1. Introduction

Second harmonic generation (SHG) was first proposed by Khokhlov et al. [1] in 1961 and demonstrated by Franken et al. [2] in the same year. It was accomplished by passing a ruby laser pulse through crystalline quartz and observing a color change at the second harmonic (SH) wavelength. Since that time, SHG has had many uses in both research and industry. The generation of an optical SH beam depends on the square of the pump intensity at fundamental frequency, and on the second order nonlinear susceptibility (χ ⁽²⁾) of the nonlinear material [3]. The requirements for achieving efficient SHG are high pump intensity, tight pump focusing, large χ ⁽²⁾ and low optical losses in the crystal. Moreover, the fundamental and SH generated beams must preserve their phase relation over the length of the nonlinear crystal. The latter requirement is known as the nonlinear phase-matching condition and is achieved experimentally by taking advantage of the birefringence of the nonlinear crystal [4].

Although it is well known that the SHG process can be accomplished with any pump mode, SHG has generally been confined to the use of a single fundamental mode, namely TEM₀₀ [5, 6]. The use of TEM_n0 modes corresponding to the higher order Hermite-Gauss (H-G) modes [7] of order n, has until now been given less attention. A full analysis of the SHG efficiency, assuming a Gaussian spatial TEM₀₀ profile of the beams, has been described in the work by Boyd and Kleinman in 1968 [8], but no general extension of this analysis to higher order TEM_n0 has - to our knowledge - been proposed. While the efficiency of the SHG with higher order pump modes has been mentioned [9], the influence of phase matching on the SH mode profiles has not been detailed.

In this paper, we describe the use of SHG for spatial mode control. The experimental setup is illustrated in Fig. 1. We use two quantitative models to account for our experimental results ; the first one is analytical and is based on generalized Boyd-Kleinman factors including higher order H-G modes SHG. The second one is a numerical model describing the wave propagation step by step in the nonlinear medium [14, 15]. These two approaches complement each other. The first one provides a good physical understanding and focuses on a H-G modes properties, whereas the latter model provides a flexible engineering tool that can be adapted to any given optical system and any pump spatial distribution. We present results for the TEM₀₀, TEM₁₀, and TEM₂₀ pump modes, and compare the predictions with the SHG produced in a single pass experiment operated with a TEM_n0 pump and find good agreement. In particular we can predict that for a TEM_n0 pump beam, with n > 0, the SH transverse profile can be altered continuously by adjusting the wavelength, nonlinear phase-matching or pump focusing. This phenomenon is a consequence of Gouy phase shifts acquired during beam propagation between higher order H-G modes [7]. This phase shift is named after L.G. Gouy [10, 11] who discovered in 1890 a π phase shift across the focus of a beam of light using an interferometer.

 

Fig. 1. Scheme for single pass SHG measurement illustrated for the case of a TEM₁₀ fundamental pump mode.

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A detailed understanding of the influence on Gouy phase shifts - and hence on multi-mode transverse SHG - of pump focusing, pump wavelength and crystal temperature provides the opportunity for applications ranging from simple temperature and wavelength sensors to quantum information and communication devices and techniques [12].

2. Theoretical analysis of the SH mode decomposition

In a given plane of the nonlinear crystal, or in a regime for which a thin crystal approximation can be used, the SHG process combines two photons from the pump field to generate one SH photon with twice the energy. The pump photons can originate anywhere within the spatial profile of the TEM_nm pump mode. However, we will restrict the discussion to TEM_n0 modes and will only describe their transverse variation along the x axis. In this case we can define {u_n} as the fundamental TEM_n0 pump mode, whose first mode has a waist of w ₀. We can also define {v_n} as the SH mode, whose first mode has a waist of w ₀/√2. With these definitions, TEM₀₀ modes at both wavelengths have maximal spatial overlap inside the crystal [3]. The normalized profile of the generated SH field for a TEM_n0 pump can be written, in the thin crystal approximation, as

