Robust squeezed light against mode mismatch using a self imaging optical parametric oscillator

We present squeezed light that is robust against spatial mode mismatch (beam displacement, tilt, and beam-size difference), which is generated from a self-imaging optical parametric oscillator below the threshold. We investigate the quantum properties of the generated light when the oscillator is detuned from the ideal self-imaging condition for stable operation. We find that the generated light is more robust to mode mismatch than single-mode squeezed light having the same squeezing level, and it even outperforms the single-mode infinitely squeezed light as the strength of mode mismatch increases.


Quantum properties of multimode squeezed light
Spatial properties of parametric down conversion. To generate squeezed light, we consider a degenerate type-I parametric down conversion in a χ (2) nonlinear crystal in the low gain regime 26 . The interaction Hamiltonian can be described as where q s(i) is the transverse component of the wave vector for the signal (idler) field, â † (� q) is the creation operator at q and t = 0 , and g is the gain parameter proportional to the crystal nonlinearity χ (2) , the length of the nonlinear crystal l c , and the maximum pump amplitude A p . The kernel K( q s , q i ) determines the spatial properties of the generated light, which depends on the pump beam distribution and the phase matching condition. For a monochromatic Gaussian pump beam focused at the center of the crystal, the kernel is given by 23,27 where k p and w p are the wavenumber and the waist of the pump beam, respectively. The Gaussian function originates from the pump distribution (making the anticorrelation between q s and q i ), and the sinc function is by the phase matching condition (making the correlation between q s and q i ).
The Hamiltonian Ĥ in Eq. (1) can be described by the transverse position operator â † (� x)(= 1 2π d 2 � q e −i� q·� xâ † (� q)) as well, using the inverse Fourier transform, The associated kernel K( x s , x i ) is decomposed with HG mn functions ( ψ H mn (x, y) ) by approximating the sinc function in K (� q s , � q i ) to the Gaussian function ( sinc(x 2 ) ≈ exp(−αx 2 )) 28,29 , where The pump beam property has been expressed in terms of the Rayleigh range z p = k p w 2 p /2 , and H n (x) is the n-th order Hermite polynomial. The key parameters determining the kernel are a modified (by α ) focusing parameter of the pump ξ and the waist size w H . ξ determines the eigenvalues µ m+n , where |µ| m+n < 1 , and w H determines the width of HG modes ψ H mn (x, y) . The Schmidt number M, quantifying the average number of modes 30 , can be also expressed as a function of the focusing parameter ξ, M has the minimum value of one at ξ = 1 , and M increases as ξ deviates from one.
With this kernel decomposition, the Hamiltonian in Eq. (3)  mn from different HG mn modes, which means that it generates squeezed light in each of the multiple spatial modes. The eigenvalues µ m+n are associated with the relative squeezing levels in different modes, which will be discussed in the following section.
(1) H = − i g 2 d 2 � q s d 2 � q iK (� q s , � q i )â † (� q s )â † (� q i ) + h.c., www.nature.com/scientificreports/ Self-imaging OPO. Let us construct a cavity that is compatible with the spatially multimode Hamiltonian in Eq. (7). While a typical cavity resonates on a single HG 00 mode 13 , for our purpose, we require a degenerate cavity resonant on all HG mn modes. The cavity degeneracy is determined by the Gouy phase shift ( θ G ) accumulated by the HG 00 mode along one cavity round-trip 31,32 . A confocal cavity has θ G = π mod(2π) , which is resonant only on half of HG mn modes having either an even or an odd number of m + n 18 . On the other hand, a self-imaging cavity exhibits θ G = 0 mod(2π) : it is a fully resonant cavity for all HG mn modes 21,22 . Figure 1 describes a self-imaging OPO, which consists of a plane mirror M 1 , a lens of focal length f, a nonlinear crystal of length l c , and a curved mirror M 2 of a radius of curvature R. The lengths l 1 and l 2 can be expressed as When l 1 = l 2 = 0 , the OPO becomes fully degenerate for all HG mn modes. This