Abstract

We present a simple and powerful method for mode generation and transformation in a ridge-type periodically poled lithium niobate (PPLN) waveguide by the use of second-order nonlinear effect and local-temperature-control technique. We show that a Hermite-Gaussian (HG) mode wave (among HG00 to HG22) can be selectively generated via the quasi-phase-matching (QPM) nonlinear process in a PPLN waveguide by tuning the wavelength of fundamental wave or the temperature of the waveguide. As well, it is demonstrated that HG mode can be transformed into Laguerre-Gaussian (LG) one via combination of HG modes which are simultaneously generated in a single PPLN waveguide with local-temperature-control technique.

© 2010 OSA

1. Introduction

Mode generation and transformation of the optical beam are being currently spotlighted in various research subjects such as higher-order spatial optical solitons [1], atom manipulation [2], micro-fluidics [3], optical tweezers [4], multi-mode quantum information [58], and stellar separation in astronomy [9]. To obtain Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes suitable to the applications, many different methods have been studied. In general a single low- or high-order HG mode is generated in the laser resonators which have intracavity elements such as metal wires [10], phase-shifting mask or phase gratings [11], specially-designed mirrors [12], and binary phase elements [13]. On the contrary, LG mode is usually converted from HG mode by using a spiral phase plate [14], a hologram [15], and cylindrical lens [10,16,17] because that those two modes can be transformed into each other. Up to now, all the proposed methods need optical alignment and additional elements for mode generation or transformation. In this letter, we propose a simple method for a specific mode generation (HG mode) and transformation (HG → LG mode) in a periodically poled lithium niobate (PPLN) waveguide by using the second-order nonlinear effect and the local-temperature-control technique [18].

2. Hermite-Gaussian (HG) modes generation in a PPLN waveguide

A Typical quasi-phase-matching (QPM) condition for the efficient second-harmonic generation (SHG) process is

kSH2kfKQPM=0 (1),
where KQPM is QPM grating vector and kf, kSH are the wave vectors of the fundamental and second-harmonic (SH) waves, respectively. Equation (1) shows QPM condition is determined with refractive indices at both fundamental and SH waves. Thus, in SHG using a bulk PPLN at a fixed temperature, there is a single set of the fundamental and SH wavelengths which satisfies the QPM condition. However, it’s not the case in SHG using a PPLN waveguide, because even a single mode PPLN waveguide at fundamental wavelength [e.g., HG00(ω)] supports multiple modes at the SH wavelength with different wave vector. Therefore, a PPLN waveguide can generate multi-mode SH beam [e.g., HGnm(2ω), where n and m are the mode numbers] [19]. It means that we can observe various SH peaks even in a PPLN waveguide by changing the fundamental wavelength or the temperature of the waveguide. Moreover a multi-mode PPLN waveguide at fundamental wavelength shows not only SHG process but also sum-frequency generation (SFG) process depending on the fundamental modes, because the refractive indices of wave vary with its modes as well as wavelengths. In other words, in a multi-mode waveguide, the SFG process results from mixing of fundamental modes which have the same optical frequency but different wave vectors. QPM condition for SFG process is given as follow:
kSF,mkf,m1kf,m2KQPM=0 (2),
where subscripts m, m1, m2 indicate orders of HG mode (HGnm) for sum-frequency (SF) wave and fundamental waves (m1m2). Therefore, in the case of SFG process, nonlinear interaction between two different fundamental modes generates a SF mode (e.g., HG00(ω)+HG10(ω)→HG10(2ω)). By using these second-order nonlinear processes, we can simply generate various low- and high-order HG modes.