where V _2i denotes the even SH modes and Γ_ni describes the spatial overlap between the squared pump and the SH modes in the transverse plane. It is given by ${\Gamma }_{\mathrm{ni}}={\int }_{-\infty }^{\infty }\frac{u_n^2\left(x\right)}{{\alpha }_n}{\nu }_{2i}\left(x\right)\mathrm{dx},$ where α_n corresponds to the normalization of the squared pump, that is ${\alpha }_n^2={\int }_{-\infty }^{\infty }{u}_n^4\left(x\right)\mathrm{dx}$ . The generated SH components are all even since the TEM_n0 pump squared profile is always even, with the 2n mode being the highest present in the SH profile. In the special case of a TEM₀₀ pump profile [5, 6] the SH profile is also a TEM₀₀. For all other cases the fundamental and SH profiles differ [13]. The field profile expression given in Eq. 1 has been used to fit the experimental profiles given in Fig. 6(c), which were obtained with defocused pumps, for which the thin crystal approximation is valid, i.e. z_R ≫ l, where z_R is the pump beam Rayleigh range and l is the crystal length.

In the regime where the latter approximation is not valid anymore, i.e. when the pump is focused or when the crystal length is of the order of the Rayleigh range, phase shifts become important in the phase-matching process. Indeed, different TEM_n0 modes accumulate different Gouy phase shifts during propagation along the z axis, which are determined by the following equation : ϕ_n(z) = (n + 1)arctan$\left(\frac{z}{z_R}\right)$[7]. Consequently the SH mode components cannot be simultaneously phase matched to the pump mode for the full length of the nonlinear crystal. The evolution of each SH H-G mode component therefore requires a local field description when propagating through the crystal.

In order to do such an analysis, we use two methods. The first one is based on a generalized calculation of Boyd-Kleinman factors denoted H_n,2p(ξ,∆k) [8]. These terms integrate the nonlinear effects along the crystal length and take into account the Gouy phase shifts of higher order modes. They are defined by

where τ = (z - f)/z_R, z = f corresponds to the position of the beam waist. The indices n and 2_p refer to the TEM_n0 pump mode and to the even TEM_2p0 generated SH mode, respectively. Generalized Boyd-Kleinman factors describe the relationship between the fundamental TEM_n0 and the SH TEM_2p0 modes, and provide an accurate prediction for the SH beam propagation as a function of the beam focusing parameter ξ, and for pump wavelength and crystal temperature dependence through the phase matching parameter ∆k. In our calculations, we make four approximations. We first consider that losses in the crystal are negligible for both the fundamental and second harmonic. This approximation is just to simplify the calculations as we expect to have similar losses for all transverse modes. This allows us to focus more specifically on the differences introduced by the use of higher order H-G pump modes. Secondly, we consider that the beam propagation axis corresponds to the optical axis of the Type I nonlinear crystal and therefore omit any beam walk-off effect. Moreover, we consider that the beam waist of the pump beam is centered in the crystal, as this optimum case is generally adopted in the experiments. Finally, we make the weak pump depletion approximation when using this model.

It is important to note that the generalization to a multi-mode pump is, although feasible, not straightforward with this model, due to the nonlinearity of the process. The calculation needs to be derived for each pump profile and quickly becomes difficult to handle analytically when the number of transverse modes in the pump increases.

The second method we use is a numerical model where the SHG is modeled using fast transform techniques [14, 15]. This wave propagation analysis takes into account diffraction and depletion of the pump beam. It can thus be used when the pump depletion is important, i.e. when the pump intensity dependence of the phenomenon cannot be neglected. Moreover, as this model is based on general field evolution equations and is not restricted to a modal decomposition of the pump, it can handle an arbitrary super-position of different pump modes. The equations for SHG in the paraxial approximation for a type-I nonlinear medium are given by:

 

Fig. 2. SHG efficiency as a function of the focusing parameter ξ, in optimal phase matching conditions, for i) TEM₀₀, ii) TEM₁₀ and iii) TEM₂₀ pump modes. All curves are normalized to the best conversion efficiency obtained for the TEM₀₀ case. We have represented the three most relevant regimes to our analysis : ’thin crystal approximation regime’ corresponding to collimated beams, ’optimal focusing regime’, and ’tight focusing regime’. Two experimental points are represented in the optimal focusing region.

