1. Introduction

In the field of nonlinear optics, available lasers are mixed to produce new frequencies, using three-wave-mixing processes in quadratic nonlinear optical crystals. For example, in sum frequency generation (SFG), two frequencies ω₁ and ω_p1 are summed to produce a higher frequency ω₂ = ω₁ + ω_p1. In order for this process to have significant efficiency, momentum conservation needs to be maintained. For the given example, this would require that k⃗₂ = k⃗₁ + k⃗_p1 where k⃗_i is the momentum (wave-vector) of the wave with frequency ω_i, a requirement also known as phase-matching. Three-wave-mixing processes can also be cascaded [1], e.g. the generated sum-frequency can be further combined with another frequency in another SFG process, which produces ω₃ = ω₂ + ω_p2 = ω₁ + ω_p1 + ω_p2.

Recently, a wideband and robust frequency conversion method was developed, by using an analogy between the dynamics of sum or difference frequency generation (DFG) process and a two state quantum system induced by electromagnetic (EM) field [2, 3].

Here, a method to perform frequency conversion via simultaneous three-wave-mixing (STWM) is developed and demonstrated experimentally. Most notably, phase-matching is found to be dependent on light intensity. This is in contrast to the hitherto accepted view that phase-matching depends only on the dispersion properties of the nonlinear medium [4]. Moreover, the intermediate frequency remains dark, hence conversion can be preformed through frequencies at which the nonlinear medium is opaque. The method presented here relies on an analogy drawn between an EM-excited three state quantum system and two STWM processes. This analogy leads to mapping of atomic phenomena such as dark states and Stark shifts to the field of nonlinear optics.

2. Theoretical analysis

2.1. Dynamical equations and atomic analogy

We consider two simultaneous three wave mixing processes (STWM), where the generated frequency is further combined with another frequency, ω_p2, in an additional process, generating a new frequency, ω₃. For the case of two SFG processes, we obtain ω₃ = ω₂ + ω_p2 = ω₁ + ω_p1 + ω_p2. Generally, two distinct pumps can drive the interaction: the first transfers energy between ω₁ and ω₂, while the second connects ω₂ with ω₃. We consider the case where the pumps are much more intense than the other waves and thereby are negligibly affected by the interaction (undepleted pumps approximation). The procedure of adiabatic elimination [5, 6] is used to reduce the three wave system into an effective two coupled waves system. Notably, a work conducted in parallel to ours applied this procedure to the case termed ’direct third harmonic generation’ [7]. Notwithstanding, here we analyze STWM by analogy to a three level quantum system, following the approach of Suchowski et al. [2, 3].

Let us start by writing the coupled equations that govern the evolution of the two STWM processes along a nonlinear crystal:

The same set of equations describes the dynamics of a three level atom interacting with two EM fields [5, 8, 9]. In the context of atomic physics, the complex amplitude of each field A_i (z) corresponds to a_i (t), the probability amplitude of population in each state. Each effective coupling coefficient σ_ij represents the strength of the dipole interaction between levels i and j, usually described by the Rabi frequency Ω_ij = d_ij · ε(t)/h̄, where d_ij is the dipole moment between states i and j, ε(t) is the induced EM field, and h̄ is Planck’s constant. Respectively, the terms Δk₁ and Δk₂ correspond to the detunings, Δ₁ and Δ₂, with the two atomic transitions.

We note that, for the optical case, the notation used here assumes quasi-monochromatic laser beams, which in the visible and near infrared means pulses longer than about 1ps [2]. However, since the interaction is coherent, femtosecond pulses can also be used. Transform limited femtosecond pulses can be stretched before entering the crystal, and the generated frequency pulse can be compressed to a transform limited duration [10].

2.2. Adiabatic elimination

Utilizing the adiabatic elimination procedure is done by assuming that each of the SFG or DFG processes exhibits a very large phase mismatch, but their sum is rather small, i.e. |Δk₁| L ≫ 1, |Δk₂| L ≫ 1 while |Δk₁ + Δk₂| L ≪ 1, where L is the length of the nonlinear crystal. The sum of the phase-mismatches, which is a central notion here, will be defined as the two-photon phase mismatch, and denoted Δk_TP = Δk₁ + Δk₂. Under these conditions, it is reasonable to assume that the oscillating terms in Eq. (1) vary much faster than the fields amplitudes at ω₁ and ω₃, allowing us to write an approximation for the intermediate frequency complex amplitude:

