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Abstract
Non-planar image-rotating OPO ring resonators necessitate polarisation compensation in contrast to their planar counterparts. This is essential for maintaining phase matching conditions for non-linear optical conversion in the resonator during each cavity round trip. In this study, we examine the polarisation compensation and its impact on the performance of two types of non-planar resonators: RISTRA with a $\frac {\pi }{2}$ image rotation and FIRE with a fractional image rotation ($\frac {\pi }{2}$ fraction). The RISTRA is insensitive to mirror phase shifts, while the FIRE has a more complex dependence of polarisation rotation on mirror phase shifts. There has been debate over whether a single birefringent element can provide adequate polarisation compensation for non-planar resonators beyond RISTRA-type. Our results show that under certain experimentally feasible conditions, even FIRE resonators can achieve adequate polarisation compensation with a single half-wave plate. We validate our theoretical analysis through numerical simulations and experimental studies of OPO output beam polarisation using ZnGeP2 non-linear crystals.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical parametric oscillators (OPOs) can generate coherent radiation in spectral regions inaccessible to lasers. Tunable high-power coherent sources covering the mid-infrared range find numerous applications in remote sensing, free-space communications, spectroscopy or defence countermeasures [1–4]. Such applications require a wide tuning range and good beam quality. Among the mid-infrared laser sources for defence applications, the OPO technology is the only one that provides widely tunable high-power or high-energy radiation. Quantum cascade lasers and other direct-emitting sources in this spectral region are limited in power and pulse energy [5–7]. For sources covering the 3 – 5 $\mu$m region, optical parametric oscillators and amplifiers (OPA) with non-linear optical (NLO) crystals are the primary approaches [8]. Zinc germanium phosphide (ZGP) has favourable properties for mid-IR conversion among NLO crystals suitable for pumping near 2 $\mu$m [9].
OPOs consist of an optical resonator and a non-linear optical medium. To realise a basic OPO device, a simple linear resonator consisting of two plane mirrors is sufficient [10]. However, ring resonators are more commonly used in practical high-power and high-energy realizations [11], where the back reflection of a pump laser beam is undesirable. Pulsed OPO devices in the mid-infrared part of the spectrum are particularly useful for many defence-related applications [12]. The beam quality of such sources is important to enable long-range propagation in free space. Therefore, beam quality improvement has been one of the main driving forces in this research area over the years [13–26].
The class of ring resonators in which the mirrors are not arranged in a plane, the non-planar resonators with field rotation, was first proposed for lasers [27,28]. Subsequently, the cavity designs with a non-planar configuration of mirrors were investigated for the improvement of the beam quality in nanosecond OPOs [29]. These studies led to the development of a specific design with a 90° image rotation, known as a Rotated Image Singly-Resonant Twisted RectAngle (RISTRA) cavity [30]. The RISTRA OPO was then demonstrated with ZGP [31–33], KTP [29,34,35], KTA [36], CSP [37], PPLN [38] and HGS [39] crystals. Another type of non-planar OPO resonator with fractional image rotation (known under the acronym FIRE) has been proposed [40]. Unlike RISTRA, it uses fractional image rotation to improve beam quality. Both types of resonators, the RISTRA and the FIRE, take advantage of image rotation to spatially average over pump inhomogeneities, leading to better mode discrimination. The mid-IR OPOs with good beam quality for both high pulse energy [23–25] and high power [41] regimes have been demonstrated. The performance of ZGP OPOs with both types of non-planar resonators has been compared for operation at high average pump power (60 W) [41] and high pump pulse energy (up to 60 mJ) [42]. In both cases, the FIRE provides better OPO output beam quality at similar conversion efficiencies.
Regardless of the beam quality benefits, fractional image rotation has an undesirable consequence: even if a cavity does not introduce polarisation anisotropy, the vector of linear polarisation rotates at the same angle as the beam image as it passes through the cavity. This compromises the optimum phase matching condition within the non-linear optical crystal required for efficient OPO operation. If polarisation evolution during cavity travel is not compensated, conversion efficiency deteriorates and the output polarisation of the signal OPO beam is affected. In the case of the RISTRA, compensation is straightforward. A half-wave plate (HWP), properly placed inside the resonator, is sufficient to maintain the polarisation of the signal beam unchanged after the cavity round trip. Furthermore, the design of the cavity ensures that the phase shifts of the individual mirrors are cancelled out, so that the polarisation state does not depend on them. This ability to compensate for polarisation does not occur automatically in the FIRE cavity because it is more complex and depends on the phase shifts experienced by the resonating light. To date, the polarisation characteristics of non-planar OPO resonators have not been studied in detail. The results presented here are the first investigations to focus specifically on polarisation compensation in non-planar image-rotating cavities with a field rotation other than 90°.