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where E is the electric field and the subscripts 1 and 2 indicate the fundamental and SH fields, respectively. d is the effective nonlinearity. These equations describe the coupling of different higher order modes through the nonlinear susceptibility (χ⁽²⁾) for both SHG’s and OPA’s. More details on the two methods can be found in [13]. These two models have been compared extensively with each other and with the experimental results and show very good agreement. We can thus use these models to make predictions about the behavior of the SHG process under different conditions.

2.1. Optimum focusing

In order to find the best SHG achievable experimentally, i.e. the optimum regime, we have plotted the SHG power as a function of the beam focusing in Fig.2. These curves are all normalized to the maximum achievable SHG power in the case of a TEM₀₀ pump. Moreover, this representation corresponds to optimal phase matching conditions, i.e. for the value of ∆k that maximizes the SHG power. Identical curves have been obtained with both theoretical models.

The maxima occur at ξ_opt0 = 2.84, ξ_opt1 = 2.7 and ξ_opt2 = 2.5 for the TEM₀₀, TEM₁₀ and TEM₂₀, respectively. The value of 2.84 exactly corresponds to the one reported in reference [8] with a TEM₀₀ pump. Latska et al. [16] recently proposed that this optimal focusing parameter value is modified when a position dependent refractive index is generated inside the crystal in order to improve the conversion efficiency . The optimal focusing parameter in this case is ξ_opt = 3.32. The maximum theoretical conversion efficiency, normalized to the TEM₀₀ efficiency is 0.50 for a TEM₁₀ and 0.40 for a TEM₂₀ pump mode. These drops in conversion efficiency relative to the TEM₀₀ case are easily explained by the smaller local intensity in higher order modes.

 

Fig. 3. Modal decomposition of the SH field as a function of the focusing parameter ξ, in optimal phase matching conditions, for fundamental pump modes a) TEM₁₀ and b) TEM₂₀. Traces i), ii) and iii) correspond to the TEM₀₀, TEM₂₀ and TEM₄₀ SH components, respectively. These results have been obtained with the analytical model.

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All efficiency curves, shown in Fig. 2, are rather flat around the optimal value, allowing an operating range between ξ = 2 and ξ = 4 in which close to maximum conversion efficiency can be obtained experimentally. This ’optimal focusing regime’ is hence comprised between z_R values of l/4 and l/8.

The case ξ ≪ 1 corresponds to z_R ≫ l, i.e. to the ’thin crystal approximation regime’. This is obviously not the optimum regime as it corresponds to a large waist w ₀ and therefore to a smaller local intensity. In the opposite regime corresponding to a very tight focusing, i.e. to ξ ≫ 1, the efficiency drops because of de-focusing and because of the importance of the phase shift between fundamental and SH beams at both ends of the crystal. When this phase shift exceeds π/2, the parametric interaction is indeed partly leading to a down-conversion of the SH power [3].

2.2. The modal decomposition

We now focus on the modal decomposition of the SH field predicted by our analytical model, first when the focusing parameter is varied and secondly when the crystal temperature is varied. Fig. 3 shows theoretical curves for the modal decomposition of the SH fields as a function of the focusing parameter, ξ. Due to different accumulated Gouy phase shifts for different focusing cases, the SH mode components are differently phase matched to the pump mode when the focusing parameter is changed. It can been seen on Fig. 3 that in the case of TEM₁₀ pump mode and with a focusing parameter of approximately ξ = 3 the generated SH fields is almost a pure SH single TEM₂₀ mode, whereas by using a focusing of ξ ≪ 1, the mode decomposition is 0.34 and 0.66 for the SH TEM₀₀ and TEM₂₀ components, respectively. In the case of a TEM₂₀ pump mode, the SH field is always a tunable linear combination of all three components. Nevertheless, it cannot be purely one of these and mainly converts into a SH TEM₄₀ component. The results presented here have been used to fit the profiles presented in Fig. 6.

 

Fig. 4. Mode components of the SH fields as a function of the phase matching temperature, in optimum focusing conditions, for a) TEM₁₀3 and b) TEM₂₀ pump modes. Traces i), ii) and iii) correspond to the TEM₀₀, TEM₂₀ and TEM₄₀ SH components, respectively. These results have been obtained with the analytical model.

