes coincide at the crystal. Another lens (L2) is used as a Fourier translator that maps the spatial frequency components to the spatial point on the detection plane. Therefore, measuring the photon number of each spatial point is equivalent to measuring the photon number of each spatial frequency component, then, considering Eq. (1), we have In Eq. (2), the independent variable j k can be also expressed as   based on the ideal energy and transverse momentum entanglement under the conditions of narrow pump linewidth and large pump beam width [30], where the delta function, representing the conditional probability of the idler photon to have wave vector of i k given the signal photon has a wave vector of s k , is defined by   are onedimension and two-dimension Dirac delta function respectively; are concerned in this work and they are corresponding to three cases of interference: (1) the two arms are both perfect 4-f system and well aligned; (2) the 4-f system is defocused and equal-inclination interference occurs; (3) the mirror in one of the arms is slightly tilted and equal-thickness interference occurs. In the first case, all the spatial modes have equal optical paths and the phase difference is approximately independent of the spatial modes. In other words, the phase difference is the same as that in the case of single-mode assumption and In this case, the interference pattern is a uniform bright or dark spot. In section 2.3, the mean photon number on the detection plane and the expression for interference visibility are presented based on case (1); case (2) and (3) are studied in section 2.4 and 2.5. In addition to these three cases, the function P also induce interference fringes when the arm difference is not zero, which is introduced in the supplementary materials.

Mean photon number on the detection plane and interference visibility
Considering  Pk defined in Eq. (2) respectively. Considering the Michelson interferometer shown in Fig. 1(b), we assume that the arm of the signal and pump photons has a length of 1 l , i.e. s p 1 l l l , another arm has a length of 2 l , i.e.
2 i ll  . In this case, the phase difference is given by where the phase 2 2 s0 2s 23 2 b l fc    can induce ring-like fringes, which is referred as angularspectrum-dependent (ASD) interference and is introduced in Supplement 1; the triangle function is defined as Then, the visibility is given by where   represents the full linewidth of photons. When the OPD is equal to 4/ c   the visibility becomes zero, therefore, the coherence length of both the signal and idler photons in the QNI is 8/ c   , which is equivalent to To observe interference phenomena, the coherence length of photons should be longer than the OPD, i.e. c si l l l   ; the coherence length is usually short enough, therefore, we assume that the arm difference l  cannot cause defocusing of the 4-f system. Other condition for this interference should be noted is that the coherence length of pump light should be far longer than the distance between the two crystals, i.e. depends on pump wavelength, therefore, the integration of pump wavelength reduces the interference visibility when the linewidth of the pump light is not narrow enough. In this work, we assume that the pump light is an ideal monochromatic plane wave.

Equal-inclination interference
If the 4-f system is defocused, circular fringes of equal-inclination interference occur. In the case that the optical path of idler arm adds a distance d , the phase difference in Eq. (2) is given by is the incident angle of idler photons (angle between the wave vector and the light propagation direction). The first term is the equalinclination term, which reflects the transverse phase difference and creates the ring-like interference fringes. The second term is the longitudinal phase difference, which is determined by Eq. (4) and does not create interference fringes. A similar equation can be obtained if the signal arm changes.
In our QNI scheme, the photons emit from the crystal in the form of plane waves (the angular spectrum of the plane waves is shown in Fig. S2 in the supplementary materials), and the lens L1 is used as a Fourier translator; if a plane wave with a transverse wave vector i  is incident on L1, it gets converted into a spherical wave that converges to a point   ii  on the mirror M2. Therefore, the equal-inclination interference fringes are only sensitive to the distance between the crystal and L1. The OPD L  included in 0  makes almost no contribution to the equal-inclination interference term. In the experiment, we move the crystal a distance d along the beam propagation axis to obtain equal-inclination interference fringes. Under the approximation that the divergence angle of the pump beam is ignored, the introduced phase shift can be expressed as 22 2 ss i 00 i s eq

Equal-thickness interference
When the crystal is located at the position of 0 d  , i.e. no equal-inclination interference fringes appear, equal-thickness interference occurs if one of the mirrors is slightly tilted, or in an equivalent case that a gradient thickness plate is inserted in one of the two arms. The interference pattern is formed by straight fringes that are perpendicular to the gradient direction. If the thickness varies along the x-axis, the phase difference is given by h represents the thickness of the sample. The generalized expression is given by  is the coordinate on the plane of M2 and n is the refractive index of the plate. The gradient thickness plate in front of and close to M2 can be seen as a phase object, and the interference pattern is actually its phase image. This interference effect can be equivalently interpreted using the theory of imaging in Ref. [36].
By substituting Eq. (12) into Eq. (2), one can predict the interference patterns. On the other hand, one can perform equal-thickness interference to measure the thickness variation of a plate sample using Eq. (12).

