1. Introduction

Although quasi-phase-matching (QPM) predates birefringent phase matching [1], a renaissance has occurred in the last few years due in part to a better understanding of ferroelectric domain inversion [2] and theoretical analysis of manufacturing tolerances [3]. There are several different forms of QPM as shown in Fig. 1. However, almost all of the work has concentrated in copropagating forward-QPM [4,5]. Recently, counterpropagating-QPM (c-QPM) has been shown to exhibit a great deal of potential such as 100% theoretical second harmonic generation (SHG) conversion efficiency at low input intensity levels [6,7], large nonlinear phase shifts due to cascading [7,8], and highly efficient all-optical switches [9]. All preceding work addressed only the steady state analysis of c-QPM. While cw operation is possible in this device due to its very efficient characteristics, the majority of nonlinear optics applications involves pulsed operation.

 

Fig. 1. The wavevector matching diagrams for (a) forward-QPM (f-QPM); (b) backward-QPM (b-QPM); (c) counterpropagating-QPM (c-QPM); and (d) surface-emitting-QPM (se-QPM). There are two simultaneously phase-matchable processes for cases (a), (b), and (d) while there are six for case (c). All cases except case (d) are collinear. The fundamental, second harmonic, and grating wavevectors are represented by red, blue, and black arrows, respectively.

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This paper examines the mirrored device shown in Fig. 2 with a pulsed input wave. In general, the method explored in this paper can be applied to any temporal profile. However, only z-propagating Gaussian temporal envelopes will be considered with the transverse spatial profile taken to be a plane wave,

Furthermore, it is assumed that no second harmonic energy is input to the device. The propagation coordinate z is normalized to the device length L, such that ρ=z/L. A highly reflective mirror is placed at the right side of the device ρ=1. The amplitude and phase shift effects have been discussed elsewhere [7]. Although this analysis does not impose any restrictions on the mirror reflectivities, all numerical examples in this paper assume ideal dielectric mirrors (i.e., r_ω =r _2ω=1).

 

Fig. 2. A schematic of the counterpropagating quasi-phase-matched (c-QPM) device under study. The input/output interface is located at ρ=0. A highly reflective mirror is located at ρ=1. The domain inversion period is Λ.

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For c-QPM four waves, two at the fundamental frequency (FF) and two at the second harmonic (SH), interact through the periodicity of the nonlinear medium. For c-QPM SHG, the domain inversion period Λ is set to the wavelength of the SH wave in the medium for perfect phase matching. These four interacting fields are normalized such their complex amplitudes are given by

where Z₀ is the impedance of free space, n_ω is the FF index of refraction, n_2ω is the SH index of refraction, {$E_{\omega }^{+}$(ρ), $E_{\omega }^{-}$ (ρ), $E_{\mathit{2}\omega }^{+}$(ρ), $E_{\mathit{2}\omega }^{-}$(ρ)} correspond to the four electric field envelopes (i.e., forward FF, reverse FF, forward SH, reverse SH, respectively), and the normalized phase mismatch is given by

One approach to analyzing pulsed operation involves applying the shooting method [10] to the set of normalized cw nonlinear coupled differential equations given in earlier work [7,8]. The initial condition is given by the temporally sampled input envelope. If the mean lifetime for the energy in the device is much smaller than the time between sample points, this approach would be applicable. However, this condition is not met by the given nonlinear system, therefore, this approach was found not to be appropriate.

The problem with the temporal sampling approach in the c-QPM configuration is that energy is temporarily stored inside the device due to the built-in feedback [7]. The field solutions at each time slice depend on the initial conditions for all four waves at every position in the device. Therefore, the time derivatives are explicitly incorporated into the differential equations, and the relaxation method [10] is used to solve them. Then, the method of lines [11] is implemented to solve the nonlinear coupled equations in each time slice by fixing time and separating it from space.

For simplicity, the input wave was assumed to be monochromatic. Because most extremely short pulses (i.e., the femtosecond regime) are created by mode-locking a large number of cavity modes, a large spectral content is contained in the envelope. Thus, the group velocity mismatch is an important consideration for these short pulses. However, in this work, group velocity effects are found to be less important for c-QPM than for common copropagating configurations. In a mirrored c-QPM device, field interaction occurs only upon reflection from the mirror. If the device length is significantly longer than the spatial pulse length, the pulse traverses a large part of the device without any nonlinear effects. Because the nonlinear interaction is confined to the region near the mirror, a device length longer than the spatial length of the pulse is unnecessary, leading to an upper limit on device length. Therefore, the group velocity mismatch is negligible, and is not included in this analysis.

The normalized input FF wave is given by

where I₀ is the peak input intensity at t=0.