ideal condition, however, leads to cavity instability 22 , and thus, small detunings ( l 1 , l 2 ) are required for stable operation. The Gouy phase shift with such a detuning is where we have assumed f = R . As a result, when the cavity is locked for HG 00 mode, a high-order HG mn mode attains a phase shift of (m + n)θ G 31 . The cavity has HG eigenmodes with the waist size of w c = R 0 /2π ( 0 : the free space wavelength) for small detunings l 1 , l 2 ≪ R , and the associated creation operators are To match the cavity modes with the eigenmodes of the Hamiltonian in Eq. (4), we position the crystal at the cavity waist and set R = 2πw 2 H / 0 , while a more general case of mismatch between the modes will be discussed in "Effect of mode mismatch inside the OPO" section. In this configuration, we obtain a decoupled quantum-Langevin-equation for each Â mn 33 , where Â i mn and Â l mn are the annihilation operators of the input and the intra-cavity loss modes, respectively, and the corresponding decay rates are given by γ i = T i /2τ and γ l = T l /2τ ( T i and T l are shown in Fig. 1, τ : the round trip time). As we consider a cavity locked for the HG 00 mode, the cavity detuning frequency of HG mn mode, mn , is given by (m + n)θ G /τ . We have used the approximation T i , T l , θ G ≪ 1 . Using Eq. (7), and then, the Fourier transform of Eq. (13) becomes Figure 1. Self-imaging OPO for generating multimode squeezed light. M 1 is a perfect dichroic mirror transmitting the pump and reflecting the squeezed light, and M 2 is a partially reflective mirror having a transmittance of T i for the squeezed light. T l is the transmittance associated with the intracavity loss. The distances between the mirrors and the lens are l 1 and l 2 , as represented in Eq. (9), where detunings l 1 and l 2 are required for stable operation of the OPO 22  www.nature.com/scientificreports/ at frequency ω.
To investigate quantum correlations of the generated light at sidebands frequency ω , we employ a vector of quadrature operators in HG mn modes, where X mn (ω) =Â mn (ω) +Â † mn (−ω) and P mn (ω) = (Â mn (ω) −Â † mn (−ω))/i . Express Eq. (15) with this vector operator as with where I kl,mn = δ kl,mn , G kl,mn = gµ k+l δ kl,mn , D kl,mn = � kl δ kl,mn . Using the input-output relation at the coupler As the input is the vacuum state, we obtain the covariance matrix generated from the OPO as follows Note that the covariance matrix V(ω) does not exhibit any coupling between different HG mn modes. We can therefore characterize it by considering each HG mn mode individually. The quantum state in each HG mn mode turns out to be a squeezed vacuum aligned with rotated quadratures X (�) mn (ω),P (�) where all the parameters are real numbers, and Note that �� 2X (�) mn (ω)� ≤ 1 and �� 2P (�) mn (ω)� ≥ 1 , and as ˜ mn → 0 , the angle is 0 if 0 ≤g mn or π/2 if g mn < 0 . Without loss of generality, we will focus on the case of 0 ≤g mn by setting 0 ≤ g and 0 ≤ µ < 1 (equivalently, 0 < ξ ≤ 1 ). One can further note that the intracavity loss T l makes reduction on the cavity escape efficiency η.
The generated multimode light from the OPO is therefore a collection of individual squeezed vacua in multiple HG mn modes, whose modal structure and quantum correlations are described by Eqs. (5) and (21), respectively. In more detail, the spectrum of quantum correlations is determined by the focusing parameter ξ , modifying g mn in Eq. (22), and the waist size is by ξ and the pump waist w p , as discussed in Eq. (5). For ξ = 1 , making µ = 0 , the generated light is a single-mode squeezed vacuum: HG 00 contains a squeezed vacuum ( �� 2X (�) 00 (ω)� < 1 ), but all high-order HG mn modes ( m, n = 0 ) are vacuum states ( �� 2X (�) mn (ω)� = 1 ). On the other hand, as ξ becomes smaller than one, µ becomes positive, and high-order modes also exhibit squeezing ( �� 2X (�) mn (ω)� < 1 ). Figure 2a compares the squeezing levels �� 2X (�) mn (ω)� for different values of ξ . The smaller ξ exhibits higher squeezing levels than the larger one for all high orders of HG mn , and the associated Schmidt numbers calculated from Eq. (6) are 8.3 ( ξ = 1/9 ) and 20.7 ( ξ = 1/81 ). The angles for X (�) mn (ω) quadratures are all zero as expected.