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In addition to the influence of the beam focusing on the SHG efficiency and modal decomposition, phase mismatch, also plays a big role, as shown in Fig. 5. In order to describe the SH modal composition according to changes in the crystal temperature, we have in Fig. 4 plotted the percentage of the TEM₀₀, TEM₂₀ and TEM₄₀ SH components in the overall generated SH field. In the case of a TEM₁₀ pump mode, the SH field can be tuned from a pure TEM₀₀ mode into a pure TEM₂₀ mode. Once again a TEM₂₀ pump field produces a temperature dependant tunable linear combination of the TEM₀₀, TEM₂₀ and TEM₄₀ modes. Close to perfect conversion into TEM₀₀ and TEM₄₀ SH components can be obtained.

3. Results

Figure 1 illustrates the SHG setup used in this study. The laser source is a diode-pumped monolithic Nd:YAG ring laser, wavelength 1064 nm. We generate the TEM_n0 pump mode by mis-aligning a ring cavity designed to prevent any transverse mode degeneracy and locked to the resonance of the TEM_n0 mode. This device is labeled as ’mode-converter’ in Fig. 1. The available TEM₀₀, TEM₁₀ and TEM₂₀ pump power is 250 mW, 80 mW and 55 mW, respectively. The SHG crystal is a bulk lithium niobate (MgO:LiNbO₃) type I second order nonlinear crystal. The transmitted fundamental beam is filtered out from the generated SH beam with a dichroic mirror (DM) and measured with a CCD camera in the far field (FF) of the center of the crystal

The experimental parameters we vary are the crystal temperature and the focusing parameter ξ= l/2z_R where l is the length of the nonlinear crystal, z_R =πω² ₀/λ is the Rayleigh length, ω₀ is the pump beam waist size and λ is the pump wavelength.

3.1. Experimental investigation of the SHG efficiency

First, the influence of beam focusing on the SHG efficiency is studied experimentally in the optimal focusing regime for TEM₁₀ and TEM₂₀ pump modes. We obtained 0.45±0.05 and 0.34±0.05, respectively, where we used a focusing parameter of ξ = 3 giving a focusing waist of w ₀ = 35μm. We find good agreement with the expected values, as shown in Fig. 2.

The temperature dependance of the SHG efficiency is plotted in Fig. 5. The plot shows the normalized SHG efficiency as a function of the nonlinear crystals temperature. The nonlinear crystal we are modeling has a birefringence that is temperature dependant and therefore it influences the phase mismatch. It yields the well known ’sinc² function’ shape for the TEM₀₀ pump case when the crystal temperature is varied [3]. The higher order H-G pump modes yield SHG efficiency curves that are more structured than the TEM₀₀ case. This is due to the contributions from the different TEM_n0 components of the multi-mode SH field. The different propagations along the crystal lead to different phase matching conditions for each H-G modes, and hence the overall SHG efficiency becomes asymmetric. It can be seen in Fig. 5 that the experimental data and theoretical plots are in very good agreement. The differences are mainly due to smaller observed SH conversion efficiency as a consequence of non-optimal focusing of the pump into the SH material. Note that the optimal phase-matching condition corresponds in each case to the generation of the SH mode of highest order in the decomposition of the SH beam. With a TEM₀₀ pump it is the TEM₀₀ SH. With TEM₁₀ and TEM₂₀ pump modes it is the TEM₂₀ and TEM₄₀ SH modes, respectively.

3.2. Experimental investigation of the SH mode decomposition

The SH 1D and 3D profiles shown in Fig. 6 are presented for different crystal temperatures and two different focusing parameters. Figure 6(a) and (b) depicts the TEM₁₀ and TEM₂₀ pump mode cases, respectively. In Fig. 6(c), the SH 1D and 3D profiles are presented for ξ ≪ 1, corresponding to z_R ≫ l, and with optimal phase matching.