Time is normalized into a dimensionless quantity by the use of the parameter g=√2σ. The intensity full width-half maximum pulse duration (t_FWHM ) is related to g and σ by

After normalization with respect to g, both time and frequency are expressed by the unitless quantities,

In terms of normalized time, the input envelope is now given by

Therefore, the input field strength at t_N =1 is 37% of its peak value.

The device length to fundamental spatial pulse length ratio Ξ_ω is the most important parameter for this nonlinear interaction. This ratio is slightly different at the fundamental frequency (FF) and second harmonic (SH) wavelengths due to dispersion. The FF value is considered as a control parameter in this analysis,

The SH ratio Ξ_2ω can be found by multiplying Eq. (8) by the ratio of the SH refractive index to the FF refractive index. If Ξ_ω is approximately one, the device is comparable to the pulse length. In the cw limit, Ξ_ω approaches zero. For Ξ_ω significantly larger than one, the majority of the device length acts as a linear medium. Because group velocity effects are neglected in this analysis, Ξ_ωwill never be considered much larger than one. This limit is consistent with most practical implementations of mirrored c-QPM due to the expense of nonlinear materials and processing.

2. The Normalized Nonlinear Time Dependent System of Equations

The nonlinear system of equations used in earlier work is not adequate for the time dependent analysis. The first and second time derivatives resulted in implicit factors of the form jω and j2ω for the FF and SH quantities, respectively. In this case, the time derivatives are more complicated, and these derivatives must be explicitly kept in the nonlinear system.

Using the same field normalizations given above, the four nonlinear equations can be written as

for the forward propagating FF wave,

for the reverse propagating FF wave,

for the forward propagating SH wave, and

for the reverse propagating SH wave. The nonlinear coupling constant is given by

for first order c-QPM where d₀ is the unmodulated second order nonlinear coefficient.

In this work, the time coordinate is sampled using N_t points over the domain of -4σ≤t+≤4σ in unnormalized time or, correspondingly, -4/√2≤t_N≤+4/√2 in normalized time. Given this sampling range, the normalized time is given by an index parameter p,

where the index p is given by

and the sampling rate is given by

For large values of Ξ_ω (i.e., Ξ_ω>0.5) the pulse width is narrow and, thus, the propagation time in the device is large relative to the time domain as discussed above. Therefore, extra time points are needed in the range t_N >4/√2 to insure that all energy has exited the device. The number of these extra points N_Et is at the same spacing Δt_N as the other time points.

The numerical first and second derivatives with respect to normalized time of an arbitrary function f(ρ, t_N ) are approximated by the common substitutions [10]

respectively. The approximate expressions of Eq. (17) are substituted into the four equations (9), (10), (11), and (12), which are then solved at each discrete time point t_Np given the two previous time points (i.e., t_Np-1 and t_Np-2 ) to calculate the approximate time derivatives.

At each time point t_Np , there is a set of four complex coupled nonlinear equations to solve as a function of one variable ρ. To solve this set of equations, the relaxation method [11] is implemented using the field data from the previous time point t_Np-1 as the initial conditions. At time point t_Np , the propagation coordinate is sampled with N_s points such that

where the index is given by

and the space between points is given by

To implement the relaxation method, the four complex field envelopes are separated into their real and imaginary parts. This numerical method involves solving the set of differential equations at every spatial sample point simultaneously. In this case, the solution vector has 8N_s elements, and an 8N_s by 8N_s matrix is inverted to obtain a correction vector for a trial solution. Two or three iterations were typically adequate to converge to a solution. Convergence is defined when the largest element magnitude in the correction vector is less than 10^-6.

3. Numerical Simulations

Several numerical simulations are presented to study the effects of sampling and of varying Ξ_ω. In all of these results presented, a 1 cm long KTP mirrored c-QPM device with ideal mirrors (i.e., r_ω =r_2ω =1) and perfect phase matching, Δκ=0, is assumed. The input peak intensity |A(0,0)|² is chosen to be Γ ² I₀ =(π/4)². As demonstrated in earlier work [8], this input intensity provides a theoretical 100% SHG conversion for a cw input. Therefore, the effects of the pulsed input are easily discernable by examining the total amount of SH energy exiting the device. The input FF wavelength is 1.064 µm with corresponding refractive indices of n_ω =1.8302 and n_2ω =1.8896 [12] assuming a z-polarized wave.

 

Fig. 3. The input (gray), output FF (red), and output SH (blue) pulse intensity envelopes for a 1 cm long KTP mirrored c-QPM device with an ideal mirror at perfect phase matching. The input wavelength is 1.064 µm with a 10 ns pulse (FWHM) and peak intensity of Γ²I₀ =(π/4)². The corresponding device to pulse ratio is Ξ_ω=0.00718. The simulation parameters are N_s =30, N_t =100, and N_Et =0.