This paper presents a theoretical analysis, supported by numerical simulations, of all cases of polarisation rotation in signal resonant non-planar ring cavities RISTRA and FIRE, followed by high-power ZGP OPO experiments. Detailed polarimetric studies of OPO output (signal) beams are performed for the first time. Recently, we have made progress in power scaling in non-planar OPOs with ZGP crystals that convert 2 $\mu$m laser radiation into the mid-IR band. We have demonstrated power levels of 30 W with good beam quality [33,41]. Here, we complement these studies by investigating the polarisation aspects that have been neglected so far in both types of non-planar ring resonators. The theoretical model and numerical simulations support our experimental results. The present study fills the literature gap by providing a complete description of the cavity mirrors’ phase-shifts effect on the performance of non-planar image rotating OPO resonators, supported by experimental studies of high-power ZGP OPOs of the RISTRA- and FIRE-type. Understanding polarisation compensation and its effect on phase matching is crucial for the further development and miniaturisation of mid-IR ZGP OPOs in non-planar resonators [43].
2. Role of cavity mirrors phase shifts in polarisation compensation
In the image-rotating cavities, the polarisation rotation is the same as the image rotation, so an appropriate correction with a retardation plate must be made to maintain the correct polarisation for phase-matching after the cavity round trip. This section presents the geometry of the systems considered and discusses methods for describing and analysing polarisation evolution in non-planar cavities.
The configuration of mirrors for the cavities considered in this work is presented in Fig. 1. As depicted in the figure, the coordinate system is linked to the ordinary ($x$ axis) and extraordinary ($y$ axis) direction of the NLO crystal. As indicated in the figure, critical phase matching is accomplished by angle tuning the crystal in the $y-z$ plane. Both cavities are singly-signal resonant. The waves are propagated along the $z$ direction in the positive uniaxial non-linear optical crystal ZGP. The optical axis of the NLO crystal lies in the $y-z$ plane and is rotated by $\theta$ with respect to the propagation direction. The extraordinary polarisation is aligned within this plane, while the ordinary polarisation of light is perpendicular. Both $e$ and $o$ polarisation directions are marked in the schemes. In the case of a ZGP, the type I phase-matching condition is fulfilled when the pump wave is linearly polarised along the x-direction ($o$), and the down-converted OPO waves signal and idler are linearly polarised along the y-direction ($e$). The RISTRA consists of 4 mirrors and provides a total of 90° image rotation at each cavity pass and the same polarisation rotation, meaning that after the round trip, the signal beam does not fulfil the phase-matching condition any more. However, polarisation compensation is straightforward with a single half-wave plate (HWP). A polarisation rotation of 90° after the OC mirror maintains phase-matching. Because $s$-polarised light at the M3 mirror swaps to $p$-polarised at the M4, the individual mirrors’ phase shifts cancel each other out and are of no concern. In the case of the FIRE, however, the fractional rotation makes things more complicated; the linear polarisation that starts a round trip becomes elliptical depending on the mirrors’ phase shifts. Thus the possibility of compensation with an HWP is not apparent any more. Instead, two different retardation plates would be necessary to bring back to the initial state of polarisation and optimize phase-matching for given mirrors’ phase delays. The usual operation with a single HWP might lead to worse performance due to insufficient phase-matching and some portion of unpolarised light in the output signal beam. To demonstrate this effect we use a matrix representation of all the cavity elements that affect the polarisation and the image rotation. The matrix for image rotation is defined as a reference frame rotation matrix with an angle defined as the angle between the planes of incidence of the neighbouring mirrors.
Fig. 1. The schematic representation of ZGP OPO realization in the two non-planar image-rotating cavities RISTRA and FIRE. The waves undergoing non-linear conversion are propagated along the z direction, and the ordinary and extraordinary polarisation planes are marked at the face of the NLO crystal and along related beams ($o$-pol, $e$-pol). The type I phase-matching condition in the positive uniaxial crystal ZGP is fulfilled when the pump beam is o-polarised (ordinary direction) and as a result signal and idler waves are e-polarised (generated in extraordinary direction). The OPO is angle tuned in the plane spanned between the optical axis and the propagation direction.
To describe and analyse the polarisation state of the OPO output beam, we use the Stokes calculus, with Stokes vectors $\textbf {S}$ describing the polarisation state of a light beam, and Mueller matrices $\textbf {M}$ used to describe polarisation altering elements. It allows for the characterization of partially polarised light beams. The Stokes parameters (4 components) are used to fully describe the polarisation state (linear, elliptical) and the unpolarised components of light obtained from the polarimetric methods described in detail in the next section.