The results in Fig. 6 show that a change of the crystal temperature causes a change in the phase matching ∆k, and modifies of the spatial profile of the SH field. In the case of a TEM₁₀ pump the SH field can be converted from a predominantly TEM₀₀ mode to a predominantly TEM₂₀ profile. Changing the crystal temperature allows reproducible control of the combination of the TEM₀₀ and TEM₂₀ modes in the SH field. This can clearly be observed on the animation presented in Fig.7. The SH mode profile was recorded with a CCD camera when the temperature was continuously varied. This corresponds to a TEM₁₀ pump used in optimal focusing conditions. Similarly, for the TEM₂₀ pump mode we find that the SH field is also a temperature dependant linear combination of the TEM₀₀, TEM₂₀ and TEM₄₀ modes. Comparing Fig. 6(a), (b) with (c), we find that changing the focusing parameter from optimal focusing, z_R = l/6, to non-optimal focusing, z_R1 ≫ l, induces a modification in the spatial profile of the generated SH field [13].

We have experimentally shown that the spatial profile of an SHG field when pumped with higher order H-G modes can be modified by adjusting the crystal temperature and the pump focusing parameter, and furthermore that the two models can describe these observations very accurately.

4. Discussion and applications

The results demonstrate that the propagation of the higher order modes in the nonlinear crystal can be understood as an intricate competition between several key parameters, namely the focussing parameter, the laser wavelength and the temperature of the nonlinear medium. The theoretical models predict both the conversion efficiencies and the spatial mode profiles very accurately. This nonlinear optical effect is a spatial mode discriminator (SMD) and is adjusted by altering the phase-matching condition which produces a mode coupling between different TEMs in the χ⁽²⁾ medium. The ratio between the generated SH components can be adjusted continuously by changing the operating parameters. The relevance of the SMD process is that we can use it as a measurement tool for the focusing parameter, the laser wavelength and the nonlinear medium’s temperature with the advantage that optical cavities or interferometers are not required. It is therefore intrinsically stable. Note that, since SHG is the reciprocal process of optical parametric amplification (OPA), we have used this effect in reverse to pump an OPA and selectively generate squeezed light in a desired higher order spatial mode [17].

 

Fig. 5. Normalized temperature dependence of second harmonic conversion efficiency for pump modes a) TEM₀₀, b) TEM₁₀ and c) TEM₂₀. The experimental data are represented by the black dots with indicative error bars. The solid lines are made with both theoretical models. The decomposition of the SH mode is also shown for each temperature with dotted/dashed lines. Traces i), ii) and iii) correspond to the TEM₀₀, TEM₂₀ and TEM₄₀ SH components, respectively. The maximum normalized theoretical conversion efficiency is 1 for a TEM₀₀ pump, 0.50 for a TEM₁₀ and 0.40 for a TEM₂₀ pump mode. The pump beams were focused in all cases to a waist of w ₀ = 35μm, inside a 20 mm long lithium niobate crystal.

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In Fig. 8(a), we illustrate a scheme for using SMD as a temperature sensor or as a device to actively control a laser frequency to pre-programmed values. The common principle first requires the conversion of a TEM₀₀ laser beam into higher order modes using mode-converters such as wave-guides [18], special phase-plates or holograms [19]. The desired TEM_nm mode is then used as a pump in a single pass SHG scheme similar to the one detailed in Fig. 1. As explained in the previous sections, the profile of the multi-mode SH beam thereby generated is strongly dependent on the pump laser wavelength and on the crystal temperature. Its spatial analysis can be performed with a CCD camera, or with a mode separation device [18] associated with individual photo-detectors as shown in Fig. 8(a). This directly provides information on the crystal temperature and laser wavelength.

In order to access the latter experimental parameters, one can for instance use a TEM₁₀ pump mode and extract the normalized SH mode intensity difference ∆A =(TEM₀₀-TEM₂₀)/(TEM₀₀+TEM₂₀). Figure 8(b) depicts ∆A as a function of crystal temperature and laser wavelength. This has been plotted using the numerical model for a 20 mm long MgO:LiNbO₃ crystal, using a TEM₁₀ pump beam around a reference wavelength of 1064 nm, under optimal focusing conditions corresponding to ξ = 2.7, and near optimal phase-matching operation around a crystal temperature of 69.95°C. Note that, at a given wavelength, this figure contains the same information as the one presented in Fig.4. In addition, Fig.8(b) shows that SMD produces a repetitive signal that spans a wide wavelength and temperature range. The spacing between the peaks is simply set by the phase shifts of the nonlinear process and is hence a constant set by the physical geometry of the system. Note that a value of ∆A = 1 and ∆A = - 1 corresponds to a pure TEM₀₀ or TEM₂₀ mode, respectively.

