APPLIED PHYSICS LETTERS 97, 191113 共2010兲
Continuous control of spatial mode rotation using second harmonic generation Jun-Hee Park,1,2 Woo-Kyung Kim,1,a兲 Woo-Jin Jeong,2,3 Myung-Gun Song,3 Hun-Hwa Kim,3 Kyung-Hwan Koo,3 Yeung-Lak Lee,4 Yena Kim,5 Ju-Han Lee,2 Woo-Seok Yang,1 and Han-Young Lee1 1
Korea Electronics Technology Institute, 68 Yatap-dong, Bundang-gu, Seongnam-si, Gyeonggi-do 463-816, Republic of Korea 2 School of Electrical and Computer Engineering, University of Seoul, 13 Siripdae-gil, Dongdaemun-gu, Seoul 130-743, Republic of Korea 3 Commax Corporation, 513-11 Sangdaewon-dong, Jungwon-gu, Seongnam-si, Gyeonggi-do 462-120, Republic of Korea 4 Advanced Photonics Research Institute, Gwangju Institute of Science & Technology, 1 Oryong-dong, Buk-gu, Gwangju 500-712, Republic of Korea 5 Department of Materials Engineering, Korea Aerospace University, 200-1 Hwajeon-dong, Deogyang-gu, Goyang-si Gyeonggi-do 412-792, Republic of Korea
共Received 11 May 2010; accepted 5 October 2010; published online 12 November 2010兲 This paper reports a continuously rotating spatial mode contributed from the interference between two second harmonic 共SH兲 high-order modes, TM10 and TM01, which were generated in z-cut MgO:LN ridge waveguides with a periodically poled structure. The TM10 and TM01 SH modes were generated from the mixed first harmonic 共FH兲 modes, the phase-matching conditions of the two modes were controlled to be close and to overlap so that the two modes can interference each other. The experiment results showed the interference of the two modes created an odd mode distribution with a specific angle and the distribution was continuously rotating according to the operation temperature. © 2010 American Institute of Physics. 关doi:10.1063/1.3507267兴 Second harmonic generation based on periodically poled lithium niobate 共PPLN兲 has been used in many studies to demonstrate various useful wavelength conversion applications. Those studies have focused on maximizing the conversion efficiency and have been generally confined to the generation of the fundamental mode, named TM00. However, researches get more interests in high-order mode generation have been increasing due to its unique physical properties and promising applications, including the orbital angular momentum1–4 and the optical frequency mixer.5–7 In particular, the interference between two orthogonal optical modes generated by nonlinear optics has been studied for encoding of quantum communication.8–10 In this paper, the interference between the TM10 and TM01 SH modes was studied. Two spatially orthogonal SH modes were obtained using a ridge-type PPLN waveguide, and it was found that the transverse distribution angle, resulted from the interference between the two modes, could be controlled by changing the phase-matching conditions such as the temperature. Figure 1 describes how to control the rotating angle of the mode distribution. The SH modes of TM10 and TM01 were generated by the incident first harmonic 共FH兲 source in the ridge-type PPLN waveguide, and the interference between the two modes formed the mode profile that was distributed toward a specific direction, which was determined by the intensity ratio between the two modes. Since the phase-matching conditions for the generation of the two modes were close to each other and overlapped, as shown in the graph in Fig. 1, the ratio of the intensities of the two modes changed according to the temperature. This means that the interfered mode distribution a兲
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can rotate as a function of temperature change. In order to understand the SH generation in a multimode waveguide and the phase-matching conditions, a non-linear coupled-mode equation must be derived from waveguides. The electric field E2 of frequency 2 in the waveguide of an m-th guided SH mode can be expressed by 2 2 共z兲Em 共x,y兲exp共− jmz兲, E2共x,y,z兲 = Am
共1兲
2 is a normalized mode profile that satisfies where Em 2 2ⴱ = 1 and m is a propagation constant of the 共1 / 2兲Em ⫻ Hm m-th mode. Then the well-known coupled mode equation between FH and SH can be obtained from7,11
FIG. 1. 共Color online兲 Concept and experiment setup for spatial mode rotation control. The dashed and solid curves in the graph represent the SHG efficiencies for TM210 and TM201, respectively.
