Abstract

We report on the design and experimental characterization of aperiodically poled lithium niobate (APLN) crystals for use in monolithically integrated dual nonlinear-optical devices. A cascade and a single aperiodic-domain-structure designs based on simulated annealing method were constructed in LiNbO3 to simultaneously perform as 4-channel electro-optically active (EOA) filters and 4-channel frequency doublers in the telecom band. We found that we could obtain a 2.44-fold enhancement in second-harmonic-generation conversion efficiency and a 2.4-time reduction in filter transmission bandwidth with the single APLN device over the cascade one when the same device length of 2 cm and the EOA field of 1027 V/mm were used.

©2008 Optical Society of America

1. Introduction

The ingenious integration of the unique electro-optic (EO) effects of periodically poled lithium niobate (PPLN) with its capability of performing efficient nonlinear frequency conversion has led to the development of many attractive monolithic multi-function devices [1, 2]. In addition to the studies of periodic quasi-phase-matching (QPM) grating materials like PPLN, recently much attention has been given to the achievement of non-uniform grating designs in QPM materials using Fibonacci optical superlattice (FOS) [3], phase-reversal sequence (PRS) [4], phase-modulated domain structure [5], and aperiodic optical superlattice (AOS) [6] techniques. These techniques can be utilized to create spatially modulated domain lattices of QPM material which are capable of providing a number of reciprocal vectors for the simultaneous compensation of a corresponding number of wave-vector mismatches, thereby allowing for efficient multiple wave-energy conversions. Multi-wavelength devices are particularly useful in telecom, optical network, optical signal processing, and holographic data storage systems because of the increase in processing capacity and multiplexity. Furthermore, new and advanced optical processing functions can be performed for the building of multi-wavelength devices of nonlinear-optical material [4]. For example, the generation of multi-channel second-harmonic (SH) waves from telecom wavelengths has become a key process in some devices that perform the dynamic wavelength reconfiguration and broadcasting in wavelength-division multiplexed (WDM) optical networks [7].

Multi-wavelength operation can also be achieved in a QPM material without nonlinear frequency conversion by means of electro-optically active (EOA) polarization-mode (PM) conversion [8]. In prior works, we have already successfully demonstrated equal-gain multi-channel second-harmonic-generation (SHG) devices [9] and equal-transmittance EOA narrowband multi-wavelength filters [8] in LiNbO3 using the AOS technique. Since multi-channel SHG and EOA-filter devices are particularly useful in common areas such as optical signal processing and communications, it would be interesting to integrate and operate both devices on the same optical substrate. In this paper, we report on the design schemes, experimental demonstration, and performance comparison of a cascade and a single aperiodic-domain-structure LiNbO3 devices. Both devices have integrated a multi-channel EOA filter with a multi-channel SH generator in monolithic aperiodically poled lithium niobate (APLN) crystals. This work is also to the best of our knowledge the first demonstration of the construction of a single aperiodic QPM-domain structure for simultaneously phase-matching electro-optic and nonlinear-optic wave-energy couplings in LiNbO3 to achieve the high-efficiency multi-channel operation of dual optical device functions.

2. Device design and simulation

The two monolithic APLN devices designed and constructed in this work can be utilized to perform the same dual optical processing functions in multiple telecom C-band wavelengths (~1530–1570nm). The devices are all 2-cm long. The first one consists of a straightforward cascading APLN EOA PM converter and APLN SH generator of equal length fabricated on a monolithic LiNbO3 crystal sandwiched between a pair of cross polarizers that work in the telecom C-band. The APLN EOA PM converter performs as a 4-channel filter for 100% transmission of 4 International Telecommunication Union (ITU) wavelengths at λ 1=1536.2, λ 2=1542.8, λ 3=1550.1, and λ 4=1557.0 nm when an optimum driving field E y of 1027 V/mm is applied along the crystallographic y axis. The following APLN SH generator then performs as a 4-channel frequency doubler, halving the 4 fundamental ITU wavelengths λ 1λ 4. Since the design schemes (based on the AOS technique assisted by an optimization algorithm constructed using the simulated annealing (SA) method [10]) for both multi-channel devices are similar to those reported elsewhere [8, 9], it is not necessary to detail them again here. The main design parameters are included in a so-called objective function, which is a convergence criterion for guiding the SA algorithm [11] to best achieve the M designated efficiencies for the prescribed M coupling processes, given by