(16) Q(ω) = [X 00 (ω),X 01 (ω), . . . ,P 00 (ω),P 01 (ω), . . .] T , www.nature.com/scientificreports/ Figure 2b,c show the effects of the Gouy phase shift θ G on the generated light. The Gouy phase creates the detuning ˜ mn , which affects both the squeezing level and the squeezing angle. For small detunings, the squeezing level and the squeezing angle remain similar to the ideal self-imaging case, but as θ G increases, the squeezing level �� 2X (�) mn (ω)� gradually degrades to zero, and the squeezing angle increases to π/2 . Such effects are stronger for a smaller transmittance T i and a higher HG mn mode, as expected from Eq. (22). In addition, the squeezing level and the squeezing angle depend on the sideband frequency ω , as shown in Fig. 2d,e. As HG 00 mode exhibits ˜ mn = 0 , it behaves as a common OPO, where the squeezing level decreases while the squeezing angle remains constant as ω increases. On the other hand, for higher modes where ˜ mn � = 0 , both of the squeezing level and the squeezing angle depend on the sideband frequency ω . The rotated squeezing angle due to non-zero ˜ mn returns to zero as ω increases. The squeezing level, in most cases, gradually decreases to zero by increasing ω , but there is a special case showing a non-monotonic behavior (the blue dashed line for HG 02 in Fig. 2d), which is because a non-zero value of ω makes the minimum value for �� 2X (�) mn (ω)� : such a case can take place at ω = �2 mn −g 2 mn − 1 for � 2 mn >g 2 mn + 1 , which can be derived from Eq. (21).

Robustness on spatial mode mismatch
Spatial mode mismatch occurs when the mode of quantum light is different from a target mode, e.g. due to beam displacement, tilting, and beam size difference. Mode mismatch is especially detrimental for couplings with single-mode elements and processes, e.g., optical cavities, optical fibers, frequency conversion, and homodyne detection. In this section, we will show that the multimode squeezed light from the self-imaging OPO is robust www.nature.com/scientificreports/ on various types of spatial mode mismatch. When deriving the result, we will consider mode mismatch only in the x-direction, but the same result can be equally obtained for the y-direction because of the symmetry of the multimode squeezed light described in Eqs. (4,5,21).
Mode-mismatch model. To model the spatial mode mismatch, instead of fixing a target mode and varying the modes of quantum light, we will use an equivalent way for the simplicity of mathematical description: we fix the quantum light but make deviations on the target mode. We consider a target mode of HG 00 with the waist size of w t and its deviations due to mode mismatches [displacement (d), tilt ( ϕ ), and size difference (w)] are respectively. Figure 3a describes the mode mismatches on a target plane. One can expand a mismatched mode φ mis based on the HG modes φ mn stemming from Eq. (23) where mis ∈ {disp,tilt,size} and a mode-mismatching parameter p ∈ {d, ϕ, w} , and Size difference

Multimode light source
Multimode light source  www.nature.com/scientificreports/ and Figure 4 shows the coefficients β disp mn , β tilt mn (−i) m+n , and β size mn , which are all real values. As d, ϕ, and w deviate from the ideal mode-matching condition more, HG 00 contributes less, which is replaced by the contributions from high-order HG mn modes.
By defining the creation operator (B mis ) † for mode φ mis and B † mn for mode φ mn , Eq. (25) can be expressed as The effect of mode mismatch can therefore be understood as contributions from high-order HG mn modes due to the emergence of non-zero coefficients β mis mn . More specifically, a quadrature operator for the mismatched mode is written as When the coefficients are real ( β mn ∈ R ) and no correlation exists between different HG mn and HG kl , i.e., ��(B mn +B † mn )�(B kl +B † kl )� = 0 , the quadrature variance in the mismatched mode is which is the weighted mean of the quadrature variances in the HG mn modes with the weighting factors of β 2 mn . As the mode mismatch increases, the weight for HG 00 decreases, and the noises from high-order HG mn come in. Since single-mode squeezed light exhibits a squeezed noise in HG 00 and the vacuum noise in HG mn , the squeezing level quickly degrades to the vacuum noise due to mode mismatch. On the other hand, multimode squeezed light exhibits squeezed noises in high-order HG mn modes together. As a result, multimode light can show less degradation on the squeezing level, which, therefore, tolerates more mode mismatch than single-mode light does.