The Stokes vector can be related to light intensities and electric field amplitudes used in the Jones matrix description [44]:
The $S_1$ parameter represents how much the light is horizontally or vertically polarised, $I_x$($I_y$) is the amount of light transmitted through an ideal polariser with the transmission axis aligned along the $x$($y$) axis. The values $I_{45}$ and $I_{135}$ correspond to the same measurements of orthogonal linear components when the polariser is set diagonally. The $I_{r}$ and $I_{l}$ correspond to circularly polarised components. We use $S_1$ to visualize the polarisation state changes after the cavity round trip when starting with a linearly polarised light beam. The value of $S_1 = 1$ corresponds to the ordinary polarisation (along $x$ in the coordinates frame related to the crystal) and $S_1 = -1$ to the extraordinary polarisation (along $y$). $S_2$ shows how much the light is polarised along the diagonals. $S_3$ shows the components that are circularly polarised.
2.1 RISTRA
The pump beam enters the RISTRA resonator via a dichroic input coupling mirror, tilted by 32.7°. It is highly transmitting for 2 $\mathrm{\mu}$m (T=86%) and highly reflective for the signal beam. The output coupler mirror (M2) is transparent for the pump and idler and has a reflectivity of $\approx 50~\%$ for the signal beam. The other two mirrors (M3 and M4) are identical to M1. The half-wave plate between M2 and M3 aims to compensate for the polarisation rotation in the resonator. For RISTRA, the retardation plate’s fast axis is rotated by 45° with respect to the incident signal beam polarisation direction to rotate it by 90°. This way, the polarisation stays unchanged after a cavity round trip. The physical length of the resonator is 130 mm.
The polarisation behaviour in RISTRA has been analysed in detail using standard Jones matrix methods [45]. Accounting for the cavity geometry and the angles between planes of incidence for each mirror, as well as the reflectivity for $s$ and $p$ polarisations, the round trip can be described by the matrix:
The data plotted in Fig. 2 show the final polarisation state after the round trip inside RISTRA, assuming an initial linearly polarised state ($o$-direction aligned with $x$-axis) resulting from phase matching. The $S_1$ Stokes parameter visualizes how the polarisation plane rotates depending on the HWP rotation; however note that the polarisation changes from linear to slender elliptical to linear e.g. between HWP angle positions 0° and 45°. The calculation is based on the matrix defined in Eq. (2). The variable in the equation is the rotation of the fast axis of the HWP placed after the M2 mirror. When the fast axis is aligned with the incoming polarisation (HWP rotation angle is 0°, equivalent to the absence of the wave plate), the $S_1=1$, so the polarisation after travelling through the cavity rotates by 90° to the orthogonal state. The perfect compensation occurs when the HWP rotation is 45°, bringing the state of the polarisation to the initial one, maintaining the phase-matching condition after every round trip. The important property of this design is the insensitivity to the individual mirrors phase shifts for $s$ and $p$ polarisation.
Fig. 2. Change of the polarisation state represented by the $S_1$ Stokes parameter after the round trip in the RISTRA when the half-wave plate inside the cavity is rotated.
2.2 FIRE
The second non-planar cavity is the Fractional Image Rotation Enhancement cavity (FIRE), composed of 6 flat mirrors [40]. Like in the RISTRA, the M1 mirror (T=84%) allows the pump beam to enter and pass through the NLO crystal while reflecting the signal beam. Again, the M2 has a reflectivity of $\approx 50~\%$ for the signal beam. The other four mirrors (M3-M4-M5-M6) that make the round trip in the resonator are all identical to the M1. A HWP can be placed between M3 and M4 or M6 and M1 to compensate for the polarisation rotation. In our experiment, we use half-wave plates for the chosen operation wavelengths. The physical length of this cavity is 222 mm, corresponding to an optical round trip of 265 mm. Using the same formalism presented above for the RISTRA, the equivalent matrix to that given in Eq. (2) can be constructed for the FIRE cavity. The difference is the image rotating angle for the pass from each mirror to the next one, that is $\gamma _1=$107.06$^{\circ }$ in this case. The total net image rotation in case of the round trip in the FIRE equals to 77.64$^{\circ }$. The round trip matrix for the FIRE is
The extraordinary direction of the crystal is assumed to be perpendicular to the plane defined by the mirrors M1, M2, M4 and M5. Starting the round trip at the crystal exit, the first rotation has to be $\gamma /2=53.53^{\circ }$ to adapt to the plane of incidence of the OC (M2). Finally, leaving mirror M1, one further half-angle rotation completes the round trip and the signal polarisation thus again meets the extraordinary crystal direction.
It is not possible to describe the evolution of polarisation in the FIRE in a simple way, as can be done for the RISTRA above, because it depends on the phase shifts of the individual cavity mirrors. Therefore, the full range of mirror phase shifts must be considered for the FIRE.