OF={Σi=1M[η0(λi)ηEO,SHG(λi)]wEO,SHG(λi)}
+βEO,SHG{max[ηEO,SHG(λi)]min[ηEO,SHG(λi)]},

where η 0(λ i), η(λ i), and w(λ i) are the expected efficiency, the calculated efficiency for an AOS structure generated during the SA process, and the weighting factor at wavelength λ i, respectively; the equalization factor β is an adjustable parameter for equalizing peak efficiencies and generally has a value falling in between 0 and 1 in our design; and the operators max[…] and min[…] yield the maximum and minimum values among all the values of the variable in the square brackets. The suffix EO or SHG denotes the coupling process for the EOA filter or SH generator, respectively. The efficiency η EO represents the optical-power transmittance calculated for the APLN EOA filter [8], while η SHG is the SHG conversion efficiency for the APLN SH generator [11] normalized to that of an ideal PPLN SHG device under the same pumping conditions and device length.

Beyond the straightforward cascading scheme, the second device is designed to construct a single QPM-domain-structure APLN crystal for simultaneously performing the same dual device functions, i.e., the multi-wavelength filtering and frequency doubling. The design scheme for such a novel APLN device is similar to that for the cascade one, except the objective function has to be modified to be

OF={Σi=1M[ηt(λi)ηEO(λi)ws,EO(λi)ηSHG(λi)ws,SHG(λi)]}
+βs,EO{max[ηEO(λi)]min[ηEO(λi)]}
+βs,SHG{max[ηSHG(λi)]min[ηSHG(λi)]},

where η t, w s,EO, w s,SHG, β s,E O, β s,SHG are the expected overall efficiency for both EO and SHG processes, the weighting factor for the transmittance η EO, the weighting factor for the efficiency η SHG, the equalization factor for the peak transmittance of η EO, and the equalization factor for the peak efficiency of η SHG, respectively. Obviously, with such an objective function, one can arbitrarily choose an efficiency weighting ratio w s,EO/w s,SHG between the two device functions for the designated wavelengths in the SA algorithm to refine for an optimum QPM domain sequence for best performance of the desired conversions. Hereafter, we will refer to the cascade and single QPM-domain-structure APLN devices as the cascade and single APLN devices, respectively.

In the AOS technique, a unit domain block of thickness Δx is chosen as the building block for composing an optimal domain structure with the SA algorithm per the prescribed objective function. Since the 1st-order QPM domain thicknesses (≡ halves of the domain periods) required for operating a PPLN EOA Šolc-type filter and a PPLN SH generator in the telecom C band are around 10.6 and 9.5 µm [12, 13], respectively, a unit domain thickness Δx sufficiently shorter than those domain thicknesses is desired to enhance the SA optimization result to work out the present APLN devices. In this study, after considering the current technical limit for LiNbO3 crystal poling, we chose Δx=5 µm. To illustrate and compare the performance of the two aforementioned monolithically integrated APLN devices, we let the two devices have the same operation efficiency when they function as EOA filters. This is done by properly setting the objective function in Eq. (2) to guide the SA algorithm to generate a single APLN structure optimized at performing the 4-channel SH generator given the prerequisite of achieving 100% transmission of the 4 designated fundamental wavelengths λ 1λ 4 provided a field of E y=1027 V/mm is applied to the structure to perform the EOA filter operation. Figure 1(a) shows the calculated transmission spectra of the two APLN devices when they function as 4-channel EOA filters. As mentioned before, this function is activated by supplying E y=1027 V/mm to the devices. Note that for the single APLN device the field E y is applied to whole crystal length because of the single aperiodic-domain-structure design. The inset shows the schematic arrangement of the devices. A pair of polarizers are installed at the input and output ends of the crystals to transmit the transverse-magnetic (TM) and transverse-electric (TE) waves in the telecom C band. It can be seen from Fig. 1(a) that all 4 designated ITU wavelengths reach ~100% transmittance for both the cascade (solid green line) and single (dashed red line) APLN devices. In particular the spectral transmission linewidths of the single APLN device can be reduced by a factor of up to 2.4 compared to those of the cascade one. This remarkable linewidth reduction, which is obviously an important advantage in an optical filter, can be attributed to the use of the full 2-cm device length as the nonlinear interaction length with the single APLN design. It is known that in a nonlinear energy conversion process, the conversion bandwidth is inversely proportional to the conversion length [13].

