## Pulse simulations of a mirrored counterpropagating-QPM device

Optics Express, Vol. 5, Issue 8, pp. 176-187 (1999)

http://dx.doi.org/10.1364/OE.5.000176

Acrobat PDF (124 KB)

### Abstract

A basic mirrored counterpropagating quasi-phase-matched device is studied with a pulsed input fundamental plane wave using the method of lines and the relaxation method. Several examples are given under varying spatial pulse length to device length ratios. An approximate upper bound on the device length is established from this study for practical pulsed applications; the largest usable length is approximately the same as the spatial length of the pulse.

© Optical Society of America

## 1. Introduction

1. J. A. Armstrong, N. Bloembergen, J. Ducuino, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. **127**, 1918–1939 (1962). [CrossRef]

2. G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass CW second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. **22**, 1834–1836 (1997). [CrossRef]

3. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. **28**, 2631–2654 (1992). [CrossRef]

4. M. Houe and P. D. Townsend, “An introduction to methods of periodic poling for second- harmonic generation,” J. Phys. D - Appl. Phys. **28**, 1747–1763 (1995). [CrossRef]

6. Y. J. Ding and J. B. Khurgin, “Second-harmonic generation based on quasi-phase matching: a novel configuration,” Opt. Lett. **21**, 1445–1447 (1996). [CrossRef] [PubMed]

7. G. D. Landry and T. A. Maldonado, “Second harmonic generation and cascaded second order processes in a counterpropagating quasi-phase-matched device,” Appl. Opt. **37**, 7809–7820 (1998). [CrossRef]

7. G. D. Landry and T. A. Maldonado, “Second harmonic generation and cascaded second order processes in a counterpropagating quasi-phase-matched device,” Appl. Opt. **37**, 7809–7820 (1998). [CrossRef]

8. G. D. Landry and T. A. Maldonado, “Efficient nonlinear phase shifts due to cascaded second order processes in a counter-propagating quasi-phase-matched configuration,” Opt. Lett. **22**, 1400–1402 (1997). [CrossRef]

9. G. D. Landry and T. A. Maldonado, “Switching and second harmonic generation using counterpropagating quasi-phase-matching in a mirrorless configuration,” Journal of Lightwave Technology **17**, 316–327 (1999). [CrossRef]

*z*-propagating Gaussian temporal envelopes will be considered with the transverse spatial profile taken to be a plane wave,

*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

7. G. D. Landry and T. A. Maldonado, “Second harmonic generation and cascaded second order processes in a counterpropagating quasi-phase-matched device,” Appl. Opt. **37**, 7809–7820 (1998). [CrossRef]

*r*

_{ω}=

*r*

_{2ω}=1).

*Z*

_{0}is the impedance of free space,

*n*

_{ω}is the FF index of refraction,

*n*

_{2ω}is the SH index of refraction, {

*ρ*),

*ρ*),

*ρ*),

*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

**37**, 7809–7820 (1998). [CrossRef]

8. G. D. Landry and T. A. Maldonado, “Efficient nonlinear phase shifts due to cascaded second order processes in a counter-propagating quasi-phase-matched configuration,” Opt. Lett. **22**, 1400–1402 (1997). [CrossRef]

**37**, 7809–7820 (1998). [CrossRef]

*I*

_{0}is the peak input intensity at

*t*=0.

*t*

_{N}=1 is 37% of its peak value.

_{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,

_{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

_{ω}will 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

*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.

*d*

_{0}is the unmodulated second order nonlinear coefficient.

*N*

_{t}points over the domain of -4

*σ*≤

*t*+≤4

*σ*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*,

*p*is given by

_{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 Δ

*t*

_{N}as the other time points.

*f*(

*ρ, t*

_{N}) are approximated by the common substitutions [10]

*t*

_{Np}given the two previous time points (i.e.,

*t*

_{Np-1}and

*t*

_{Np-2}) to calculate the approximate time derivatives.