Here we consider whether it is possible to obtain satisfactory compensation for the polarisation change in FIRE using only a half-wave plate, as we have shown for the RISTRA. The calculation is based on the matrix describing the evolution of the polarisation given in Eq. (4), assuming one HWP is positioned in one of the ways depicted in Fig. 1. The surface plot in Fig. 3 shows the dependency of the polarisation state on the phase shifts difference between the p and s directions of the mirrors after the cavity round trip in the FIRE. The $S_1$ parameter of the Stokes vector represents the polarisation state after the cavity round trip. The minimum value of the $S_1$ parameter is plotted, chosen from all the possible $S_1$ values corresponding to different fast axis angles of the HWP. Regardless of the choice of position of the wave plate between either mirrors M2 and M4, or mirrors M1 and M5, the results are qualitatively the same, so we restrict all our further discussion to the HWP positioned after the OC mirror as it was in the case of the RISTRA. The sufficient level of polarisation compensation manifests itself in the graph as a value of $S_1$ close to $-1$ (polarisation along the y-axis, parallel with the extraordinary direction of the NLO crystal). If such a value cannot be found among all the possible HWP rotations, it means that for a particular set of phase-shift values for the HR and OC mirrors, there is no compensation possible with a single wave plate inside a cavity. A more complex combination of wave plates would be needed to optimize the polarisation in the FIRE cavity.
Fig. 3. Change of the polarisation state after one FIRE cavity round trip represented by $S_1$ Stokes parameter. For each ($\phi ^{OC}$, $\phi ^{HR}$) pair, the HWP rotation angle was determined that gives the minimum $S_1$ value depicted in the plot.
It can be noticed that for some range of the retardance values introduced by the OC mirror ($\sim 0 - 10$°), regardless of the phase shift difference of the HR mirrors of the cavity, it is possible to maintain unchanged polarisation after round trip with a single HWP. In fact, the retardance values of the OC mirrors used in our ZGP OPO experiments happen to be in the desired range where $\phi ^{OC}$ is small. In the experiment we used the cavity HR mirrors with a phase shift difference $\phi ^{HR}$ of about 46° and the outcoupling mirror $\phi ^{OC}$ of about 4°. These values have been used for calculations and simulations presented in this paper. The above analysis suggests that the FIRE shall perform equally well with a single HWP, as is the RISTRA for our experimental conditions.
3. Experimental set-up for polarisation analysis of OPO beams
The experimental set-up used to support the polarisation compensation analysis consists of a 10 kHz pulsed high-power Ho:LLF MOPA system [46] that pumps a ZGP OPO with a non-planar image-rotating resonator geometry. The 2.06 $\mathrm{\mu}$m pump source delivers 68.7 W at 2065 nm in TEM$_{00}$ with a diffraction-limited beam quality ($M^2 \approx 1$) at a maximum pulse energy of 6.9 mJ. The pulse duration is 25 ns. The 300 mm lens focuses the pump beam from the Ho:LLF to a spot diameter of $\sim 1.1$ mm in the center of a ZGP crystal, maintaining a pump fluence < 1.5 J/cm$^2$. In the present study, we used the ZGP crystal with a size of 6 x 6 $\mathrm {mm^2}$ (aperture) x 20 mm (length), and cut at 55° with respect to the optical axis to allow type-I phase matching. The end faces are AR-coated for the pump and the output wavelengths (3-5 $\mathrm{\mu}$m). The entire OPO system is discussed in detail elsewhere [33,41].
The layout of the experimental set-up to analyse the polarisation and monitor the output power of the OPO beams is presented in Fig. 4. The output beams are directed towards the polarimeter by a set of protected silver mirrors introducing negligible phase shifts and a CaF wedge reflecting at a very small angle to minimize the Fresnel reflectivity coefficients difference for orthogonal polarisation components. A 200-mm lens collimates the beam in the polarimeter composed of the birefringent wave plate and a Wollaston prism that separates the orthogonal polarisation components (s and p polarisation). The separated beam intensities are registered with a pyroelectric camera acting as a detector. When the transmission axis of the prism is rotated with respect to the incoming polarisation plane, the intensity is recorded in the channels for the s and p polarised beam images.
Fig. 4. Experimental setup used for the analysis of the polarisation of the ZGP OPO output beams with the notation used for the geometry of the polarisation analysis system.
We have applied classical Stokesmetry to determine the polarisation state of light. The technique involves passing the light through a series of optical components, consisting of a linear retarder and a polariser. In our experiment, these components are a quarter-wave plate (QWP) and a Wollaston prism, with a pyroelectric camera serving as the light detector. The measurements are performed at least four times with different settings of the analysing optics to gather complete information and calculate the Stokes parameters [44,47,48]. To calibrate the wave plate and evaluate the potential error margins in our set-up [49], we used a stable low-power quantum cascade laser operating at the same wavelength as the OPO signal beam, with a frequency below 4 $\mu$m.
