## Abstract

Polarization independent sum frequency generation (SFG) is proposed in an electro-optic (EO) tunable periodically poled Lithium Niobate (PPLN). The PPLN consists of four sections. External electric field could be selectively applied to them to induce polarization rotation between the ordinary and extraordinary waves. If the domain structure is well designed, the signal wave with an arbitrary polarization state could realize efficient frequency up-conversion as long as a z-polarized pump wave is selected. The applications in single photon detection and optical communications are discussed.

©2010 Optical Society of America

## 1. Introduction

Recently, more and more attention has been paid to single photon detection at telecom wavelength making use of Sum-frequency generation (SFG) [1–3]. The infrared single photons are converted to the visible regime for high detection efficiency, which is very important in quantum communication. However, the nonlinear optical up conversion schemes are basically polarization sensitive [1,2]. Although some techniques could be adopted to solve this problem, such as employing a 90° rotated second nonlinear crystals [3] or inserting a wave plate between a pair of nonlinear materials, they are not monolithic thus may have stability and reliability concerns in practical applications. As a consequence, it is desired to develop a kind of compact and robust configuration to realize the polarization independent SFG [4]. On the other hand, quasi-phase-matched (QPM) SFG or cascaded SFG/difference frequency generation (DFG) in ferroelectric waveguides has been proposed for various photonic applications. For example, all-optical wavelength converters [5,6] and logic gates [6–8], which are two key functional elements in future photonic networks.

In this letter, a four-section periodically poled LiNbO_{3} (PPLN) is proposed to realize the polarization independent SFG. The signal wave with an arbitrary state of polarization could be converted to the sum-frequency wave (SFW) in this special PPLN. The z-polarized pump wave is injected into the sample then external DC electric fields along the y-axis are selectively applied to manipulate the lights’ polarization states. The polarization dependence of SFG thus could be greatly suppressed with a suitable PPLN structure. The interaction among these waves is investigated through coupling wave equations. Related mechanism and future applications are also discussed.

## 2. Theory and simulation

Figure 1
shows the schematic diagram of a polarization independent PPLN containing four sections. The first and the fourth sections are identical whose period is designed just for QPM SFG. However, the periods in the second and third sections are different; they are designed only for polarization rotation of the signal wave and the SFW, respectively [9]. External DC electrical fields are applied at the y-surfaces of the second and third sections. With a suitable electric field, the second and third sections could just act as 90° polarization rotators for signal wave and SFW, respectively. In order to utilize the largest nonlinear coefficient d_{33} of LiNbO_{3}, the lights should propagate along the crystal’s x-axis and the escorting pump wave should be z-polarized. Because the wavelength bandwidths for polarization rotation in the second and third sections are very sharp [9], the polarization state of the pump wave is not affected inside the sample.

When a z-polarized signal wave is propagated through the sample, the signal wave will be converted to the z-polarized SFW basically in the first section and then the generated z-polarized SFW is further totally converted to a y-polarized SFW in the third section. At last the SFW passes through the last section without nonlinear frequency conversion. On the other hand, when a y-polarized signal wave is injected, it passes through the first section without SFG. Then it is converted to a z-polarized signal wave in the second section. Because the third section has no impact to it, the generated z-polarized signal wave is further frequency up-converted to a z-polarized SFW after it enters the fourth section. In a word, a y-polarized signal wave generates a z-polarized SFW while a z-polarized signal wave induces a y-polarized SFW. As long as the first and the last sections have the same length, the PPLN may have the same frequency up-conversion capability for y- and z- polarized signal waves. Because all normally-incident lights could be divided into y- and z- polarized components, polarization independent frequency up-conversion thus could be expected.

To analysis the above processes in detail and obtain the numerical results, the corresponding coupling equations can be deduced under the plane-wave approximation with consideration of both SFG and electro-optic (EO) interactions [10,11].