Experimental setup
A schematic of our experimental setup is shown in Fig. 2. The 525.2-nm light beam of the CW pump laser is generated in a single-pass sum-frequency generation (SFG) process [37] (the SFG source is not shown in Fig. 2). In the SFG source, the wavelengths of the two pump beams are 1540 nm and 797 nm respectively and the linewidths are 10-kHz and 200-kHz respectively. Because both the pump beam have narrow linewidth, the 525.2-nm SHG laser beam also has a narrow linewidth (< 210 kHz). All the three beams are vertically polarized and satisfy type-0 phase-matching condition. The SFG laser beam is collected into a single-mode fiber and exits through a fiber collimator (the FC in Fig. 2). Because the idealized plane-wave pump in the SPDC is a necessary condition of strict transverse momentum correlations, the pump beam for SPDC is collimated by a lens group; its width is of order 400 μm and its power is of order 60 mW that is the maximum power we can obtain. Two PPKTP crystals are used in the experiment, one for SFG and the other for SPDC. The two crystals have the same parameter values: their dimensions are 1 mm × 2 mm × 10 mm, and their grating periods are 9.34 μm. The temperature of the crystal used during SFG is set at 24 °C, which is an optimum temperature for SFG. The temperature of the crystal used for SPDC is set at 29 °C. This temperature is determined by performing difference-frequency generation (DFG) between the 525.2-nm and 1540-nm laser beams. The two temperatures are different because the laser focusing conditions are different in the two crystals; the pump beams are tightly focused in the SFG process, while the pump beams are collimated in the DFG or SPDC process; to maximize the SFG efficiency for focused Gaussian beams, a mismatch should be introduced [32]. Because SPDC is the inverse of SFG, the central wavelength of the idler and signal photons are approximately 1540 nm and 797 nm. The pump, signal, and idler photons are split through a long-pass dichroic mirror (DM2), where the idler photons pass through DM2 while the pump and signal photons are reflected. After the second pass of the crystal, the pump photons pass through another short-pass dichroic mirror (DM1), while the idler and signal photons are reflected (DM1 used in our experiment is different from the DM in Fig. 1(b), where we assume the DM transmit signal photons to clearly illustrate the 4-f imaging system). Before detection on the ICCD, the photons pass through an 800-10 nm band-pass filter (the center transmission wavelength is 800 nm and the FWHM bandwidth is 10 nm), therefore, only the signal photons are detected while the idler photons are discarded. To control the arm difference roughly or finely in the experiment, we fix M2 on the displacement platform and M1 on the piezoelectric transducer (PZT).