 

Fig. 1. (a) Calculated transmission spectra of the cascade (solid green line) and single (dashed red line) APLN devices when they function as 4-channel EOA filters. (b) Calculated SH conversion spectra of the cascade (solid green line) and single (dashed red line) APLN devices when they function as 4-channel SH generators. The upper inset shows a schematic arrangement of the devices, where the APLN crystal is sandwiched in between a pair of cross polarizers that work in the telecom C-band.

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The two devices switch to function as 4-channel SH generators when the EOA field E y is set to zero. This occurs because an APLN requires TM-polarized waves to interact over the whole domain structure to perform the prescribed quasi-phase-matching frequency conversions. Figure 1(b) shows a further comparison of the SHG conversion efficiency (in %/W; normalized to the fundamental pumping power) of the two APLN devices calculated as a function of the fundamental wavelength. There is a clearly a prominent enhancement of the SHG conversion efficiency (up to 2.44-fold) in comparison with that of the cascade device (solid green line) which is obtained for all 4 gain-equalized SHG peaks with the single APLN device (dashed red line). The result of this efficiency enhancement reveals the superiority of adopting a single APLN structure over a cascade one for optimized integration of different QPM device functions using the SA method. This also illustrates that it is always desirable to supply as many domain building blocks as possible for the SA algorithm to enhance its optimization ability. Take the single APLN design as an example, if a smaller unit domain thickness, say, Δx=2 µm is instead used, the number of the domain building blocks increases by 2.5 times and the calculated SHG conversion efficiency is enhanced by a factor of ~1.3 compared to the current design using Δx=5 µm based on the same EOA field (1027 V/mm). In the SA process all of the unit domain blocks are available and can be manipulated to optimize the composing of a single aperiodic-domain structure for performing both the two device functions, while, for the same device length, half or part of the block numbers are obtained to build a section of a cascade structure for performing a specific device function. In Fig. 1(b) we see a narrowed spectral linewidth in the SHG peaks achieved with the single APLN device. This can be understood based on the same remark made for the analogous effect illustrated in Fig. 1(a). The relevant parameters used in the design to achieve these results are w EO=1, w SHG=1, β EO=0.5, and β SHG=0.2 for the cascade APLN device and w s,EO=0.34, w s,SHG=0.66, β s,EO=0.25, and β s,SHG=0.25 for the single APLN device for all 4 designated wavelengths. The operating temperature of all the devices is 37°C.

 

Fig. 2. Measured (solid red circles) and calculated (solid green lines) transmission spectra of the (a) cascade and (b) single APLN devices at 37°C when they functioned as 4-channel EOA filters driven at 1027 V/mm.

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Fig. 3. Measured (solid red circles) and calculated (solid blue lines) SH conversion spectra of the (a) cascade and (b) single APLN devices at 37oC when they functioned as 4-channel SH generators.

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3. Experimental demonstration and discussion