*d*(

_{33}*x*) =

*d*(

_{33}f*x*) is the modulated nonlinear coefficient. ${\epsilon}_{23}^{(s)}(x)=-{n}_{sy}^{2}{n}_{sz}^{2}{\gamma}_{51}(x)E$ for the signal wave and ${\epsilon}_{23}^{(o)}(x)=-{n}_{oy}^{2}{n}_{oz}^{2}{\gamma}_{51}(x)E$ for the SFW, where ${\gamma}_{51}(x)={\gamma}_{51}f(x)$ is the modulated EO coefficient in PPLN and E refers to the external DC electric field. As no electric field is applied to the first and the fourth section, the corresponding E is 0. $\Delta {k}_{1}{}^{\prime}={k}_{oz}-{k}_{sz}-{k}_{pz}$, $\Delta {k}_{2}{}^{\prime}={k}_{sy}-{k}_{sz}$, $\Delta {k}_{3}{}^{\prime}={k}_{oy}-{k}_{oz}$ are the wave vector mismatch for SFG and polarization rotation of the signal wave and the SFW, respectively. The asterisk denotes complex conjugation.

In a PPLN, the structure function *f*(x) changes its sign form + 1 to −1 periodically in different domains. It can be expanded as Fourier series, $f(x)={\displaystyle \sum _{m}{g}_{m}}\mathrm{exp}(-i{G}_{m}x)$ where G_{m} are the reciprocal vectors and g_{m} are the amplitudes of the reciprocal vectors. Without loss of generality, the reciprocal vector G_{1} (the subscript 1 is ignored hereinafter) is adopted to compensate the wave vector mismatch for efficient conversion. For example, in the first and fourth section, G_{1, 4} (here the second subscript denotes which section the reciprocal vector is in) is adopted as $\Delta {k}_{1}{}^{\prime}={k}_{oz}-{k}_{sz}-{k}_{pz}={G}_{1,4}$ to compensate the nonlinear phase mismatch for SFG and the wave vector mismatch becomes $\Delta {k}_{1}={k}_{oz}-{k}_{sz}-{k}_{pz}-{G}_{1,4}=0$. In the same way, G_{2} in the second section is adopted to compensate the phase mismatch for the signal wave’s polarization rotation as $\Delta {k}_{2}{}^{\prime}={k}_{sy}-{k}_{sz}={G}_{2}$ (*i.e.,*
$\Delta {k}_{2}={k}_{sy}-{k}_{sz}-{G}_{2}=0$). And G_{3} in the third section is adopted to compensate the phase mismatch for SFW’s polarization rotation as $\Delta {k}_{3}{}^{\prime}={k}_{oy}-{k}_{oz}={G}_{3}$ ($\Delta {k}_{3}={k}_{oy}-{k}_{oz}-{G}_{3}=0$). In this case, the coupling Eqs. (1) can be simplified as:

_{i}(i = 1, 2, 3) represents the coupling coefficients for SFG and EO effect.

Numerical solutions to Eqs. (2) have been done to verify our design. A 1064 nm pump wave is employed and the injected signal wave has the wavelength of 1550 nm. In order to convert the signal wave to the SFW completely, we set the intensity of the signal wave at 0.2 MW/cm^{2}, which is far weaker than the intensity of the pump wave at 10 MW/cm^{2}. The PPLN period in the first and fourth section is designed at Λ = 11.62 *μm* to satisfy the SFG QPM condition at room temperature 25 °C. The corresponding periods of these two sections are 500 for complete up-conversion. Λ = 20.48 *μm* in the second section and Λ = 7.27 *μm* in the third section are selected for polarization rotation of the signal wave and the SFW, respectively. *d _{33}* = 25.2

*pm*/

*V*and γ = 32.6

*pm*/

*V*are used for simulation. To realize the complete conversion between the ordinary and extraordinary waves, we set the electric field at 710

*V*/

*mm*and the periods of the second and the third sections at 250 and 260 respectively. The sample length is about 18.6

*mm*.