The transmitted light intensity ($I_T$) can be determined using the following equation [47]:
It represents the light passing through a retardation plate with retardance $\delta$ and its fast axis rotated by angle $\beta$, as well as a linear polariser at angle $\alpha$. The configuration is shown in the Fig. 4. $I, C, M$, and, $S$ are four parameters of Stokes vector. The first three of them can be determined from the intensities recorded in channels S and P after the Wollaston prism: $I_S(\alpha )$ and $I_P(\alpha )$. The fourth parameter can be obtained by incorporating a quarter-wave plate with a phase retardation of $\delta =90^{\circ }$. The intensities measured using the QWP are represented as $I_{P/S}(\alpha =45^{\circ },\beta =90^{\circ })$, where $\beta$ represents the angle between the fast axis and the $x$-axis.
The correction ${S_3'=[S_3+S_2\cos \delta ]/\sin \delta }$ must be made if the QWP is not perfect ($\delta \neq 90^{\circ }$), as it can be directly derived from the Eq. (5).
4. Numerical simulation of parametric conversion with OPODESIGN
The simulation code OPODESIGN solves the paraxial wave equations with the Fast Fourier Transformation (FFT). It propagates the signal, idler and pump waves in the crystal, the cavity with image rotation and outside the cavity to establish the caustic [50,51]. The split-step method is applied to integrate the OPO equations inside the NLO crystal by a fourth-order Runge-Kutta scheme; thus, the conversion and back conversion are accounted for. The simulation uses only single frequencies for all waves and thus neglects effects of spectral widths (line broadening). The temporal pulse evolution is realized by sampling time slices of cavity round trip time duration. The spatial walk-off effect is considered. Computations are done for a lateral area of 25.6 mm x 25.6 mm with a sampling rate of $2^8$ x $2^8$ mesh points. This numerical tool has been previously proven to reproduce the experimental results of ZGP OPOs in non-planar image-rotating cavities with respect to conversion efficiency and beam quality [24,33].
For the purpose of this work, the OPODESIGN code has been extended to take into account the polarisation state of the resonant signal wave. The signal beam is split into an ordinary and an extraordinary wave with respect to the crystal and these two beams are propagated in the cavity. The ordinary signal wave passes the crystal without being amplified. The signal beam may be regarded as composed of a large variety of beamlets ($2^8$ x $2^8$ lateral mesh points), their individual polarisation states being perfect. Hence, for every round trip and every sampling point in the lateral plane at the crystal entrance, the ordinary and extraordinary field components are transformed by the 2 x 2 - Jones matrix of the cavity, representing the effects of the half-wave plate and reflections, phase shifts and image rotations of the mirrors. The polarisation of the total (output) signal beam is then analysed according to the Stokes formalism to account for unpolarised parts of the beam. For the HWP, in OPODESIGN, the matrix is
By this definition, it is manifest that we find a period of 180° in the simulation (and in the measurements) when the fast axis of the HWP is rotated, since $\theta _{HWP}$ and $\theta _{HWP}+180$ yield the same Jones-matrix.
5. Dependence of the OPO conversion efficiency on the polarisation state of the signal beam
The parametric conversion efficiency drops if the polarisation state of the resonant signal beam is not maintained during the cavity round trip. It is the result of a polarisation mismatch between the pump and the OPO beam. Typically for ns-pulsed operation, there are tens of round trips during the transient of a pump pulse. Without proper polarisation compensation, the performance of the OPO is degraded because the resonant signal beam is not parametrically amplified on each pass. The best angular position of the wave plate in the non-planar cavity is usually determined empirically. However, there has been no quantitative study of the dependence of OPO conversion efficiency on cavity polarisation mismatch. To this end, we perform a series of experimental studies comparing the performance of OPO cavities with and without internal HWP. We also look at how, as a function of the angle of rotation of the HWP, the OPO output power (conversion efficiency) and the polarisation state of the signal beam evolve.
The slope-efficiency curves to compare operating with and without HWP in both resonator types are plotted in the top row of Fig. 5. The slope and conversion threshold differences stand out in the graphs. In both cases, the conversion efficiency drops significantly. The absence of the compensation plate has a slightly greater effect on the RISTRA (from 50% to 36%), where the slope efficiency deteriorates relatively more than for the FIRE (from 46% to 42%). This is not surprising, since in the case of the RISTRA, the absence of the HWP leads to the polarisation switching between s and p on each round trip. In the case of the FIRE, the absence of the HWP leads to an elliptical state, which, averaged over the pump pulse, fulfils the phase matching condition to a higher degree. This is also reflected in the observed conversion thresholds. The conversion threshold is usually comparable for both resonators when the polarisation compensation is optimal. As can be seen from the graphs, in the case of this experimental realisation with a pump focusing to a diameter of about 1 mm, the parametric conversion starts at about 1 mJ pump pulse energy. Thus, the difference for the case where the non-planar resonators are operated without the internal HWP is clearly visible (red solid lines and data points in the graphs). The corresponding threshold values for RISTRA and FIRE are 1.7 mJ and 1.9 mJ, respectively.