Mode-mismatch tolerance of multimode squeezed light. We will use the multimode squeezed light in "Quantum properties of multimode squeezed light" section to investigate its robustness on mode mismatch. We first consider the ideal mode matching of the multimode light with a target mode and then, to account for mode mismatch, we will make deviations on the target mode, as discussed in "Mode-mismatch model" section. Figure 3b,c depicts linear optical elements through which the multimode light propagate from the OPO to a target plane. The optical elements transform the HG modes ψ mn in Eq. (11) into new modes I [ψ mn ] , which can be obtained by Huygen-Fresnel's integral I through the associated ABCD matrix 31 : where A, B, C, and D are the matrix elements, and  We thus obtain the expression of a quadrature variance at the mismatched mode at sideband frequency ω: where the sideband quadrature operator X mis (ω) is B mis (ω) + (B mis ) † (−ω) , the covariance matrix V(ω) is given in Eqs. (20,21), and As V(ω) contains X -quadrature squeezed vacua in HG mn modes when θ G = 0 and g mn > 0 , if γ mn ∈ R , only squeezed-quadrature noises are coupled into the mismatched mode, which makes the multimode squeezed light robust on mode mismatch. In the image plane, such a condition is satisfied for mode mismatches by displacement and beam-size difference, Second, we investigate the mode mismatch in the Fourier plane (FP), described in Fig. 3c. The focal length of the lens is chosen as f 1 = w t w c π/ 0 . Denoting the unitary operation for transforming into the Fourier plane by Û FP , the associated creation operators are related as and thus, the variance by the sideband operator X mis (ω) =B mis (ω) + (B mis ) † (−ω) is where Like the case of the image plane, the condition ζ mn ∈ R makes the multimode squeezed light robust on mode mismatch. In the Fourier plane, mismatches by tilt and beam-size difference with an additional π/2-phase shift satisfy the condition, Figure 5 shows the robustness of the multimode squeezed light V(ω) on mode mismatch, compared with the result of a single-mode squeezed light in HG 00 . We first consider the multimode light by the ideal self-imaging condition ( θ G /2π = 0 , black solid line), and more general cases will be discussed later. As shown in Fig. 5a, when mode mismatch by displacement (in the image plane) or tilt (in the Fourier plane) occurs, the squeezing level by the single-mode light (original squeezing of 9.5 dB, blue dashed line) quickly degrades, e.g., less than 3 dB for d/w t > 1 or πw t sin ϕ/ 0 > 1 . On the other hand, the multimode light with the same squeezing in HG 00 maintains the squeezing level very well by tolerating the mode mismatch, exhibiting more than 7 dB in the same condition. It is noteworthy that, at sufficiently large mode mismatch, the multimode light even outperforms single-mode light with infinite squeezing (black dashed line). Furthermore, the multimode squeezed light is (32) I IP [ψ mn ] = (−1) m+n+1 φ mn (x, y), (41) www.nature.com/scientificreports/ robust on beam-size mismatch on both the image plane and the Fourier plane, as shown in Fig. 5b. Similar to the previous case, the multimode light maintains the squeezing level very well in the influence of mode mismatch, even outperforming the single-mode infinitely squeezed light.

Effect of loss.
Here we investigate the effect of loss on the multimode light in terms of the mode mismatch.
In Eqs. (21,22), the escape efficiency η accounts for the intracavity loss, but it can be generalized to incorporate the total loss in the system, 1 − η , e.g. propagation and detection losses. Figure 6 shows the squeezing level by mode mismatch for different amounts of losses. η = 1 corresponds to no loss in the total system ( 1 − η = 0 ), which is identical with the black solid lines ( θ G /2π = 0 ) in Fig. 5a,b. As the loss increases by reducing η , the squeezing level decreases for all the three cases of infinitely squeezed single-mode light, single-mode squeezed light (9.5 dB), and the multimode light (9.5 dB in HG 00 mode). Although such losses exist, we still find that the multimode light is more robust on mode mismatch than the single-mode light (9.5 dB), and for a sufficiently large mismatch, it again outperforms the infinitely squeezed light.