Figures 2
and 3
show the numerical simulation results. Figure 2 describes the light intensity evolution of the signal wave and the SFW inside our devised PPLN. It can be seen that no matter with what kind of polarization states, the injected signal wave is almost totally converted to SFW. The light intensity of the SFW remains about 0.5 MW/cm^{2} for the y-polarized (Fig. 2(a)) and z-polarized (Fig. 2(b)) signal waves. Figure 3 shows the spectral response. Figures 3(a) and 3(b) correspond to the results when a y-polarized signal wave is injected. Figures 3(c) and 3(d) correspond to the results when a z-polarized signal wave is injected. For example, when a y-polarized signal wave is injected, the light intensity of the generated z-polarized SFW reaches the highest and the light intensity of the y- and z- polarized signal wave vanishes if the phase matching condition is satisfied in accordance with Fig. 2. In addition, the light intensity of the z-polarized signal wave shows double peaks at the wavelength which has a slight offset from the phase matching point. The results are consistent when a z-polarized signal wave is injected but the light intensity of the z-polarized signal wave has small fluctuations near the phase matching point. On the other hand, the y-polarized SFW is negligible at side wavelengths when a y-polarized signal wave is injected. This is because when a y-polarized signal wave near the phase matching point is injected, the z-polarized SFW will not be generated in the first section so that there is no y-polarized SFW being converted in the third section. As a result, the y-polarized sum-frequency wave will not be observed at the side wavelengths. However, when a z-polarized signal wave at side wavelengths is injected, the signal wave will not be fully converted into the z-polarized SFW in the first section and the left signal wave will not be totally converted into the y-polarized signal wave. The remaining z-polarized signal wave is partly frequency up-converted into SFW in the forth section so we observe a non-negligible generation of z-polarized SFW at side wavelengths. Because the total energy of involved lights should be conserved and there are some z-polarized SFWs at side wavelengths, the light intensity of the z-polarized signal wave thus shows small side fluctuations (*e.g.*, double peaks) at corresponding wavelengths.

Our previous results only show the SFG behaviors of pure y- and z- polarized signal waves, which is not enough. To well testify the polarization independency, an arbitrarily polarized signal should be considered. We calculated the SFW intensities when the signal waves have different intensity ratio and phase difference between the y- and z- polarized components. Figure 4
shows the simulation results. The signal wave with arbitrary polarization state could be converted to the SFW with basically the same intensity at 0.5 MW/cm^{2}. The intensity fluctuation is less than 1%. We can conclude that our multi-section PPLN really may realize the polarization independent SFG perfectly. Considering the different photon energies of the signal and SFW, it is found that the signal photons almost have been totally converted into SFW photons in visible regime. Although the calculated quantum efficiency is not exactly at 100%, it might be due to the intrinsic slowly varying approximation of the coupled wave approach. As a consequence, a “fully” polarization independent SFG conversion is still expectable if no propagation loss is considered. This would be very attractive in the single photon detection and other applications. In comparison with other techniques, our approach has many advantages. For example, it’s monolithic and easily applied in future photonic systems. In addition to the quantum detection, we believe our technique is also applicable for classical optical communications, because it may open a new window in eliminate the polarization dependent issues in a fiber-optic link with wavelength conversion.

## 3. Discussion

In addition to bulk PPLN, Lithium niobate waveguide also has been utilized in many optical devices and applications. In comparison with bulk crystal, the waveguide has non-uniform mode distribution while with high field confinement. Efficient fiber to waveguide coupling has been realized for years so that a PPLN waveguide could be well compatible with most optical communication devices. Although our simulation above is based on the plane-wave approximation, the key coupling equations, *i.e.*, Eqs. (2), still can be used in the waveguide case with minor revision about the coupling coefficients and wave vector mismatches. The overlap integral of the mode fields also should be added in the coupling coefficients. The propagation constants and effective refractive indices take place of the wave vectors and bulk wave’s refractive indices, respectively. We take a typical Ti-diffused PPLN waveguide as an example.