We next perform proof-of-principle experiments for the two proposed monolithic APLN devices. We fabricated the two devices in LiNbO3 based on the calculated cascade and single aperiodic QPM-domain structures, both 2 cm in length, 1 mm in width, and 0.5 mm in thickness using the standard electric-field poling technique [14]. The domain grating structure of the fabricated APLN crystals was revealed by HF etching and inspected by a 500× power microscope, showing an average overpoled domain error [9] of ~14% and random domain errors [9] of <5%. The cascade one is comprised of a 1-cm-long APLN structure in the front section, which is the 4-channel EOA PM converter, followed by another 1-cm-long APLN structure, the 4-channel SH generator. The two y faces of the front 1-cm-long section of the cascade APLN crystal and the whole length of the single APLN crystal were sputtered with NiCr alloy to form side electrodes. The end faces of the two APLN crystals were optically polished without the application of an optical coating. These fabricated APLN devices were characterized using an external cavity laser (ECL) tunable in the telecom C band followed by an erbium-doped fiber amplifier (EDFA) as the pumping source. An in-line polarizer whose input and output were coupled with polarization maintaining (PM) fibers was connected to the EDFA output to decide and fix the polarization direction of the pump beam. The pump beam was then collimated and size-reduced to a radius of ~108 µm in the crystal. The APLN crystal to be measured was installed in a temperature controlled oven. A polarizer whose transmission axis was aligned parallel to the APLN y axis to transmit TE-polarized waves in the telecom C band was mounted at the output end of the crystal as the analyzer. Figures 2(a) and (b) show the measured transmission spectra of the cascade and single APLN devices (solid red circles), respectively, at 37°C when they functioned as 4-channel EOA filters under the same driving field. The calculated transmission spectra of the two devices are plotted (solid green lines) for comparison. The experimental data are in good agreement with the theoretical calculations. The SHG efficiency of the two APLN devices was measured after removing the driving field. Figures 3(a) and (b) show the measured SHG spectra of the cascade and single APLN devices (solid red circles), respectively, at 37°C. The SHG efficiency is expressed in %/W, which is normalized to the fundamental pumping power. The measured results are in reasonable agreement with the calculated results plotted as solid blue lines in the figures.

An interesting SHG spectrum can be obtained from the cascade APLN device when the transmission axis of the in-line polarizer at the input end of the crystal is switched to transmit TE waves in the telecom C band and the EOA PM conversion section is activated by applying the E y. Figure 4 shows the measured (solid red circles) and calculated (solid blue line) SHG spectra for the cascade APLN device with this system configuration (refer to the inset). We clearly obtained peak-narrowed telecom C-band SHG signals with highly suppressed sidelobes in comparison with the spectra obtained in Fig. 3(a). The peak narrowing and sideband suppression in the obtained SHG spectrum resulted from the cascaded second-order nonlinear conversion process (i.e., cascaded EOA-PM:SHG conversions).

 

Fig. 4. Measured (solid red circles) and calculated (solid blue line) SH conversion spectra of the cascade APLN device. The transmission axis of the in-line polarizer at the input end of the crystal is switched to transmit TE waves in the telecom C band and the EOA PM conversion section is activated. The inset illustrates such system arrangement.

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A survey of Figs. 2 and 3 shows that the measured SHG spectra deviated more from the theoretical predictions than did the measured EOA filter transmission spectra for the same APLN devices. This could be attributable to the relatively lower tolerance of the QPM SHG device (in terms of output performance) to fabrication or/and measurement errors. In QPM devices, fabrication errors mainly originate during the electric-field poling process. To understand the impact of fabrication errors on the present APLN devices, we introduced the same measure for the domain engineering errors as adopted in Ref. [9] for the aperiodic QPM-domain structures of the present two APLN devices. Figures 5(a) and (b) respectively illustrate the simulated EOA filter transmission and SHG spectra of the single APLN device when its engineered domain-width is overpoled by an average error of 20% (solid red lines) [9]. The calculated spectra without any device error are again plotted (solid green lines) for comparison. The results indeed show that decreases in SHG peak efficiencies due to device fabrication error are more apparent than that of the EOA filter peak transmittances. Lin et al. has analyzed possible measurement errors for an aperiodic QPM device [8] and concluded that the major error could come from non-uniform temperature distribution along the crystal arising from the non-ideal performance of the crystal oven. To estimate this temperature effect, we further simulated the output spectra of the single APLN device for a temperature gradient of -0.1oC/cm descending from the crystal center to both ends, but without domain engineering errors, as indicated by the dashed blue lines shown in Figs. 5(a) and (b), for the EOA filter transmission and SHG conversion, respectively. The results indicate that such a temperature gradient in the crystal has a minor effect on the performance of the present APLN devices, which is consistent with the observations made in Ref. [8] for a relatively short device. In addition, one can observe from Fig. 5(b) that the peak values of the output SHG spectra become rippled due to errors. These simulation results can be readily understood by looking at the characteristic efficiencies of QPM EOA and SHG devices. To illustrate this, we simplify the expressions of the conversion efficiencies of a PPLN EOA PM converter [12] and a PPLN SH generator [13] to some proportional normalized-forms as