Effect of mode mismatch inside the OPO. In "Self-imaging OPO", we assumed that the eigenmodes of the interaction Hamiltonian (4) perfectly match with the cavity modes (11), i.e., the same waist size, w H = w c . However, mode mismatch can take place inside the OPO due to waist size difference ( w H = w c ) or the Gaussian approximation ( sinc(x 2 ) ≈ exp(−αx 2 ) ) used for the Kernel. We investigate how the mode mismatch inside the OPO affects the robustness of multimode light on mode mismatch to a target mode. At first, we consider the waist size difference ( w H = w c ) while keeping the Gaussian approximation. To deal with the size difference, we employ a change of basis from the eigenmodes of the interaction Hamiltonian to the cavity modes Figure 5. Robustness of multimode squeezed light on mode mismatch. The squeezing level coupled into a target mode is plotted by varying (a) displacement or tilt and (b) beam size. We use T i = 0.1 , T l = 0 , ω = π/25 , and g 00 = 1/2 . The black dashed line is for the single-mode infinitely squeezed light, and the blue dashed line is for single-mode 9.5-dB squeezed light, and the solid lines are for multimode squeezed light with ξ = 1/81 for different Gouy phase shifts of θ G . Figure 6. Effect of loss on the squeezing level with mode mismatch. 1 − η corresponds to the total optical loss (e.g. by including the detection inefficiency). For different amounts of η = 1 (black), η = 0.95 (red), and η = 0.9 (blue), the performances of three different squeezed lights are compared (dot dashed: single-mode infinitely squeezed light, dashed: single-mode 9.5-dB squeezed light, and solid: multimode squeezed light with ξ = 1/81 ). We use the following parameters for the plots: ω = π/25 , g 00 = 1/2 , ˜ mn = 0. www.nature.com/scientificreports/ where the basis change matrix U(w c , w H ) is given as 34 By describing the interaction Hamiltonian in the cavity mode basis, one obtains a modified gain matrix G ′ where G is the original gain matrix in Eq. (18). Differently from G , G ′ is a non-diagonal matrix in general. One can use G ′ instead of G for calculating the covariance matrix (20) and the squeezing levels in target modes (35,39). Figure 7a,b shows that, even with a large difference in waist sizes ( w H = 1.4w c ), the light from the self-imaging OPO still exhibits robustness on mode mismatch: displacement or tilt in Fig. 7a and beam size in Fig. 7b. This robustness is due to the multimode nature of the interaction Hamiltonian: although the waist size of the interaction Hamiltonian varies, the interaction Hamiltonian can still provide a multimode gain ( G ′ ) in the multiple cavity modes, which in turn generates multimode squeezed light required for robustness on mode mismatch.
Second, we consider the interaction Hamiltonian without using the Gaussian approximation. Let us rewrite the associated kernel (2) using w p , ξ , and α: wc , n, n ′ � |m − m ′ | and |n − n ′ |: even 0 else www.nature.com/scientificreports/ We decompose the kernel numerically since analytical expression is unknown due to the inclusion of the sinc function 28 . The Schmidt number solely depends on ξ/α because w p just acts as the scaling factors of q s and q i . To compare the properties of the original Hamiltonian (2,46) and those by the approximated one (4), we find, for a given ξ , the coefficient α that gives the same Schmidt number as the Gaussian approximation (6); this way of choosing α is justified because the robustness on mode mismatch originates from the occupation of squeezed light in multiple modes, depending highly on the Schmidt number. For ξ = 1/81 , the corresponding α is 0.46. In addition, w p is determined by maximizing the overlap between the first eigenmode of Eq. (46) and the HG 00 cavity mode. A modified gain matrix G ′ is then obtained by is the inverse Fourier transform of Eq. (46), and ψ mn (x, y) are the cavity modes defined in Eq. (11). G ′ , being a non-diagonal matrix, is used instead of G to find the covariance matrix (20) and the squeezing levels in target modes (35,39).
In Fig. 7c,d, we compare the robustness of mode mismatch by the original Hamiltonian and by the approximated Hamiltonian. It is evident that both cases exhibit robustness on mode mismatch by outperforming the single-mode squeezed light. In Fig. 7c with θ G /2π = 0 , one can find that the original Hamiltonian shows a slightly better performance than the approximated one. It is because, while the Schmidt numbers are the same, the eigenvalue spectrum of the original Hamiltonian is distributed more toward lower-order eigenmodes than that of the approximated Hamiltonian, which is advantageous for a small amount of mode mismatch. At θ G /2π = 0.002 , their difference becomes negligible because of the squeezing angle rotation in high-order modes. In Fig. 7d with θ G /2π = 0 , the original Hamiltonian shows better squeezing level for w/w t > 1 but worse for w/w t < 1 when compared with the approximated Hamiltonian: the asymmetry comes from the negative correlations between even-order HG mn modes for the original Hamiltonian (e.g., G ′ 00,04 , G ′ 00,22 < 0 in Eq. (47)). At θ G /2π = 0.001 , the difference in the performances of the two Hamiltonians is negligible as in the case of Fig. 7c. Effect of Gouy phase shift. Until now, we have explored the robustness on mode mismatch in the ideal self-imaging condition. For stable operation of the OPO, however, a small detuning by the Gouy phase shift is necessary 22 . The detuning degrades the squeezing level and rotates the squeezing angle for high-order HG modes, as discussed in Fig. 2. Here we investigate whether the robustness on mode mismatch can still be sustained with detunings from the ideal condition. Figure 5 compares the squeezing levels coupled in a target mode with non-zero Gouy phase shifts, θ G /2π = 0.002, 0.004 , and 0.006 for the displace/tilt mismatching and θ G /2π = 0.001, 0.002 , and 0.003 for the size mismatching. As expected, the squeezing level becomes degraded as more Gouy phase shift is introduced. For θ G /2π = 0.002 in (a) and θ G /2π = 0.001 in (b), the generated multimode light can still beat the performance of the infinitely squeezed single-mode light for a sufficiently large mismatch. The cases of θ G /2π = 0.004 in (a) and θ G /2π = 0.002 in (b) exhibit a squeezing level worse than the infinitely squeezed light but better than 9.5 dB squeezed single-mode light. However, we find no advantage in using the multimode light with θ G /2π = 0.006 in (a) and θ G /2π = 0.003 in (b), being worse than the 9.5 dB single-mode light; in this regime, due to the rotations of the squeezing angles of high-order HG modes in Fig. 2c, X mn (ω) quadratures exhibit larger noises than the vacuum noise. To take advantage of using multimode light, keeping a small-enough Gouy phase shift is required.