In the first and fourth sections for frequency up-conversion, the coupling coefficient K_{1} in Eqs. (2) should be modified by an overlap factor $~{\displaystyle \iint {e}_{sf}(y,z){e}_{s}^{*}(y,z){e}_{p}^{*}(y,z)dydz}$ [12], where *e _{sf}* (

*y*,

*z*),

*e*(

_{s}**y*,

*z*) and

*e*(

_{p}**y*,

*z*) are the normalized field profiles of the SFW, signal wave and pump wave, respectively. This modification may result in a decrease of the coupling coefficient, so the power density of the input light should be elevated accordingly to compensate the coupling coefficient reduction. As a result, the conversion efficiency in these two sections still may be kept in a desired level. On the other hand, the TE/TM mode conversion realizes in the second and the third sections, which correspond to the polarization rotation in a bulk PPLN. In this case, the coupling coefficients K

_{2}and K

_{3}should be multiplied by the overlap factors of the involved mode fields and the applied electric field [13]. We take K

_{2}for the second section as an example. It changes to ${K}_{2}{}^{\prime}=\frac{i{\gamma}_{51}{g}_{1}{\omega}_{s}E}{2c}\sqrt{{n}_{sy,eff}^{3}{n}_{sz,eff}^{3}}\nu $. Different from the original coefficients in Eqs. (2), the refractive indices of the ordinary and extraordinary signal waves are replaced by the effective counterparts of the TE and TM modes respectively. An overlap factor of the mode fields and the modulating field is added, defined by $\nu ={\displaystyle \iint {e}_{o}^{*}(y,z){e}_{y}(y,z){e}_{e}(y,z)dydz}$, where

*e*(

_{o}*y*,

*z*),

*e*(

_{e}*y*,

*z*) and

*e*(

_{y}*y*,

*z*) are the normalized field profiles of the ordinary wave, extraordinary wave and the applied electric field, respectively. As the overlap factor is less than 1 so that the mode conversion efficiency will be affected. Unlike the SFG in the first and fourth sections, the TE/TM mode conversion is a linear process so it is not influenced by the light intensity. However, a 100% mode conversion is still easily achievable by extending the length of the corresponding second section or just simply increasing the applied field. In the third section, the situation is similar for the mode conversion of the SFW. A word, same results could be obtained in all four PPLN sections, thus the simulation curves shown in Fig. 2–4 should also be valid in PPLN waveguides.

Unlike the bulk PPLN with negligible inherent loss, the waveguide normally has some propagation loss that has to be taken into account. For a typical Ti-diffused PPLN waveguide, the propagation losses of SFW and signal waves are ~0.2 dB/cm and ~0.1 dB/cm, respectively [14,15]. Even if these losses are considered, the SFG photon to photon quantum efficiency still could reaches ~94%. Because the commercial detectors in telecom wavelength only has 1/3~1/4 single photon detection probability than that in visible band, our up-conversion scheme are still very favorable for single photon detection operating at infrared wavelengths [1].

## 4. Conclusion

In summary, we proposed a polarization independent PPLN containing four well-designed sections. External DC electric fields are applied to the given sections to induce the polarization rotation between the ordinary and extraordinary waves. A five-wave-coupling approach is proposed to study the optical power transferring among these waves. The SFG dependency on the signal wave’s polarization states are investigated, which exhibits great polarization-insensitivity. The light intensity of the converted SFW keeps at the same level with arbitrary signal wave’s polarization sates. The related mechanism and future applications in the single photon detection and optical communications were also discussed.

## Acknowledgements

This work is supported by National 973 program under contract No. 2011CBA00205 and 2010CB327803, and the National Science Foundation of China (NSFC) program No. 60977039 and 10874080. The authors also acknowledge the support from New Century Excellent Talents Program and the Specialized Research Fund for the Doctoral Program of Higher Education.

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