*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

*N*

_{s}elements, and an 8

*N*

_{s}by 8

*N*

_{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

_{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}

*I*

_{0}=(π/4)

^{2}. As demonstrated in earlier work [8

8. G. D. Landry and T. A. Maldonado, “Efficient nonlinear phase shifts due to cascaded second order processes in a counter-propagating quasi-phase-matched configuration,” Opt. Lett. **22**, 1400–1402 (1997). [CrossRef]

*n*

_{ω}=1.8302 and

*n*

_{2ω}=1.8896 [12] assuming a

*z*-polarized wave.

_{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}.

_{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.

*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.

*(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

**37**, 7809–7820 (1998). [CrossRef]

*(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.

_{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.

*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.

*Γ*

^{2}

*I*

_{0}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

*I*

_{0}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

_{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.

## Acknowledgments

## References and links

1. | J. A. Armstrong, N. Bloembergen, J. Ducuino, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. |

2. | G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass CW second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. |

3. | M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. |

4. | M. Houe and P. D. Townsend, “An introduction to methods of periodic poling for second- harmonic generation,” J. Phys. D - Appl. Phys. |

5. | J. Pierce and D. Lowenthal, “Periodically poled materials & devices,” Lasers & Opt. |

6. | Y. J. Ding and J. B. Khurgin, “Second-harmonic generation based on quasi-phase matching: a novel configuration,” Opt. Lett. |

7. | G. D. Landry and T. A. Maldonado, “Second harmonic generation and cascaded second order processes in a counterpropagating quasi-phase-matched device,” Appl. Opt. |

8. | G. D. Landry and T. A. Maldonado, “Efficient nonlinear phase shifts due to cascaded second order processes in a counter-propagating quasi-phase-matched configuration,” Opt. Lett. |

9. | G. D. Landry and T. A. Maldonado, “Switching and second harmonic generation using counterpropagating quasi-phase-matching in a mirrorless configuration,” Journal of Lightwave Technology |

10. | W. H. Press, S. A. Teukolsky, and W. T. Vetterling, |

11. | D. Zwillinger, |

12. | V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(230.4320) Optical devices : Nonlinear optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 24, 1999

Published: October 11, 1999

**Citation**

Gary Landry and Theresa Maldonado, "Pulse simulations of a mirrored counterpropagating-QPM device," Opt. Express **5**, 176-187 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-5-8-176

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### References

- J. A. Armstrong, N. Bloembergen, J. Ducuino, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962). [CrossRef]
- G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, "42%-efficient single- pass CW second-harmonic generation in periodically poled lithium niobate," Opt. Lett. 22, 1834-1836 (1997). [CrossRef]
- M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second harmonic generation: tuning and tolerances," IEEE J. Quantum Electron. 28, 2631- 2654 (1992). [CrossRef]
- M. Houe and P. D. Townsend, "An introduction to methods of periodic poling for second- harmonic generation," J. Phys. D - Appl. Phys. 28, 1747-1763 (1995). [CrossRef]
- J. Pierce and D. Lowenthal, "Periodically poled materials & devices," Lasers & Opt. 16, 25-27 (1997).
- Y. J. Ding and J. B. Khurgin, "Second-harmonic generation based on quasi-phase matching: a novel configuration," Opt. Lett. 21, 1445-1447 (1996). [CrossRef] [PubMed]
- G. D. Landry and T. A. Maldonado, "Second harmonic generation and cascaded second order processes in a counterpropagating quasi-phase-matched device," Appl. Opt. 37, 7809-7820 (1998). [CrossRef]
- G. D. Landry and T. A. Maldonado, "Efficient nonlinear phase shifts due to cascaded second order processes in a counter-propagating quasi-phase-matched configuration," Opt. Lett. 22, 1400-1402 (1997). [CrossRef]
- G. D. Landry and T. A. Maldonado, "Switching and second harmonic generation using counterpropagating quasi- phase-matching in a mirrorless configuration," Journal of Lightwave Technology 17, 316-327 (1999). [CrossRef]
- W. H. Press, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, New York, 1992).
- D. Zwillinger, Handbook of Differential Equations, 2nd ed. (Academic Press, San Diego, 1992).
- V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, 2nd ed. (Springer, Berlin, 1997).

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