We, therefore, investigate how small Gouy phase shift is required to exhibit advantages of using multimode squeezed light. For the quantification, we define an enhancement factor F in decibels, where � 2X(sin) is the squeezing level by single-mode squeezed light, and � 2X(mul) is the squeezing level by multimode squeezed light (having the same initial squeezing level in HG 00 as the single-mode light). A positive value of F indicates that the multimode squeezed light is more robust to mode mismatch than the single-mode squeezed light. As the Gouy phase shift is determined by the detuning ratios �l 1 /R and �l 2 /R , as given in Eq. (10), we calculate the enhancement factor as varying the detunings. Figure 8 shows the enhancement factor of using the multimode squeezed light when the mode overlap by mismatch is 50% (corresponding to the dashed lines at d/w t = π w t sin ϕ/ 0 = 0.83 and w/w t = 2.45 in Fig. 4). A broad range of �l 1 /R and �l 2 /R exhibits enhancements compared with the single-mode case. Comparing (a) and (b) in Fig. 8, mode mismatch by size difference requires more stringent conditions for the enhancement; this is because size difference involves higher-orders of HG modes than those by displacement and tilt, as shown in Fig. 4, and the squeezed lights in higher-order modes are more susceptible to the Gouy phase shift as shown in Fig. 2b,c. However, these stringent conditions are still achievable using only off-the-shelf positioning devices: a typical OPO employs a curved mirror with the radius of the curvature in the order of R = 100 mm, and position controllability in the order of 100 µ m (e.g. by using a linear stage) can readily achieve very small values of �l 1 /R, �l 2 /R = 0.001. |� q s + � q i | 2 sinc w 2 p 4 ξ α |� q s − � q i | 2 .

Conclusion
In this paper, we have shown that multimode squeezed light generated from a self-imaging OPO is robust on spatial mode mismatch. First, we found an analytic form of the quantum properties of the multimode light at sidebands frequency by taking into account the Gouy phase shift (required for OPO stability) and the intracavity loss. By decomposing mode mismatches of displacement, tilt, and size difference into a HG mn mode basis, we found that the mode mismatches induce contributions from high-order HG mn modes, which makes the multimode squeezed light robust on mode mismatch. We showed that the multimode light from the self-imaging OPO with a small Gouy phase shift can even outperform the single-mode infinitely squeezed light, in terms of displacement and size difference in the image plane and of tilt and size difference in the Fourier plane. Such robustness on the multiple cases of mode mismatch is made possible because of the fully degenerate nature of the self-imaging OPO, which cannot be accomplished by using a confocal OPO 18 or two single-mode OPOs 16 . Our work of mitigating the mode mismatching loss will have broad applications to quantum technologies based on squeezed light, e.g., quantum-enhanced gravitational-wave detection 11,12 , deterministic quantum teleportation 7 , measurement-based quantum computing [1][2][3][4] , and Gaussian boson sampling 5,6 . . When mode mismatch is 50% by (b) displacement or tilt and by (c) size difference, we compare the squeezing levels in a target mode by single-mode squeezed light (9.5 dB) and multimode squeezed light (9.5 dB at HG 00 and ξ = 1/81 ). l 1 and l 2 are the detunings applied for cavity stability. F > 0 indicates that the multimode squeezed light is better than the single-mode light. The area where the graph is not drawn corresponds to an unstable region of the OPO. For the plot, we use parameters of T i = 0.1 , T l = 0 , ω = π/25 , and g 00 = 1/2.