Fig. 5. Input-output power conversion graphs (top row), and Stokes vector representation of the output signal (bottom row). The solid line ellipse represents the experimental Stokes vector in the electric field coordinates. The dashed line circles represent the unpolarised components. In each case, the label indicates the value of the degree of polarisation (DOP).
The results of the polarisation measurements of the signal beam for both cases, with and without the presence of the HWP, are shown in the bottom row of the figure. Polarisation ellipses are visualized with py-pol python library [52,53] to show the corresponding Stokes vectors for the signal beam. Optimising the angular position of the compensating wave plate results in near-perfect linear polarisation of the signal beam, with negligible unpolarised component (represented by the dashed circles). When the HWP is absent in the cavities, thus not compensating for polarisation evolution, we observe a more elliptical polarisation state for both resonators with significant unpolarised components. The degree of polarisation, defined as
is close to 1 for optimized operation, but without the HWP, it drops to 0.35 for the RISTRA and 0.59 for the FIRE.The idler beam does not resonate in the OPO cavity. Therefore, its polarisation state is not affected. It was observed that the idler beam is always perfectly linear s polarised.
Taken together, these results indicate that the lack of proper polarisation compensation has a profound effect on the performance of the OPO in the non-planar resonators, imposing an elliptical polarisation on the signal beam with a significant proportion of a randomly polarised component. The following section takes a closer look at the power scaling that occurs when the amount of polarisation compensation is affected by the angular position of the internal HWP.
6. Change of the OPO performance due to the HWP rotation
The polarisation state of the signal beam and the total output power were monitored to assess how the rotation of the HWP, and hence the level of polarisation compensation, affects the performance of the ZGP OPO. The pump pulse energy and repetition rate were kept constant throughout. Only the intra-cavity half-wave plate was rotated over the angular range of 180°. The pump pulse energy ($E_p$) was chosen to allow stable operation above the OPO conversion threshold ($E_p>2.5$ mJ) for the whole range of HWP angular positions in both experiment and numerical simulations.
6.1 RISTRA
Figure 6(a) shows the experimental and numerical results of the power modulation as a function of the wave plate rotation in the RISTRA cavity. The experimental data points are represented by blue squares and red stars. The latter represent the data points obtained without the HWP. In the graph, the green diamonds represent the numerical simulation data from OPODESIGN corresponding to the experimental conditions.
Fig. 6. (a) The change in the ZGP OPO output power when the HWP in the RISTRA cavity is rotated. The OPO is pumped at a constant pulse energy of around 2.6 mJ and a repetition rate of 10 kHz. The two values measured without the HWP inside the cavity are marked with red stars. In case of single-frequency operation (the case of simulation), the points are equivalent to the 0° and 90° angular positions of the HWP in RISTRA. (b) The change of the Stokes parameter $S_1$ with HWP rotation angle as simulated and measured outside the cavity.
During the power modulation measurements, the polarisation state of the signal beam was also monitored. For each angular position of the HWP inside the OPO cavity, the polarimetric analysis was performed and the corresponding Stokes vector was retrieved in accordance with the procedure described in detail in the proceeding section. The change in the polarisation of the OPO signal beam, represented here by the $S_1$ Stokes vector component, is shown in the graph in Fig. 6(b). The experimental data are compared with the numerical results.
For the optimal alignment of the HWP (around -45°, 45°, 135°), the total output power from the OPO reaches more than 8 W. The angular positions 0° and 90° of the HWP correspond to the rotation of the polarisation by 0° and 180°, respectively. In these cases, the HWP has no effect on the polarisation of the signal, it is equivalent to the case of a polarisation rotation of 90° after each round trip in the RISTRA. The OPO total output power drop below 4 W. For a monochromatic beam, the case is equivalent to the RISTRA cavity operated without the wave plate; the polarisation is not altered and thus not compensated at all. The two data points marked by the stars are added to the graph for comparison. These points are the measurements of the output power when there is no HWP inside the cavity, and the output power is around 3.2 W. The OPODESIGN code gives lower values for these points compared to the experimental data. OPODESIGN also predicts different values for consecutive periods of 90°, which are not observed in the experimental data. In the experiment, rotating the HWP by 90°, i.e. rotating the polarisation by 180°, does not change the output power at all.