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冉 冊冕冕
d 2 Am 共z兲 = − j dz 2
ⴱ
2 Em 共x,y兲exp共jmz兲P2共x,y,z兲dxdy,
共2兲
where P2 is the nonlinear polarization for SHG with frequency 2 and can be written as P2 = 20deff共E兲2 .
共3兲
Next, the electric field E of frequency in the waveguide is defined as the combination of the guided FH modes and can be expressed by E共x,y,z兲 = 兺 An共z兲En共x,y兲exp共− jnz兲.
共4兲
n
Using Eqs. 共3兲 and 共4兲, Eq. 共2兲 can be transformed into a following coupled-mode equation, which describes the contribution of the FH wave 共the combination of the p-th and q-th guided modes兲 to the specific m-th SH mode generation: d 2 A 共z兲 = − jAp 共z兲Aq共z兲exp共j⌬z兲, dz m where
= 共2 − ␦ pq兲0def f
冕冕
再
1, if p = q 0, if p ⫽ q
冎
2 − p − q . ⌬ = m
,
ⴱ
2 Em 共x,y兲
⫻关Ep 共x,y兲Eq共x,y兲兴dxdy,
␦ pq =
共5兲
共6兲 共7兲 共8兲
The nonlinear coupling coefficient defined by Eq. 共6兲 is proportional to an overlapping integral of the SH optical modes and the nonlinearity. From Eq. 共6兲, it is certain that 2 , can be generthe fundamental mode of the SH wave, TM00 ated with high conversion efficiency from the contribution of 2 兩2, because the profiles of ETM00 and 兩ETM00 兩2 are simi兩ETM00 lar. On the other hand, the maximum conversion efficiency 2 2 and TM01 , as shown in Fig. 1, is for the odd modes of TM10 · TM10 兲 and obtained when they are phase-matched to 共TM00 共TM00 · TM01兲, respectively, which are odd-even mixed modes. To obtain the blue light from the mean of SHG, z-cut MgO:PPLN ridge waveguides were fabricated with a period of 4.3 m which satisfies the quasi-phase-matching condition of ⌬ = 2 / ⌳ in Eq. 共8兲.12 The SHG experiments were performed using a continuous-wave diode-pumped solidstate Ti:sapphire tunable ring laser as an FH wave source, which was coupled to a single-mode polarization-maintained fiber 共Nufern Model PM980-XP兲 with a mode-field diameter of 6 m using an object lens. Next, the FH wave was transmitted to the fabricated PPLN ridge waveguide, and the output beam was focused on a beam analyzer. The FH wave was removed using an IR-cut filter, and finally, the near-field images of the generated SH wave were observed. With respect to the FH wavelength, sample temperature, and input fiber position, the variation in the SHG was measured. The detailed experiment set-up is described in Fig. 1. Figures 2共a兲–2共c兲 show the cross-sectional view of the fabricated ridge waveguide and the observed CCD images of 2 2 and TM01 modes were genthe SH modes. Both the TM10
FIG. 2. 共Color online兲 Waveguide cross section, measured SH mode image, and conversion efficiencies. 共a兲 Cross-section of the fabricated ridge waveguide. 关共b兲 and 共c兲兴 Generated SH mode images for TM210 and TM201 at 923 nm, respectively. 共d兲 Conversion efficiencies of the TM210 and TM201 modes as functions of the temperature at the FH wavelength of 923 nm.