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Fig. 4. The input (gray), output FF (red), and output SH (blue) pulse intensity envelopes for the same parameters of Fig. 3 except the input pulse width is 100 ps pulse (FWHM). The corresponding device to pulse ratio is Ξ_ω=0.718. The sampling rates are (a) {N_s, N_t, N_Et }={30, 100, 0} giving a numerical leakage of 8.38% and (b) {N_s, N_t, N_Et }={30, 300, 0} giving a numerical leakage of 3.08%.

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The simulation results for a quasi-cw configuration are shown in Fig. 3. For this 10 ns long pulse (FWHM), the corresponding device to pulse ratio is Ξ_ω=0.00718. The number of points used in this simulation is modest: {N_s, N_t, N_Et }={30, 100, 0}. Using the rectangular rule, the conservation of energy density can be checked by calculating the energy densities in the input and output pulses. The percentage of the energy lost to the numerical simulation is defined to be the numerical leakage. In this example, the numerical leakage is only 0.092%. The percentage of energy in the output FF and SH pulses are found to be 6.76% and 93.1%, respectively. As shown in Fig. 3, the wings of the pulse are not completely converted to the SH. In these regions, the input intensity and subsequent nonlinear response are too small for efficient interaction. However, the overall SHG conversion efficiency at this pulse length is still near the cw limit. Additionally, a slight delay in the output pulse with respect to the input pulse is seen in Fig. 3. This shift is due to the transit and interaction times of the c-QPM device. This simulation successfully demonstrates that the cw analyses are good approximations for small Ξ_ω.

As the value of Ξ_ω is increased, however, less efficient effects are discovered. Fig. 4 shows the simulation results for a 100 ps long pulse (FWHM), with the same input peak intensity, in the same KTP sample. This pulse length is two orders of magnitude smaller than the previous case. The corresponding pulse to device ratio is approaching unity, Ξ_ω=0.718. Because the numerical values vary more rapidly in time and space than the previous case, more sample points are warranted. This fact is clearly demonstrated in Fig. 4. The example shown in Fig. 4 (a) has the same number of points as the example in Fig. 3, {N_s, N_t, N_Et }={30, 100, 0}. The output SH and FF energies as percentages of the input energy are found to be 14.9% and 77.3%, respectively. The corresponding numerical leakage is large at 8.38%. Therefore, more simulation points are needed to reduce the numerical leakage. The addition of 30 extra time points (i.e., N_Et =30) reduces the numerical leakage to 8.33%. The largest reduction of the numerical leakage relies on significant increases in the number of time points N_t . Using the sample point set {N_s, N_t, N_Et }={30, 200, 0}, the numerical leakage is reduced to 4.50%. This error is reduced to 3.08% if N_t is increased to 300 as shown in Fig. 4 (b). The SHG efficiency found for this example is only 82%. Additional increases in the time sampling rate provides reduction in the numerical uncertainty while simultaneously increasing the demands on the computational resources. In comparison with the more accurate results shown in Fig. 4 (b), the simulation shown in Fig. 4 (a) demonstrates a typical characteristic of under sampling. Due to the lack of resolution in time, the response spreads out in time and the peak amplitude is reduced.

In an effort to reduce the numerical leakage, the number of space points was increased to N_s =50 while the number of time points remained the same. An improvement of only 0.01% was achieved while requiring approximately four times as long to compute. This example demonstrates a general trend that the temporal sampling rate is more important than the spatial sampling rate. For Ξ_ω larger than unity, however, the field profiles change rapidly in space, and the spatial sampling frequency must be increased.

The shape of the output FF envelope in Fig. 4 (b) is significantly different than the envelope in Fig. 3. The first FF peak (at t_N =-1 in Fig. 3) is reduced in the larger Ξ_ω case. Because the pulse is narrower with respect to the device length in this case, the energy from the leading edge of the input pulse is still in the device when the higher intensity energy from the middle of the input pulse arrives at the input interface. Therefore, the low intensity of the leading pulse tail contributed to SH conversion. This characteristic is unique to the c-QPM configuration. In traditional configurations, the leading and trailing edges of the pulse are not converted due to the low intensity levels. Another interesting attribute of c-QPM is demonstrated by the location of the SH peak. The peak of the SH pulse is located at 1.277±0.189 in normalized time. This value corresponds to 108 ps±1.6 ps in unnormalized time. For a linear device, the FF and SH pulses would have exited the device in 122 ps and 125 ps, respectively. As discussed elsewhere [7], the nonlinear interaction captures a significant portion of the energy before it reaches the mirror. This process concentrates the intensity and significantly enhanced the nonlinear behavior.