In experiment, a 22-mm-long ridge-type zinc doped PPLN (ZnO:PPLN) waveguide (Z-cut type, NTT Electronics) of about 6.7 μm QPM period was used to demonstrate mode generation. Both end faces of a ZnO:PPLN waveguide were angle-polished to prevent the internal multi-reflection. The physical area of the ridge waveguide was measured by microscope to be 9.2 μm × 6.8 μm, which is large enough to guide multi-mode waves. A fundamental wave from the a wavelength-tunable diode laser (DL100, Toptica Photonics) was coupled into the ZnO:PPLN waveguide with the polarization of transverse-to-magnetic field (TM). The temperature of the ZnO:PPLN waveguide was controlled to be 25 °C by a Peltier device. The measured SH and SF curves as a function of fundamental wavelength are shown in Fig. 1 . Insets are the mode profiles of the SH or SF waves. The full-width at half maximum (FWHM) of the SH curve and the SH conversion efficiency of lowest mode (HG00) were measured to be 0.1 nm and 87%/W. Such a narrow bandwidth and high conversion efficiency indicate that a good QPM-grating (duty cycle near 50:50) and a homogeneous waveguide were fabricated through the whole length of the waveguide [20]. Since the used ridge waveguide allows several modes of the fundamental wave to be guided, we can observe a large number of SH (SF) modes from a set of fundamental modes satisfying the QPM condition while tuning the wavelength (Eq. (1) or (2)). The first peak is originated from SHG process (HG00(ω)+HG00(ω)→HG00(2ω)) and the other peaks are due to the SFG process by mixing of two fundamental modes (HG00(ω)+HG10(ω)→HG10(2ω), HG00(ω)+HG01(ω)→HG01(2ω), HG00(ω)+HG11(ω)→HG11(2ω)).

 

Fig. 1 SH and SF curves as a function of fundamental wavelength in a ZnO:PPLN waveguide (@25 °C). Each peak shows different HG mode (from HG00 to HG11). 1st peak: HG00(ω)+HG00(ω)→HG00(2ω), 2nd peak: HG00(ω)+HG10(ω)→HG10(2ω), 3rd peak: HG00(ω)+HG01(ω)→HG01(2ω), 4th peak: HG00(ω)+HG11(ω)→HG11(2ω).

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Figure 2 shows various HG modes which are generated in a ZnO:PPLN waveguide via second-order nonlinear processes. The low-order HG modes (HG00, HG10, HG01, HG11) show much higher optical powers than the high-order HG modes because our waveguide allows higher guiding confinement to the low-order fundamental modes than the high-order fundamental ones. Figure 3 shows SH (SF) wavelength as a function of the operation temperature for different HG modes. The wavelength turning rate is measured to be about 0.05 nm/°C. All the generated HG modes show red shift as increase of operation temperature and one fixed fundamental wave can make different HG mode of SH (SF) depending on the operation temperature (see dotted arrows in Fig. 3). If we make more than two different-temperature sections in a PPLN waveguide [18], more than two HG modes can be generated simultaneously in the waveguide. The generated modes can interact with each other and make new mode such as LG mode.

 

Fig. 2 HG modes which are generated in a ZnO:PPLN waveguide by second-order nonlinear process.

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Fig. 3 Wavelengths of different HG modes (HG00, HG10, HG01, HG11) as a function of the operation temperature. The wavelength tuning rate is about 0.05 nm/°C.

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3. Laguerre-Gaussian (LG) modes generation in a PPLN waveguide

The experimental setup to generate LG modes in a ZnO:PPLN waveguide is shown in Fig. 4 . An optical wave from the wavelength-tunable diode laser (λ=1066.07 nm, P=50 mW) was used two-fold by a fiber-optic 10:90 power splitter. The optical wave in the lower branch was coupled into the ZnO:PPLN waveguide which is placed in a sample holder consisting of two separated Peltier devices to obtain the localized temperature in the sample [18]. Temperatures of the first- and the second-section in the sample holder were set at 26.6 °C and 42.6 °C to generate HG10 and HG01 mode SF waves, respectively. The generated SF modes in the ZnO:PPLN waveguide were guided by a 10× objective lens and then observed by a charge-coupled device (CCD) camera. The optical wave in the upper branch of the power splitter was served as a fundamental wave for SHG which has lowest HG mode (HG00). To generate the single mode SH wave (HG00), a fiber-pigtailed ridge-type magnesium (MgO)-doped PPLN waveguide (Λ=6.8 μm, ηSHG=250%/W, Commax Co., Ltd.) which was fabricated for a single mode (HG00) SHG process through the precise dry etching technique [21] was used as wavelength converter. The generated SH beam was superimposed on the SF beams from the ZnO:PPLN waveguide at a screen by a beam combiner (BS2). The polarizations of both fundamental waves were adjusted to TM-polarizations, before launched into the PPLN waveguides, for the maximum nonlinear interaction with nonlinear coefficient d33.