6.2 FIRE
The effect of HWP rotation in the FIRE cavity was analysed in the same way as for the RISTRA, cf. Figures 7(a), b. The experimental power and polarisation data points were acquired with steps every 11.25° in the rotation of the HWP between 0 and 180°. The performance is characterised by a similar 90 degree modulation period. However, the minima of the conversion efficiency are shifted to positions of 62.5° and 152.5°. As in the case of RISTRA, we observed a slightly less deep and more regular power modulation in experimental data compared to numerical predictions (Fig. 7(a)). The data points corresponding to the OPO without HWP, marked by blue stars, are given along with the two minima recorded by scanning the waveplate angular position. We do not observe much of a difference between the performance of the OPO without the HWP and when it is inserted, but tuned to the most unfavourable position. In these cases, however, the numerical simulation gives a much lower value than observed experimentally.
Fig. 7. (a)The change in the ZGP OPO output power when the HWP in FIRE cavity is rotated. The OPO is pumped at a constant pulse energy of around 2.6 mJ and a repetition rate of 10 kHz. Marked with blue stars are the two values measured without the HWP inside the cavity. The points correspond to the 62.5° and 152.5° angular position of the HWP in the FIRE. (b) The change of the Stokes parameter $S_1$ with HWP rotation angle as simulated and measured outside the cavity.
The values of $S_1$ parameter for the FIRE cavity associated with the HWP rotation steps are plotted in the Fig. 7(b). The experimental data in this case spans over 120° covering both power minima. There is less agreement between the numerical prediction and the experimental data contrary to the RISTRA, but the general trend observed is reproduced by the numerical simulation. The different is the most apparent for the points with the weakest polarisation compensation, where the numerical results indicate worse performance than observed experimentally.
7. Role of OPO phase distribution (phase noise) on the polarisation states
As presented above, the simulations with the code OPODESIGN predicts the power modulation and the $S_1$ parameter of Stokes vector describing the polarisation states in RISTRA and FIRE, but there is severe discrepancy in the Stokes parameters $S_2$ and $S_3$ that is manifested in the $DOP$ observed. Typically, the measurements yield for the RISTRA without internal half-wave plate, as an example, a normalized Stokes vector ${(1, -0.310, -0.042, 0.160)}$, while OPODESIGN gives ${(1, -0.310, -0.285, -0.777)}$. In what follows, it is shown that strong phase noise in the signal beam could be the reason for the discrepancy in the Stokes parameters $S_2$ and $S_3$.
Let us focus on any finite area in the cross section of the signal beam leaving the output coupler M2(OC), so small that the intensity is nearly constant within. At any point, the extraordinary and ordinary (reference to crystal frame) components of the Jones vector may be written as $a \exp (i\phi )$ and $b$, respectively. Here, $a$ and $b$ are real and we have $S_0 = a^2 + b^2$ and $S_1 = a^2-b^2$, being independent of $\phi$. Applying a linear polariser in 45° and -45° positions, we end up with the two Jones vectors having the e / o-components $\frac {1}{2} [a \exp (i \phi ) + b]$, $\frac {1}{2} [a \exp (i \phi ) + b]$ and $\frac {1}{2} [a\exp (i \phi )-b]$, $-\frac {1}{2} [a \exp (i \phi )-b]$, respectively. The Stokes parameter $S_2$ can now be calculated as $S_2 = I_{+45}-I_{-45} = 2 a b \cos \phi$ and, by introducing a quarter-wave plate at zero angle position in front of the linear polariser (90° phase shift of the e-component of the Jones vector), in an analogous way, $S_3 = 2 a b \sin \phi$.
It is now assumed that in the area under consideration the random phase angles $\phi$ are uniformly distributed in a range $\Delta \phi$ around a certain central angle $\phi _0$. The average value of $S_2$ then follows as the integral over $\phi$ of $S_2$:
The average of $S_3$ is obtained by the same procedure. We thus find
Note that $S_{2_\textrm {average}}$ and $S_{3_\textrm {average}}$ are proportional to the sinc-function $\sin ({\Delta \phi }/{2})/({\Delta \varphi }/{2})$. If we further let the width of the phase interval $\Delta \phi$ approach $2\pi$, $S_{2_\textrm {average}}$ and $S_{3_\textrm {average}}$ can always be made very small compared to $S_0$. In fact, this is exactly what is observed in the measurements. An example of the comparison between the numerical data for $DOP$ obtained according to the procedure described above and the experimental data for the RISTRA cavity is presented in the Fig. 8.