Figure 5 is an animation of the normalized field magnitudes in the device as a function of normalized time for the simulation shown in Fig. 4 (b). No SH appears until the reflected FF field becomes appreciable near t_N =-0.7662. Both SH fields continue to grow as more FF energy enters the device. However, the reverse SH field |D(ρ)| is always larger than the forward propagating SH field |C(ρ)|. This discrepancy is due in part to the reflection of the SH energy at the mirror. The forward FF envelope peaks near the middle of the device at approximately t_N ≅0.5. After t_N =1.1635, the magnitude of the reverse propagating FF envelope is larger than the corresponding forward FF envelope. This point corresponds to a reversal in the net FF energy flow at the ρ=0 interface. Also observable at this time, the SH envelope peaks near the ρ=0 interface. The delay between the FF and SH peak energies is clearly demonstrated in this animation.

 

Fig. 5. Normalized field profiles inside the device for the simulation parameters used in Fig. 4(b). The field magnitudes are |A(ρ)| (red), |B(ρ)| (orange), |C(ρ)| (blue), and |D(ρ)| (cyan). (363 KB)

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As the value of Ξ_ω becomes larger, the temporal sampling requirements increase rapidly. Figure 6 shows the simulation results for a 50 ps pulse (FWHM) incident on the same KTP sample used in the previous examples. When compared to the example of Fig. 4, this pulse width corresponds to a modest increase in the pulse length to device ratio (i.e., Ξ_ω=1.437). However, the large number of points used in this example produces only marginally acceptable results. The point set used in producing Fig. 6 is {N_s, N_t, N_Et }={40, 600, 300}. The large number of extra time points N_Et was necessary due to the long latency of the device in normalized time. Although a fairly large number of points was used, a modest numerical leakage of 3.3% was incurred. Compared to the previous case, the length of the device is larger with respect to the pulse length and the nonlinear interaction is significantly reduced. The energy in the output pulses was found to be 50.5% and 46.2% for the FF and SH, respectively. Another indication of the reduced interaction is observable by the temporal location of the output peaks, 121 ps for the FF and 114 ps the SH. These results approach the linear propagation values discussed above.

Figure 7 shows an animation of the normalized field magnitudes in the device for the simulation shown in Fig. 6. As shown in the animation, the pulse propagates across the device unaltered until it is reflected from the mirror near t_N =-0.6. Both SH waves appear at the region near the mirror. Therefore, a significant portion of the nonlinear material is superfluous. Similarly, after the nonlinear interaction occurs near the mirror, the resulting envelopes propagate without interaction until exiting the device at ρ=0. This example demonstrates the lack of confinement by the mirror causing a considerable decrease in conversion efficiency for Ξ_ω greater than one.

These examples effectively demonstrate the device length restrictions for c-QPM. In designing copropagating devices, a longer length device is generally better than a shorter one. For ultra-short pulses with a high spectral content, however, the group velocity mismatch imposes an upper limit on the length. For mirrored c-QPM devices, the largest length usable is approximately the same as the spatial length of the pulse. This restriction is a drawback of c-QPM in some ultra-short pulse applications. However, the increased effectiveness of the nonlinear interaction compensates for this disadvantage in many cases.

 

Fig. 6. The input (gray), output FF (red), and output SH (blue) pulse intensity envelopes for a 1 cm long KTP mirrored c-QPM device with an ideal mirror at perfect phase matching. The input wavelength is 1.064 mm with a 50 ps pulse (FWHM) and peak intensity of Γ²I₀ =(π/4)². The corresponding device to pulse ratio is Ξ_ω=1.437. The simulation parameters are N_s =40 N_t =600, and N_Et =300.

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Fig. 7. Normalized field profiles inside the device for the simulation parameters used in Fig. 6. The field magnitudes are |A(ρ)| (red), |B(ρ)| (orange), |C(ρ)| (blue), and |D(ρ)| (cyan). (1.19 MB)

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In all examples presented in this paper, the normalized intensity Γ²I₀ was kept constant while Ξ_ω was varied. Because Γ is proportional to the device length, an increase in Ξ_ω for a given pulse length requires a significant increase in the input intensity I₀ in the above analysis. If the only design parameter is taken to be the length of the device, it was found that the optimum length occurs for Ξ_ω near unity. The reductions in efficiency stem from a lack of confinement for Ξ_ω greater than one and too small of a nonlinear reaction for Ξ_ω less than one.

4. Conclusion

This paper examined the mirrored configuration of counterpropagating-QPM with a pulsed FF input wave. The field solutions to the complete set of c-QPM differential equations with explicit time dependence were found numerically using the method of lines. Several examples were presented that demonstrated the sampling frequency in time was considerably more important that the sampling frequency in space. The ratio of the spatial pulse length to the length of the device Ξ_ω was found to be the principal parameter governing device behavior. If Ξ_ω is significantly less than one, the cw analysis presented in earlier work was found to be applicable. Because of significantly diminished nonlinear interaction when Ξ_ω is greater than one, an upper bound was established for practical pulsed applications; the largest usable length is approximately the same as the spatial length of the pulse.