 

Fig. 4 Schematic of experimental setup to generate LG mode in a ZnO:PPLN waveguide by the use of second-order nonlinear process and local-temperature-control technique. LD, wavelength tunable diode laser; PS, power splitter, PC1, PC2, polarization controller; PPLN1, Ridge-type Zn:PPLN waveguide; PPLN2, Ridge-type MgO:PPLN waveguide; Lens1, Lens2, 10× objective lens; Filter1, Filter2, infrared cut-off filter; BS1, BS2, beam splitter; CCD, charge-coupled device camera.

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Any LG mode can be described as a combination of HG modes with appropriate amplitude and phase because both modes can form complete sets of solutions to the paraxial wave equation [10,16]. Figure 5 shows results by combination of HG10 and HG01 modes with different phase delays and amplitudes. To induce the phase difference between two generated modes in the ZnO:PPLN waveguide, we have fixed the temperature of first-section in the sample holder at 26.6 °C, and then slightly changed the temperature of second-section. The change of the temperature in a ZnO:PPLN induces the change of the refractive index, resulting in the phase difference between two modes. In such a way, we have achieved up to π phase shift between two modes within the temperature change amount of 0.96 °C. We have only a small variation in the SF power of HG01 mode during phase control because that the temperature bandwidth (~1.2 °C) of the mode mixing-type SF (HG01) wave is larger than the changing amount of temperature in the second-section. To control the relative amplitude between both HG modes, we have changed the physical length of the sections. If the physical lengths of two sections are different, HG modes which are generated from each section have different amplitude. On the contrary, both HG modes have same amplitude when the physical lengths of two sections are equal. The experimental and theoretical results [10] of the former case are shown in Fig. 5 (a) and (b), the case of the latter is shown in Fig. 5 (c). The color and gray images indicate theoretical calculations and experimental results, respectively. Pictures of each column show the result of the combination of both modes at different temperatures. Each temperature step of 0.12 °C (or 0.24 °C) shows π/8 (or π/4) phase shift between two modes (compare color and gray image in Fig. 5). The resultant mode shapes of two modes (HG10 and HG01) combination with different amplitude are shown in Fig. 5 (a), and (b). In the case of Fig. 5 (a), the amplitude HG10 mode is bigger than that of HG01 mode. Figure 5 (b) shows vice versa case. From Fig. 5 (a), (b), we have observed the spatial rotation of the combination mode depending on the amplitude ratio of HG10 and HG01 modes. Particularly, as shown in Fig. 5 (c), at π/2 phase shift point (42.6 °C), we achieved LG01 mode from the combination of two HG modes. The transformed LG01 mode which has angular momentum was confirmed by observation of a singularity [15] in interference pattern at a screen in Fig. 4. We have also transformed LG10 mode from the combination of HG20 and HG02 modes. In principle, we can make all kind of LG modes by using such kind of mode tailoring method in a nonlinear waveguide.

 

Fig. 5 Results of the combination between HG10 and HG01 modes with different phase delays and amplitudes. The color and gray images indicate theoretical calculations and experimental results, respectively. a, Amplitude of HG10 is larger than that of HG01. b, Amplitude of HG10 is smaller than that of HG01. c, Amplitude of HG10 and HG01 is equal. The temperatures which are written below pictures indicate the deviations from the temperature at π/2 phase shift between two modes (HG10, HG01).