 

Fig. 6. SH profiles generated in the crystal, observed in the far field (FF), for three different phase matching temperatures, with optimal focusing conditions, a) TEM₁₀ and b) TEM₂₀ pump modes. The cross-section traces contain both the experimental data and the theoretical fits. The length of the crystal was l = 5 mm and a focusing parameter of ξ = l/6 was chosen. c) SH profiles for TEM₁₀ and TEM₂₀ collimated pump modes, i.e. the thin crystal approximation. Again the cross-section traces contain both the experimental data and the theoretical fits. The cross-section traces are made with the generalized Boyd-Kleinman analysis. The experimental parameters used are a focusing of ξ ≪ 1, and with optimal phase matching, ∆k = 0.

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Fig. 7. This movie has been recorded with a CCD camera and shows the SH profiles generated by the non-linear crystal. The phase matching temperatures is swept from below optimum phase matching to above optimum. The field is a TEM₁₀ mode at 1064 nm and the SH filed moves from a predominantly TEM₂₀ mode to a predominantly TEM₀₀ mode. [Media 1]

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Fig. 8. a)Illustration of a scheme for temperature sensing and control of a laser frequency over a wide spectrum. The beam is first converted into a desired higher order mode, in this case a TEM₁₀ mode, with a mode-converter. Pumping the nonlinear material with this beam then generates a SH multi-mode beam. The detection of these modes requires spatial detectors like array or CCD detectors arrays or mode separator associated with individual detectors. b) Normalized SH mode amplitude difference dependence, ∆A, as a function of temperature and wavelength for a TEM₁₀ pump mode using the numerical model.

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As an example, if we generate 1 mW of SH power at 532 nm from a 20 mm long MgO:LiNbO₃ crystal, assuming detectors with 100% quantum efficiency, the shot noise limited sensitivity is: 2×10^-8 × 1/√Hz. It can be seen from Fig.8(b) that DA changes by approximately 1/4 °C in the full dynamical range, i.e. from ∆A = - 1 to ∆A = 1. This yields a temperature sensitivity of the order of 2 ×1/4×2.10^-8 = 10^-8 °C×1/√Hz. Moreover, ∆A changes by approximately 1/15 nm in the full dynamical range, which corresponds to a wavelength sensitivity of approximately 2×1/15×2×10^-8≃2.7×10^-9 nm×1/√Hz. This can be translated into a frequency sensitivity of 3×10⁸×2.7×10^-18/(1064×10^-9)²≃700 Hz×1/√Hz. Both sensitivities are linearly proportional to the length of the nonlinear crystal. It is important to note that a more elaborate pump shape and detection can easily improve the sensitivity. This optimization will be investigated using the numerical model that has been developed and tested.

Temperature and frequency sensitivities calculated above assume that all other parameters are fixed, and therefore correspond to an upper limit of the performance in this particular configuration. For example, for temperature sensing, the frequency of the pump laser can be stabilized at less than 1 kHz for 1 s, using the technology of commercial non planar ring oscillator (NPRO) YAG lasers. Even a beam focusing accuracy of 1 % in a bulk crystal would not limit the temperature sensitivity. One can even think of implementing such a device in a waveguide for reproducible beam focusing. This scheme could also be applied to laser frequency control, by using ∆A to generate an error signal. Good temperature stabilization could allow reproducible frequency locking over a wide range of frequencies, with up to a kHz accuracy.

The single pass scheme presented in Fig. 8(a) could be of interest in optical sensing systems. Another example is the use of the continuous variation of the SH spatial profile for trapping, un-trapping, or continuously displacing particles in an optical beam [20]. A ring-shaped pump beam would be implemented, for instance like a Laguerre-Gauss (L-G) beam [7]. When the crystal temperature is continuously varied, several rings would be generated, moving in or out from the center of the beam. Further optimization of the pump profile for best trapping performance could again be investigated using the numerical model. Similar optical manipulations are currently performed using interference between several optical beams whose frequency is continuously varied [21].