The origin of these large random phase shifts between the e- and o-components of the signal fields behind the OC may be caused by the relatively broad spectral width (around 200 nm) of the free-running ZGP-OPO radiation (incoherence). In addition, the phase distribution may vary from pulse to pulse - the experimental procedure averages over many pulses during the acquisition time, whereas OPODESIGN is a single frequency code that does not consider the effects of finite spectral width. If experimentally possible at all, a polarisation conserving narrow band filter applied to the signal beam behind the OC eventually yields results closer to the predictions of OPODESIGN and hence would lend support to the phase noise hypothesis.
8. Conclusions and summary
This study aimed to investigate the impact of polarisation compensation on high-power OPO performance in image-rotating cavities (signal resonant) and evaluate the feasibility of using a single half-wave plate in the FIRE cavity for optimization and miniaturization. A general theoretical method for determining polarisation compensation during non-planar resonator transit was presented. The focus was mainly on the FIRE cavity, which offers better beam quality compared to the commonly used RISTRA cavity. The results show that, under certain conditions that are experimentally feasible, a single HWP at an optimal angle can in fact provide polarisation compensation in FIRE and RISTRA cavities. This analysis can be expanded to include any image rotating resonator with arbitrary mirror phase shifts. Careful evaluation of the phase shifts of each cavity mirror (different for s and p) and the resulting changes in polarisation is essential to achieve optimum OPO performance. This can be accomplished by monitoring the polarisation state of the OPO signal beam, as demonstrated in this study.
In our high-power ZGP OPO experiment, pumped at around 2 $\mu$m, we performed experimental polarimetric studies to investigate the effect of changes in signal polarisation on the parametric conversion. The comparison between two commonly used non-planar OPO cavities showed that uncompensated polarisation can lead to a significant drop in efficiency. The evolving polarisation within the cavity leads to the accumulation of an unpolarised light component, which we quantified using classical Stokes analysis.
In our numerical tool, OPODESIGN, we have included the effects of polarisation. Unlike other available numerical codes and models, our tool considers polarisation mismatch in the parametric process. For instance, the models implemented in the SNLO software [54–56] and the SISYFOS simulation system [57–59] only account for perfectly matched polarisation in their OPO calculations.
The experimental as well as the computational output power data confirmed the expected modulation period of 90° from both cavities. The conversion efficiency in the RISTRA is optimized for the 45° rotation of the HWP, while the deep minima are observed at 0° and 90° or operation without the presence of the wave plate. The same is true for the FIRE with the minima located around 62.5° and 152.5°.
In our code for tight pump beam focusing and a long NLO crystal (around 20 mm), there is an intrinsic exaggeration of back conversion. The effect of lower signal feedback due to polarisation deterioration is partly counterbalanced by weaker back conversion. Hence, the power peaks (maxima) are broadened and flattened. The numerical remedy is to (artificially) shorten the crystal from 20 mm to 15 mm and raise the pump pulse energy from 2.6 mJ to 3.5 mJ. The agreement for the parameter $S_1$ is satisfactory, but there is a discrepancy for $S_2$ and $S_3$. This discrepancy is also reflected in the simulation prediction of the $DOP$ of the OPO output beam. However, it can be reduced by artificially introducing phase noise in the numerical model. This brings the results closer to the experimental values from the ZGP OPO, being characterized by a relatively wide spectral width and, hence, imperfect temporal coherence.
The broad emission of free-running ZGP-OPOs is useful for many applications. However, this also affects polarisation, because a retardation plate acts differently on different spectral components. We showed that a single half-wave plate can provide a sufficient degree of compensation for a satisfactory performance in a broadband OPO. By positioning the intra-cavity wave plate optimally, the output from the two non-planar resonators produces nearly perfect linear polarisation with a degree close to unity and thus suppresses the unpolarised component.
Our numerical simulation, which was limited to single frequency operation, closely matched the experimental results. To improve the agreement, further corrections are needed for back conversion and possible phase noise. However, simulations of parametric amplification with finite spectral width are very time consuming. Experimentally, for better computational validation, the OPO can be operated in a spectrally restricted regime by using frequency-selective elements or injection seeding for spectral narrowing.
The polarisation of the beam from fractional image rotation OPOs had not been studied before. It was assumed that the beam could be elliptically polarised with a significant unpolarised component due to the averaging of many polarised states generated in the NLO crystal during the pulse duration. Our measurements are the first to confirm that a non-planar resonator, other than RISTRA-type, can operate efficiently with a single compensation element. This makes the performance of FIRE as good as that of RISTRA in terms of polarisation compensation. Our study was supported by experimental and numerical results using ZGP non-linear crystals. This study is crucial for further developing non-planar OPO cavities with fractional beam image rotation. The method can also be adapted and applied to any non-planar ring cavity to analyse the impact of mirror phase shifts on polarisation rotation.
Funding
Direction Générale de l’Armement; Bundesamt für Ausrüstung, Informationstechnik und Nutzung der Bundeswehr.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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