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4. Conclusion

In this letter, we have demonstrated easy and powerful mode generation and transformation in a PPLN waveguide by using the second-order nonlinear effect and local-temperature-control technique. This new method allows active-mode generation and transformation which is not possible in former methods. We can also realize the local-temperature-control effect in a uniformly temperature controlled PPLN waveguide by using multi-section QPM-gratings which have different periodicity [22]. We believe that this new mode generation technique gives easy access of various laser beam modes to researchers who work in photonics research field.

Acknowledgments

We would like to thank Dr. H.-Y. Lee and W.-K. Kim (KETI) for their helpful discussion about a mode mixing-type sum-frequency generation in a ridge-type PPLN waveguide. This work was partially supported by National Research Foundation of Korea Grant funded by the Korea Government (2009-0076197), by Ministry of Knowledge Economy (MKE) through “Leading Edge R&D Program” and “Industrial Core Technology Development Program”.

References and links

1. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98(5), 053901 (2007). [CrossRef]   [PubMed]  

2. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006). [CrossRef]   [PubMed]  

3. R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco, and M. J. Padgett, “Multipoint holographic optical velocimetry in microfluidic systems,” Phys. Rev. Lett. 96(13), 134502 (2006). [CrossRef]   [PubMed]  

4. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]   [PubMed]  

5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef]   [PubMed]  

6. K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J.-F. Morizur, P. K. Lam, and H.-A. Bachor, “Entangling the spatial properties of laser beams,” Science 321(5888), 541–543 (2008). [CrossRef]   [PubMed]  

7. J. Janousek, K. Wagner, J.-F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagation modes,” Nat. Photonics 3(7), 399–402 (2009). [CrossRef]  

8. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007). [CrossRef]  

9. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006). [CrossRef]   [PubMed]  

10. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]  

11. J. R. Leger, D. Chen, and K. Dai, “High modal discrimination in a Nd:YAG laser resonator with internal phase gratings,” Opt. Lett. 19(23), 1976–1978 (1994). [CrossRef]   [PubMed]  

12. S. D. Silvestri, V. Magni, O. Svelto, and G. Valentini, “Lasers with super-Gaussian Mirrors,” IEEE J. Quantum Electron. 26(9), 1500–1509 (1990). [CrossRef]  

13. G. Machavariani, “Effect of phase imperfections on high-order mode selection with intracavity phase elements,” Appl. Opt. 43(34), 6328–6333 (2004). [CrossRef]   [PubMed]  

14. K. J. Moh, X.-C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007). [CrossRef]   [PubMed]  

15. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef]   [PubMed]  

16. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]   [PubMed]  

17. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90(20), 203901 (2003). [CrossRef]   [PubMed]  

18. Y. L. Lee, Y.-C. Noh, C. Jung, T. J. Yu, B.-A. Yu, J. Lee, D.-K. Ko, and K. Oh, “Reshaping of a second-harmonic curve in periodically poled Ti:LiNbO3 channel waveguide by a local-temperature-control technique,” Appl. Phys. Lett. 86(1), 011104 (2005). [CrossRef]  

19. M. L. Sundheimer, Ch. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes,” Opt. Lett. 18(17), 1397–1399 (1993). [CrossRef]   [PubMed]  

20. Y. L. Lee, N. E. Yu, C.-S. Kee, D.-K. Ko, J. Lee, B.-A. Yu, W. Shin, T. J. Eom, and Y.-C. Noh, “Wavelength filtering characteristics of Solc filter based on Ti:PPLN channel waveguide,” Opt. Lett. 32(19), 2813–2815 (2007). [CrossRef]   [PubMed]  

21. S. W. Kwon, W. S. Yang, H. M. Lee, W. K. Kim, H.-Y. Lee, W. J. Jeong, M. K. Song, and D. H. Yoon, “The ridge waveguide fabrication with periodically poled MgO-doped lithium niobate for green laser,” Appl. Surf. Sci. 254(4), 1101–1104 (2007). [CrossRef]  

22. G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with Fourier synthetic quasi-phase-matching gratings,” Opt. Lett. 23(11), 864–866 (1998). [CrossRef]  