By substituting the outcome of Eq. (2) into Eq. (1) and moving to the rotating frame using A₁ = Ã₁ (z) exp [−i(Δk_eff/2 – σ₁₂σ₂₁/Δk₁)z] and A₃ = Ã₃ (z) exp [i(Δk_eff/2 – σ₂₃σ₃₂/Δk₂)z], a reduced two coupled wave equations set for ω₁ and ω₃ is obtained,

The physical interpretation of Eqs. (2) and (3) is as follows: since each of the two STWM processes is greatly phase-mismatched, the amplitude of the intermediate frequency, |A₂|, will oscillate very rapidly in comparison to variation in the amplitudes of ω₁ and ω₃. As a result, |A₂| will never build up to a significant value and thus remain low throughout the interaction. Nevertheless, energy can still be transferred between ω₁ and ω₃ via ω₂, effectively imitating a four-wave-mixing process with undepleted pumps, where ω₃ = ω₁ + ω_p1 + ω_p2 in the case of two simultaneous SFG processes. We emphasize that the efficiency of conversion from ω₁ to ω₃ does not change even if the material is opaque at ω₂.

The rate of energy transfer and the amount of energy participating in this process are determined by the effective phase-mismatch Δk_eff. In this mechanism, two additional contributions to the conventional phase mismatch parameters appear. These contributions, which depend on the coupling coefficients (and thus on the intensities of the pumps, I_p1 and I_p2), are the equivalent of the atomic Stark shift [5]. In the nonlinear optics context, we choose to define it as δk_S = σ₁₂σ₂₁/Δk₁ + σ₂₃σ₃₂/Δk₂.

This term is critical where significant energy transfer between ω₁ and ω₃ is desired, which occurs when the effective phase-matching parameter is equal to zero, i.e. when Δk_eff = Δk₁ + Δk₂ +δk_S = 0. Since δk_S ∝ I_p it introduces a pump intensity dependence into the phase-matching condition. This is contrary to the conventional wave-mixing phase mismatch, which is considered to depend solely on the dispersion properties of the nonlinear material. Such dependence exists since the rate of each of the two STWM processes depends on pump intensity. To obtain a net flow of energy from one frequency to another, these rates must be compatible over a significant interaction length. The mechanism here is thus fundamentally different from phase-modulation in four-wave-mixing processes, which stems from intensity dependent refractive index.

The analytical solution of Eq. (3), for the initial condition A₃ (0) = 0, is

The effective phase-matching condition can be satisfied using the quasi-phase-matching (QPM) technique [13]. For a periodically poled crystal with period Λ, the phase-mismatch of each of the two STWM processes should be modified with the addition of the periodicity pattern, i.e. in the first order Fourier approximation, Δk̃₁ = Δk₁ + 2π/Λ and Δk̃₂ = Δk₂ + 2π/Λ. Substituting these expressions into the phase-matching condition produces a cubic equation with one solution that adheres to the conditions of adiabatic elimination.

3. Numerical simulation

A QPM solution as described above was used to numerically simulate the interaction. A single pump at λ_p1 = λ_p2 = 1064nm was used to drive both processes. We have used a periodically poled KTiOPO₄ (PPKTP) crystal with a poling period of Λ = 8.6μm, which was calculated to phase-match conversion from λ₁ = 3010nm to λ₃ = 452nm, via the two implicit SFG processes 3010nm + 1064nm → 786nm and 786nm + 1064nm → 452nm, when the crystal temperature is 125°C. The simulation does not assume undepleted pump or adiabatic elimination.

The resulting evolution of the intensities of the interacting waves, with pump intensity of I_p = 10GW/cm², is plotted in Fig. 1(a). Energy oscillations between the λ₁ = 3010nm input and the λ₃ = 452nm output are clearly seen. Since each photon converted from λ₁ to λ₃ is paired with two pump photons, the peak output intensity at λ₃ is higher than that at λ₁. The inset shows the fast oscillations of the low intensity at the adiabatically eliminated intermediate wavelength λ₂ = 786nm, which are over 3 orders of magnitude lower than the peak output intensity.

 

Fig. 1. (a) Numerically simulated intensities of the 3010nm input wave (red dashed line), 452nm output wave (blue dash-dotted line) and 786nm intermediate wave (green solid line). Introducing a 10cm⁻¹ absorption at 786nm decreases the 452nm intensity by just 0.5%. Inset: close-up view of the intermediate wave over 34.4μm. Shading indicates poling. (b) Analytically calculated photon conversion efficiency vs. the two-photon phase-mismatch. The difference between the peak efficiencies phase-mismatches is the Stark shift.