ηPPLNEOsin2(F(D)π2L2+(ΔβEO2)2L)

and

ηPPLNSHGF2(D)sinc2(ΔβSHG2L)

respectively, where F(D)≡sin(πD) originates in the first-order Fourier coefficient of the periodic domain structure with domain duty cycle D; L is the device length; and Δβ EO and Δβ SHG are the wave-vector mismatches of the two nonlinear coupling processes. The domain engineering error introduced during the lithographic poling process actually results in the form of an asymmetric domain duty cycle rather than the alteration of the domain period [9]. In this case, the peak efficiencies of the two QPM devices occur at Δβ EO and Δβ SHG=0, corresponding to η PPLLN EO,peak∝sin2 (πsin(D)/2 and η PPLN SHG,peak∝sin2 (πD). For an asymmetric duty-cycle value (i.e., 0<D<0.5), η PPLN SHG, peak is always smaller than η PPLN EO, peak. For example, for a 20% domain-width error, D=0.4/0.6, leading to η PPLN EO, peak~0.994 and η PPLN SHG, peak~0.90, which is similar to the descent ratios for the peak efficiencies of the APLN EOA-filter and SHG devices presented in Figs. 5(a) and (b) due to the domain-width error. If temperature error exists, the wave-vector mismatch is not compensated for. With wave-vector mismatches for Δβ EO and Δβ SHGπ/L, we can approximate Eqs. (3) and (4) to be

ηPPLNEOcos2(1π(ΔβEO2L)2)

and

ηPPLNSHGsinc2(ΔβSHG2L),

respectively, if D=0.5 is assumed. Again η PPLN SHG in Eq. (6) is always smaller than η PPLN EO in Eq. (5) for Δβ EO and Δβ SHG<~π/L, but they approach each other and reach ~1 when Δβ EO and Δβ SHG→0, which is consistent with the results obtained in Figs. 5(a) and (b) when a minor temperature error is introduced (dashed blue lines).

 

Fig. 5. Simulated (a) EOA filter transmission and (b) SH conversion spectra of the single APLN device when no error (solid green lines), an average 20% domain-width overpoled error (solid red lines) [9], or a temperature gradient of -0.1°C/cm descending from the crystal center to both ends (dashed blue lines) is introduced to the device.

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It is possible to implement the present multi-channel APLN devices in LiNbO3 waveguides supporting both TE and TM modes. Take a Ti:LiNbO3 waveguide as an example. The required EOA field can be reduced by a factor of >50, due to the possibility of reducing the electrode separation to the order of the waveguide width [12]. The SHG peak conversion efficiency can be increased by a factor of >100 due to an increase in the pump intensity resulting from the highly reduced mode size in the waveguide [15].

4. Conclusion

We have to the best of our knowledge first achieved constructing monolithic APLN crystals capable of integrating and performing two nonlinear-optical device functions. These devices were designed using the simulated-annealing optimization method. The performance of a cascade and a single QPM-domain-structure APLN devices, both capable of functioning as 4-channel EOA filters and 4-channel SH generators in the telecom C band, was characterized and compared. With the same 2-cm device length, we obtained enhancement of the SHG conversion efficiency by a factor of ~2.44 and reduction in the spectral bandwidth of ~2.2 with the single APLN device for all the 4 ITU wavelengths when compared to the cascade one. When driven with an optimum field of 1027 V/mm along the crystallographic y axis, the two APLN devices switched to function as EOA filters, working to simultaneously transmit the same 4 wavelengths with ~100% transmittance. With the single APLN device we again obtained a ~2.4-fold narrower transmission bandwidth than with the cascade one. We attribute the superior performance of the conversion efficiency and output spectral width of the single APLN device to the fact that both the optimization process for device design and the nonlinear energy conversions performed in this device in effect use twice the device length of the cascade scheme. We found the APLN SHG device to be more susceptible to the fabrication and measurement errors than the APLN EOA filter device, even when they are built with the same aperiodic QPM-domain structure.

Acknowledgments

This work was supported by the National Science Council (NSC) of Taiwan under NSC Contract Nos. 96-2120-M-001-005 and 96-2221-E-008-042 and partially supported by the Technology Development Program for Academia (TDPA) under Project Code 95-EC-17-A-07-S1-011. The authors thank the support of measurement instruments from HOPE lab at National Tsing-Hua University, Taiwan.