References

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  1. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98(5), 053901 (2007).
    [CrossRef] [PubMed]
  2. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
    [CrossRef] [PubMed]
  3. R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco, and M. J. Padgett, “Multipoint holographic optical velocimetry in microfluidic systems,” Phys. Rev. Lett. 96(13), 134502 (2006).
    [CrossRef] [PubMed]
  4. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [CrossRef] [PubMed]
  5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
    [CrossRef] [PubMed]
  6. K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J.-F. Morizur, P. K. Lam, and H.-A. Bachor, “Entangling the spatial properties of laser beams,” Science 321(5888), 541–543 (2008).
    [CrossRef] [PubMed]
  7. J. Janousek, K. Wagner, J.-F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagation modes,” Nat. Photonics 3(7), 399–402 (2009).
    [CrossRef]
  8. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
    [CrossRef]
  9. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
    [CrossRef] [PubMed]
  10. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
    [CrossRef]
  11. J. R. Leger, D. Chen, and K. Dai, “High modal discrimination in a Nd:YAG laser resonator with internal phase gratings,” Opt. Lett. 19(23), 1976–1978 (1994).
    [CrossRef] [PubMed]
  12. S. D. Silvestri, V. Magni, O. Svelto, and G. Valentini, “Lasers with super-Gaussian Mirrors,” IEEE J. Quantum Electron. 26(9), 1500–1509 (1990).
    [CrossRef]
  13. G. Machavariani, “Effect of phase imperfections on high-order mode selection with intracavity phase elements,” Appl. Opt. 43(34), 6328–6333 (2004).
    [CrossRef] [PubMed]
  14. K. J. Moh, X.-C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007).
    [CrossRef] [PubMed]
  15. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
    [CrossRef] [PubMed]
  16. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [CrossRef] [PubMed]
  17. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90(20), 203901 (2003).
    [CrossRef] [PubMed]
  18. Y. L. Lee, Y.-C. Noh, C. Jung, T. J. Yu, B.-A. Yu, J. Lee, D.-K. Ko, and K. Oh, “Reshaping of a second-harmonic curve in periodically poled Ti:LiNbO3 channel waveguide by a local-temperature-control technique,” Appl. Phys. Lett. 86(1), 011104 (2005).
    [CrossRef]
  19. M. L. Sundheimer, Ch. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes,” Opt. Lett. 18(17), 1397–1399 (1993).
    [CrossRef] [PubMed]
  20. Y. L. Lee, N. E. Yu, C.-S. Kee, D.-K. Ko, J. Lee, B.-A. Yu, W. Shin, T. J. Eom, and Y.-C. Noh, “Wavelength filtering characteristics of Solc filter based on Ti:PPLN channel waveguide,” Opt. Lett. 32(19), 2813–2815 (2007).
    [CrossRef] [PubMed]
  21. S. W. Kwon, W. S. Yang, H. M. Lee, W. K. Kim, H.-Y. Lee, W. J. Jeong, M. K. Song, and D. H. Yoon, “The ridge waveguide fabrication with periodically poled MgO-doped lithium niobate for green laser,” Appl. Surf. Sci. 254(4), 1101–1104 (2007).
    [CrossRef]
  22. G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with Fourier synthetic quasi-phase-matching gratings,” Opt. Lett. 23(11), 864–866 (1998).
    [CrossRef]

2009

J. Janousek, K. Wagner, J.-F. Morizur, N. Treps, P. K. Lam, C. C. Harb, and H. A. Bachor, “Optical entanglement of co-propagation modes,” Nat. Photonics 3(7), 399–402 (2009).
[CrossRef]

2008

K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J.-F. Morizur, P. K. Lam, and H.-A. Bachor, “Entangling the spatial properties of laser beams,” Science 321(5888), 541–543 (2008).
[CrossRef] [PubMed]

2007

K. J. Moh, X.-C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007).
[CrossRef] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[CrossRef]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98(5), 053901 (2007).
[CrossRef] [PubMed]