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Using the same parameters, the Stark effect in frequency conversion was analyzed. In Fig. 1(b) the photon conversion efficiency from λ₁ to λ₃ is calculated for two cases: low pump intensity of 1GW/cm² and high pump intensity of 12.4GW/cm². The results are plotted as a function of the two-photon phase-mismatch Δk_TP = Δk₁ + Δk₂. For low pump intensity the Stark shift is negligible, so the peak efficiency is obtained near the two-photon resonance condition Δk_TP = 0. However, when the pump intensity is high, the Stark shift can no longer be neglected and the efficiency peak shifts by δk_S to Δk_TP = −δk_S, which is the exact shift amount predicted analytically using Eq. (4).

4. Experiment

Following the numerical simulation, STWM with adiabatic elimination had been experimentally realized. The experimental system is depicted in Fig. 2. A 4.4nsec Nd:YAG laser beam with λ_p = 1064nm, which served as the strong pumps, was passed through a temperature controlled PPKTP crystal with a Λ = 8.6μm period. A laser with λ₁ = 3010nm and an estimated 10nm FWHM bandwidth, which was produced from the same pump through an optical parametric oscillator, served as our input. The emerging power at the λ₃ = 452nm output and the intermediate wavelength λ₂ = 786nm were independently measured, using power detectors or a spectrometer, and peaks of both wavelengths were observed, corresponding to the expected output and intermediate wavelengths. When either the pump or the input were prevented from entering the crystal, both peaks would vanish.

 

Fig. 2. Experimental apparatus.

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Next, we have chosen to examine the conversion efficiency of the process as a function of the two-photon phase mismatch. This was done by tuning the crystal temperature from 90°C to 175°C, affecting the refractive index [14], and thus the phase-mismatch parameter. The input and pump intensities were kept constant at 71kW/cm² and 44.6MW/cm², respectively. Figure 3(a) shows the experimentally measured 452nm output power vs. the PPKTP crystal temperature, after decrementing the background noise, and accounting for the transmission and reflection of the optical components in the apparatus. The detected noise, defined as the power measured when only the pump beam was incident on the PPKTP crystal, was found to be around 152nW and thus negligible compared to the measured output power which reaches 38μW. As seen, optimum two-photon phase matching was obtained at crystal temperature of 133°C. The small deviation from the expected value of 125°C can be accounted for by the inaccuracy in the Sellemeier equations that were used to calculate the wavelength and temperature dependence of the crystal’s refractive index [14, 15]. The power at λ₂ was below the noise level and therefore could not be measured. Its upper bound was found to be 51.8nW, which is 733 times lower than the peak 452nm output power, agreeing with the theoretical prediction of adiabatic elimination.

 

Fig. 3. Blue dots are experimentally measured 452nm output power vs. (a) crystal temperature (b) c-polarized pump power. The green line was analytically calculated by Eq. (5).

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Finally, the predicted dependence of the 452nm output power on pump power was experimentally checked. A half wave plate was placed in the path of the pump beam before it entered the PPKTP crystal, and gradually rotated through 45 degrees. Due to the fact that KTiOPO₄ is a biaxial birefringent crystal, its phase-matching condition is polarization-dependent: only the pump power polarized along the PPKTP c axis contributes to the STWM processes. Figure 3(b) shows the experimentally measured 452nm output power obtained with various values of c-polarized pump power. In this experiment the noise (defined above) was measured at each wave plate angle and decremented from the detected power. The experimental measurements fit well on the analytically calculated output power (using Eq. (5)), showing a quadratic nature, where the nonlinear coefficient χ⁽²⁾ was taken to be 9% higher than the value reported for other processes and wavelengths [16]. This is consistent with the low conversion efficiency limit of Eq. (5), ${\left|{\tilde{A}}_3\left(z\right)\right|}^2\approx {\left|K_3{\tilde{A}}_1\left(0\right)z\right|}^2\propto I_p^2$. This experimental observation is a clear indication that STWM is in action, since a single TWM process would result in a linear rather than quadratic dependence, while direct third harmonic generation would yield a cubic dependence.

5. Conclusion

We have experimentally demonstrated frequency conversion through an intermediate frequency which remains dark throughout the interaction, enabling conversion through frequencies at which the nonlinear medium is absorptive. Additionally, phase-matching was shown to depend not only on dispersion, but also on pump intensity. This phenomenon enables new methods of all optical switching. We presented an analogy with EM-excited three state quantum systems which lead to these new effects. The same mathematical treatment can also be applied to cases involving three or more STWM processes, analogous to multi-state quantum systems [17]. Other atomic physics schemes, involving temporal variation of the induced EM fields [5, 18], can be transferred to frequency conversion using aperiodically poled structures.