References and links

1. Y. H. Chen, Y. Y. Lin, C. H. Chen, and Y. C. Huang, “Monolithic quasi-phase-matched nonlinear crystal for simultaneous laser Q-switching and parametric oscillation in a Nd:YVO4 laser,” Opt. Lett. 30, 1045–1047 (2005). [CrossRef]   [PubMed]  

2. S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007). [CrossRef]  

3. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997). [CrossRef]  

4. M. H. Chou, K. R. Parameswaran, and M. M. Fejer, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999). [CrossRef]  

5. M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched LiNbO3 wavelength converter with a continuously phase-modulated domain structure,” Opt. Lett. 28, 558–560 (2003). [CrossRef]   [PubMed]  

6. B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999). [CrossRef]  

7. M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5 µm band wavelength conversion based on difference frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998). [CrossRef]  

8. C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007). [CrossRef]  

9. Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett. 27, 2191–2193 (2002). [CrossRef]  

10. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983). [CrossRef]   [PubMed]  

11. B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000). [CrossRef]  

12. C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007). [CrossRef]  

13. 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. 23, 2631–2654 (1992). [CrossRef]  

14. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phasematched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995). [CrossRef]  

15. G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001). [CrossRef]  

References

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  1. Y. H. Chen, Y. Y. Lin, C. H. Chen, and Y. C. Huang, “Monolithic quasi-phase-matched nonlinear crystal for simultaneous laser Q-switching and parametric oscillation in a Nd:YVO4 laser,” Opt. Lett. 30, 1045–1047 (2005).
    [Crossref] [PubMed]
  2. S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
    [Crossref]
  3. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
    [Crossref]
  4. M. H. Chou, K. R. Parameswaran, and M. M. Fejer, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999).
    [Crossref]
  5. M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched LiNbO3 wavelength converter with a continuously phase-modulated domain structure,” Opt. Lett. 28, 558–560 (2003).
    [Crossref] [PubMed]
  6. B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
    [Crossref]
  7. M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5 µm band wavelength conversion based on difference frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998).
    [Crossref]
  8. C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
    [Crossref]
  9. Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett. 27, 2191–2193 (2002).
    [Crossref]
  10. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [Crossref] [PubMed]
  11. B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
    [Crossref]
  12. C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
    [Crossref]
  13. 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. 23, 2631–2654 (1992).
    [Crossref]
  14. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phasematched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
    [Crossref]
  15. G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
    [Crossref]

2007 (3)

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

2005 (1)

2003 (1)

2002 (1)

2001 (1)

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

2000 (1)

B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
[Crossref]

1999 (2)

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

M. H. Chou, K. R. Parameswaran, and M. M. Fejer, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999).
[Crossref]

1998 (1)

1997 (1)

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

1995 (1)

1992 (1)

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. 23, 2631–2654 (1992).
[Crossref]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Arbore, M. A.

Asobe, M.

Bosenberg, W. R.

Byer, R. L.

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phasematched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
[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. 23, 2631–2654 (1992).
[Crossref]

Chang, C. L.

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

Chang, G. W.

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Chang, J. Y.

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

Chen, C. H.

Chen, Y. H.

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

Y. H. Chen, Y. Y. Lin, C. H. Chen, and Y. C. Huang, “Monolithic quasi-phase-matched nonlinear crystal for simultaneous laser Q-switching and parametric oscillation in a Nd:YVO4 laser,” Opt. Lett. 30, 1045–1047 (2005).
[Crossref] [PubMed]

Chiang, A. C.

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Chou, M. H.

Dong, B. Z.

Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett. 27, 2191–2193 (2002).
[Crossref]

B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
[Crossref]

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

Eckardt, R. C.

Fan, F. C.

Fejer, M. M.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Grundkötter, W.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Gu, B. Y.

Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett. 27, 2191–2193 (2002).
[Crossref]

B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
[Crossref]

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

Hauden, J.

Huang, C. Y.

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

Huang, Y. C.