erated at around 923 nm, and the measured maximum output powers were 0.34 mW and 0.86 mW at 66 mW of pumping power, respectively. The significant difference between the conversion efficiencies of the two modes should be due to 2 2 the higher overlap integral of TM01 than that of TM10 . The 2 fundamental SH mode, TM00 , showed the maximum output power of 1.2 mW at 921.5 nm, with the same pumping power as that of the higher modes. 2 and The dependency of the SHG efficiency for TM10 2 TM01 on the temperature is also plotted in Fig. 2共d兲. The temperature FWHM 共full width at half-maximum兲 of the two modes showed a same value, 1.2 ° C. Since the phasematching conditions of the two modes is overlapped, as shown in Fig. 2共d兲, the characterization of one mode should have been performed while minimizing the other mode by adjusting the input fiber position, which determines the amplitudes of the FH mode propagating within the waveguides. 2 Figure 2共d兲 shows that the amplitude ratio of the TM10 2 and TM01 modes can be determined from the temperature. Thus, the spatial distribution of the total SH modes can be written as a function of the temperature, which is given by 2 2 2 2 共T兲ETM10 共x,y兲 + ATM01 共T兲ETM01 共x,y兲, E2共x,y兲 = ATM10
共9兲 2 2 where ATM10 and ATM01 are the complex amplitudes of the 2 2 TM10 and TM01 modes, respectively. To predict the variation in the spatial mode distribution 2 2 and TM01 within a owing to the interference between TM10 waveguide structure shown in Fig. 2共a兲, the phase-matching conditions of each mode were calculated using Eq. 共8兲. Figure 3共a兲 shows the normalized conversion efficiencies of the two modes according to the temperature, which agree well with those of the experiment shown in Fig. 2共d兲. The experimental output shows an asymmetric shape on both sides, however, which deviates from the sinc2 functions predicted by the coupled-mode theory. This might be caused by the optical inhomogeneities, especially the nonuniformity of the refractive index along the propagation axis.13 Figure 3共b兲 shows the calculated phase difference between the two modes, which was constant at 23.4– 25.5 ° C; but the phase difference was delayed at 23.0 and 25.5 ° C, 2 was minimum due to a at which the amplitude of TM01
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FIG. 3. 共Color online兲 Simulation results of the conversion efficiencies, phase difference, and rotation angle according to the temperature. 共a兲 SHG conversion efficiencies. 共b兲 Phase difference between TM210 and TM201. 共c兲 Rotation angle caused by the interfered mode pattern.
phase mismatch. Analogously, the phase difference returned to the previous value of 23.4 and 25.7 ° C, at which the am2 was minimum due to a phase mismatch. plitude of TM01 The interfered mode distribution was calculated by substituting the amplitude of the higher modes at each temperature in Figs. 3共a兲 and 3共b兲 into Eq. 共9兲. Figure 3共c兲 shows the temperature dependence of the rotation angle. The result reveals that the rotation angle continuously changed as the temperature varied. Also, the sudden variation in the rotation angle at around 23.4 and 25.7 ° C was due to the tuning curve of the side lobe according to the temperature, as the side lobe was relatively narrower than the center lobe. To demonstrate experimentally the interfered pattern change in the SH output mode according to the temperature, the input fiber block was adjusted to a specific position to 2 2 and TM01 make the maximum SH intensity in both the TM10 modes similar. Then the SHG measurements were performed with a 923 nm SH wave. Figure 4 shows the normalized SH
FIG. 4. 共Color online兲 Continuous rotation of the SH mode distributions according to the temperature change at the FH wavelength of 923 nm.
mode patterns. The figures clearly reveal that the temperature change rotated the transverse profile, which resulted from the interference pattern change between the two basic mode profiles. The results in Fig. 4 can be understood clearly when it is considered that the transverse distribution angle is determined by the amplitude ratio between the two modes, which continuously changes with the temperature, as shown in Fig. 2共d兲. The vertically distributed mode profiles were obtained at 23.4 and 26.0 ° C, at which the conversion efficiencies of 2 the TM10 mode were nearly zero. On the other hand, the mode profile was horizontally distributed at 25.2 ° C, at 2 which the conversion efficiency of TM01 was very low. Diagonally declined profiles can be reasonably expected between the temperatures of the vertically distributed mode profile and the horizontally distributed mode profile, as confirmed from the mode profiles at 24.4 and 25.6 ° C, as shown in Fig. 4. However, the two diagonally declined profiles were 2 spatially orthogonal because the phase of TM01 was de2 layed at 25.4 ° C, at which the amplitude of TM01 was minimum due to a phase mismatch. In conclusion, z-cut MgO:PPLN ridge waveguides were fabricated for the generation of blue light by means of SHG. The TM10 and TM01 SH modes were generated from the contribution of mixed FH modes, and the phase-matching conditions of these two generated modes were close to each other and overlapped. It was observed that the interference of the two modes created an odd mode distribution with a specific angle, and the distribution rotated continuously according to the amplitude ratio of the two modes. Finally, it was explained that the angle of the mode distribution could be controlled by adjusting the temperature. This research was supported by a grant 共code under Grant No. I090903兲 from Gyeonggi-do International Collaborative Research Program. 1
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