Results and discussion
In the experiment, we first found the position of M2 where the arm lengths were equal, meanwhile, the coherence length was measured; next, we tested the stability of the interferometer because a relatively stable interferometer is the premise of the reliability of the data; after that, we observed the equal-inclination interference by changing the position of the crystal; then, we studied the influence of OPD on interference in three cases; in the third case, the refractive index of a BBO crystal was measured; finally, an equal-thickness interference was observed and used to measure the angle between the two surfaces of the BBO crystal. As we mentioned above, the equal-thickness interference fringes in our setup are equivalent to the phase imaging of the phase object in the idler arm, therefore, the clear imaging results can prove the reliability of the equal-thickness interference fringes; the imaging results are shown in the supplementary materials. Fig. 3 (a) shows the relation between interference visibility and the position of M2. The inserts are some interference patterns recorded by the ICCD with a 10 s exposure time. To improve the signal-noise ratio and obtain accurate visibilities, we set such a long exposure time, although the exposure time could be much shorter. In order to show the visibility directly, we measure the visibility and record the fringe patterns for equal-inclination interference (corresponding to 6 d  in Fig. 4), instead of the interference where the patterns are uniform spots. The background is taken before the data acquisition and is subtracted by the ICCD camera automatically when signals are taken. Nevertheless, given that the lowest stable temperature of our ICCD is -25 °C, there were remain dark counts that reduce the interference visibility. The interference visibility defined by Eq. (8) is measured from the first-order bright ring and dark ring at the center of the pattern. When calculating the visibility, the average dark counts are removed. In order to determine the position of M2 giving an equivalent path, we approximately fit the experimental data with a triangle function based on Eq. (7) and (8). And the origin of horizontal axis has been set at the position with maximum interference visibility obtained by the fitting. The fitting result shows that the maximum interference visibility is 58.8 2.7%  and the coherence length is c exp 1.28 0.06 mm l  . The uncertainties were estimated assuming that the counts follow Poisson statistics. The visibility is mainly limited by the losses of the optical elements in the interferometer; the lens L1 is uncoated and has losses of 8.0%, 6.9%, and 7.2% for the pump, signal, and idler photons respectively. To improve the visibility, one can use an off-axis parabolic mirror with a silver coating to reduce the losses. The predicted coherence length is c pre 1.20 mm l  from Eq. (9), where the predicted bandwidth used in Eq. (9) is calculated based on quasi-phase-matching condition [32] and the Sellmeier equation we used is from Ref. [38]. The total error comprises the fitting error, the measurement error, and the Sellmeier equation not perfectly matching our crystal.

Coherence length and interference stability
We then test the interference stability and show the result in Fig. 3 (b). M2 is moved to the position with maximum interference visibility, and the ICCD works in the kinetic mode. In this mode, the ICCD cannot subtract background automatically, so the inserts are vaguer than that in Fig. 3 (a). 4000 interference patterns were continuously recorded and each has 1 s exposure time. We first average the normalized pixels value in the green region shown in the insert, then plot a curve that reflects the light intensity changes with time to assess the stability. The relative root-mean-square intensity fluctuation is 2.4%. This indicates that our interferometer is to some extent stable.

Equal-inclination interference
After the equal arm position is determined, we observe the equal-inclination interference pattern with different crystal positions. The crystal position is set as the origin where the interference pattern is a uniform bright or dark spot. Then, we move the crystal away from L1 and keep the positions of mirrors unchanged. We assume that the coordinate of the crystal (the distance between crystal and origin) is d and the interference patterns with different coordinate d are shown in Fig. 4 (a), where more interference rings appear with increasing coordinate d . By substituting Eq. (10) into Eq. (2) and calculating the maximum value of the cosine function, one can obtain the relationship 2 s0 + / 2 aN     , where s s 2 f   represents the radius of the ring-like patterns and N is the constructive interference order. The quadratic coefficient given by  The equal-inclination interference could potentially be useful in metrology [2]. In a traditional Michelson interferometer, the variation of fringes can be used to measure the small displacement and changes of refraction index in one of the two arms. In this interferometer, the small displacement and changes of refraction index between the crystal and the lens L1 can be similarly measured by recording the variation of interference fringes. The results shown in Fig.  4 show the dependence of fringes on displacements. Comparing with the measurement in a traditional interferometer, the QNI has advantages as follow: firstly, no reference path is needed (the Michelson geometry is not needed when doing this) and therefore the measurement is not influenced by the errors from reference arm; secondly, the range of measurement is not limited by the coherence length of photons, which is usually very short for photons from SPDC; thirdly, the equivalent wavelength is smaller than the physical wavelengths of the pump photons and is also smaller than the equivalent wavelength in Ref. [2], which means the variation of fringes is more sensitive to the displacement and therefore the measurement sensitivity could be higher than that in traditional interferometers.