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Y. H. Chen, Y. Y. Lin, C. H. Chen, and Y. C. Huang, “Monolithic quasi-phase-matched nonlinear crystal for simultaneous laser Q-switching and parametric oscillation in a Nd:YVO4 laser,” Opt. Lett. 30, 1045–1047 (2005).
[Crossref] [PubMed]

Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett. 27, 2191–2193 (2002).
[Crossref]

Jundt, D. H.

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. 23, 2631–2654 (1992).
[Crossref]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Lee, Y. L.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Lee, Y. W.

Lin, C. H.

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

Lin, S. T.

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Lin, S. W.

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

Lin, Y. Y.

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Y. H. Chen, Y. Y. Lin, C. H. Chen, and Y. C. Huang, “Monolithic quasi-phase-matched nonlinear crystal for simultaneous laser Q-switching and parametric oscillation in a Nd:YVO4 laser,” Opt. Lett. 30, 1045–1047 (2005).
[Crossref] [PubMed]

Magel, G. A.

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. 23, 2631–2654 (1992).
[Crossref]

Ming, N. B.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

Miyazawa, H.

Myers, L. E.

Nishida, Y.

Parameswaran, K. R.

Pierce, J. W.

Quiring, V.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Ricken, R.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Schreiber, G.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Sohler, W.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Suche, H.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Suzuki, H.

Tadanaga, O.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Yang, G. Z.

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

Zhang, Y.

B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
[Crossref]

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

Zhu, S. N.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

Zhu, Y. Y.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

Appl. Phys. B (1)

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Appl. Phys. Lett. (1)

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

IEEE J. Quantum Electron. (1)

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. 23, 2631–2654 (1992).
[Crossref]

J. Appl. Phys. (1)

B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Exp. (3)

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Opt. Lett. (5)

Science (2)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. (a) Calculated transmission spectra of the cascade (solid green line) and single (dashed red line) APLN devices when they function as 4-channel EOA filters. (b) Calculated SH conversion spectra of the cascade (solid green line) and single (dashed red line) APLN devices when they function as 4-channel SH generators. The upper inset shows a schematic arrangement of the devices, where the APLN crystal is sandwiched in between a pair of cross polarizers that work in the telecom C-band.
Fig. 2.
Fig. 2. Measured (solid red circles) and calculated (solid green lines) transmission spectra of the (a) cascade and (b) single APLN devices at 37°C when they functioned as 4-channel EOA filters driven at 1027 V/mm.
Fig. 3.
Fig. 3. Measured (solid red circles) and calculated (solid blue lines) SH conversion spectra of the (a) cascade and (b) single APLN devices at 37oC when they functioned as 4-channel SH generators.
Fig. 4.
Fig. 4. Measured (solid red circles) and calculated (solid blue line) SH conversion spectra of the cascade APLN device. The transmission axis of the in-line polarizer at the input end of the crystal is switched to transmit TE waves in the telecom C band and the EOA PM conversion section is activated. The inset illustrates such system arrangement.
Fig. 5.
Fig. 5. Simulated (a) EOA filter transmission and (b) SH conversion spectra of the single APLN device when no error (solid green lines), an average 20% domain-width overpoled error (solid red lines) [9], or a temperature gradient of -0.1°C/cm descending from the crystal center to both ends (dashed blue lines) is introduced to the device.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

O F = { Σ i = 1 M [ η 0 ( λ i ) η EO , SHG ( λ i ) ] w EO , SHG ( λ i ) }
+ β EO , SHG { max [ η EO , SHG ( λ i ) ] min [ η EO , SHG ( λ i ) ] } ,
OF = { Σ i = 1 M [ η t ( λ i ) η EO ( λ i ) w s , EO ( λ i ) η SHG ( λ i ) w s , SHG ( λ i ) ] }
+ β s , EO { max [ η EO ( λ i ) ] min [ η EO ( λ i ) ] }
+ β s , SHG { max [ η SHG ( λ i ) ] min [ η SHG ( λ i ) ] } ,
η PPLN EO sin 2 ( F ( D ) π 2 L 2 + ( Δ β EO 2 ) 2 L )
η PPLN SHG F 2 ( D ) sin c 2 ( Δ β SHG 2 L )
η PPLN EO cos 2 ( 1 π ( Δ β EO 2 L ) 2 )
η PPLN SHG sinc 2 ( Δ β SHG 2 L ) ,

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