Three cases for OPD and equal-thickness interference
In the following, we study the influence of OPD on interference phenomenon, here OPD is determined by arm difference and the crystal is fixed at the position of 0 d  . Three cases for OPD are discussed: (1) the OPD changes slightly, around one or two microns; (2) the OPD changes within the coherence length; (3) the change of OPD can exceed the coherence length. In the first case, the interference patterns are bright or dark spots. We adjust the PZT voltage to record the spot patterns in the cases of different OPDs. Fig. 5 (a) shows that the average photon counts of the spot changes with PZT voltage. We fitted the data using a sine function and the fitting results show that the OPD period is 1527 8 nm  , where the uncertainty was estimated using 95% confidence bounds of the fitting. From Eq. (4), the theoretical period is equal to the idler wavelength 1540 nm . The error is from the nonlinearity and inaccuracy of the dependence of thickness to the voltage of the PZT and the measurement error of the voltage. In this case, small phase changes can be measured in practical applications, like in a traditional interferometer.
In the second case, ring-like patterns appear and the number of fringes increases with increasing OPD. The ring-like patterns are similar to the equal-inclination interference patterns, however, the patterns are not created by equal-inclination interference; they are caused by the phase 2  in Eq. (6). This phenomenon referred as ASD interference is explained briefly in section 2 in Supplementary 1. In Fig. 3 (a), the number of fringes changes with OPD, which is caused by the ASD interference. The ASD interference has much shorter equivalent wavelength than the equal-inclination interference above, which indicates that the ASD interference has potential in improving measurement sensitivity when the OPD is measured by counting fringes.
In the third case, the coherence length is used as a ruler to measure OPD. For an application example, we inserted a BBO crystal with a length of 0.5 mm h  in the idler arm to measure its slow axis refractive index. Fig. 5 (b) shows that the interference visibility curve before and after insertion of the BBO crystal. The red arrow shows the translation direction.  [39]. The error is also caused by the inaccuracy of the thickness of BBO crystal and the BBO surface being not perfectly vertical to the light propagation direction. When the BBO crystal is inserted and the PPKTP crystal is fixed at the origin ( 0 d  ), the ring patterns become straight fringes (not spot). The interference patterns, shown in Fig. 5 (c), are created by equal-thickness interference because the two surfaces of the crystal are not perfectly parallel. To ensure the accuracy of Eq. (11) and (12), the BBO crystal was very close to the mirror M2 and was about 5mm from it; the distance cannot be shorter, which was prevented by the mounting frame of the crystal (assuming the Gaussian waist for idler photons is 400 μm in the crystal, one can calculate out the waist on the mirror M2 that is of order 92 μm, which indicates that the Rayleigh length is of order 17 mm; therefore, the distance 5 mm and the slight focal shift caused by the BBO crystal is small enough to satisfy the phase imaging approximation in section 2.5). The brightness of fringes reflects the relative phase difference   s  in Eq. (12). Based on Eq. (11) and (12), one can obtain the thickness function The angle of 1.3' 0.1'  between the two surfaces is obtained by dividing the thickness difference   oi / 2 1 nk   by the distance between bright and dark strips, where the distance was measured manually on the camera and the uncertainty was derived from the variance of the measurement for the distance. Here, we only demonstrate the measurement practicability, not accuracy, because no standard value of the angle can be used as a reference.

Conclusion
In summary, a QNI with Michelson geometry is realized. The interference visibility, coherence length, equal-inclination interference, and equal-thickness interference are demonstrated theoretically and experimentally and the experimental results agree with those of our theoretical model. In stressing the study of interference pattern observed by moving crystal, the variation of the common path of the idler and signal photons can prompt equal-inclination interference and the interference fringes vary faster with the increasing common path than those for traditional equal-inclination interference with increasing path difference; this phenomenon is parameterized by an equivalent wavelength and the expression for the equivalent wavelength is presented, which is determined by the wavelengths of all three photons in the SPDC process (or other two photons, because only two wavelengths are independent). The potential application and advantages in metrology is discussed in section 4.2. To improve the parameters of the present interferometer, one can improve the pump power to reduce the exposure time and reduce the loss of elements to improve the interference visibility.
By comparing with the traditional interferometers, the main advantage of QNIs is that they can measure the infrared physical quantities through the detection of visible photons. In this work, we demonstrate that the infrared refractive index of a crystal can be measured using the QNI. Also, the surface irregularities can be measured through equal-thickness interference, like in the traditional interferometers. When the sample is an IR material that has high transmittances in infrared band and low transmittances in visible band, the nonlinear interferometer based on spontaneous down-conversion can provide infrared photons from near-IR to far-IR [40] and take advantage of visible detectors to record the interference fringes. For example, this type of interferometer can be used in the application of microscopes for silicon chips [19]. We believe that the above research for the quantum nonlinear Michelson interferometer and its advantage demonstrated here may not only enrich the knowledge regarding optical interference but also be helpful for the possible applications of infrared metrology and imaging.