Y. L. Lee, N. E. Yu, C.-S. Kee, D.-K. Ko, J. Lee, B.-A. Yu, W. Shin, T. J. Eom, and Y.-C. Noh, “Wavelength filtering characteristics of Solc filter based on Ti:PPLN channel waveguide,” Opt. Lett. 32(19), 2813–2815 (2007).
[CrossRef] [PubMed]

S. W. Kwon, W. S. Yang, H. M. Lee, W. K. Kim, H.-Y. Lee, W. J. Jeong, M. K. Song, and D. H. Yoon, “The ridge waveguide fabrication with periodically poled MgO-doped lithium niobate for green laser,” Appl. Surf. Sci. 254(4), 1101–1104 (2007).
[CrossRef]

2006

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[CrossRef] [PubMed]

R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco, and M. J. Padgett, “Multipoint holographic optical velocimetry in microfluidic systems,” Phys. Rev. Lett. 96(13), 134502 (2006).
[CrossRef] [PubMed]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[CrossRef] [PubMed]

2005

Y. L. Lee, Y.-C. Noh, C. Jung, T. J. Yu, B.-A. Yu, J. Lee, D.-K. Ko, and K. Oh, “Reshaping of a second-harmonic curve in periodically poled Ti:LiNbO3 channel waveguide by a local-temperature-control technique,” Appl. Phys. Lett. 86(1), 011104 (2005).
[CrossRef]

2004

2003

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90(20), 203901 (2003).
[CrossRef] [PubMed]

2001

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

1998

1994

1993

M. L. Sundheimer, Ch. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes,” Opt. Lett. 18(17), 1397–1399 (1993).
[CrossRef] [PubMed]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[CrossRef]

1992

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco, and M. J. Padgett, “Multipoint holographic optical velocimetry in microfluidic systems,” Phys. Rev. Lett. 96(13), 134502 (2006).
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[CrossRef]

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[CrossRef]

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Williams, R. E.

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S. W. Kwon, W. S. Yang, H. M. Lee, W. K. Kim, H.-Y. Lee, W. J. Jeong, M. K. Song, and D. H. Yoon, “The ridge waveguide fabrication with periodically poled MgO-doped lithium niobate for green laser,” Appl. Surf. Sci. 254(4), 1101–1104 (2007).
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S. W. Kwon, W. S. Yang, H. M. Lee, W. K. Kim, H.-Y. Lee, W. J. Jeong, M. K. Song, and D. H. Yoon, “The ridge waveguide fabrication with periodically poled MgO-doped lithium niobate for green laser,” Appl. Surf. Sci. 254(4), 1101–1104 (2007).
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Y. L. Lee, N. E. Yu, C.-S. Kee, D.-K. Ko, J. Lee, B.-A. Yu, W. Shin, T. J. Eom, and Y.-C. Noh, “Wavelength filtering characteristics of Solc filter based on Ti:PPLN channel waveguide,” Opt. Lett. 32(19), 2813–2815 (2007).
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Figures (5)

Fig. 1
Fig. 1

SH and SF curves as a function of fundamental wavelength in a ZnO:PPLN waveguide (@25 °C). Each peak shows different HG mode (from HG 00 to HG 11 ). 1st peak: HG 00 (ω)+HG 00 (ω)→HG 00 (2ω), 2nd peak: HG 00 (ω)+HG 10 (ω)→HG 10 (2ω), 3rd peak: HG 00 (ω)+HG 01 (ω)→HG 01 (2ω), 4th peak: HG 00 (ω)+HG 11 (ω)→HG 11 (2ω).

Fig. 2
Fig. 2

HG modes which are generated in a ZnO:PPLN waveguide by second-order nonlinear process.

Fig. 3
Fig. 3

Wavelengths of different HG modes (HG 00 , HG 10 , HG 01 , HG 11 ) as a function of the operation temperature. The wavelength tuning rate is about 0.05 nm/°C.

Fig. 4
Fig. 4

Schematic of experimental setup to generate LG mode in a ZnO:PPLN waveguide by the use of second-order nonlinear process and local-temperature-control technique. LD, wavelength tunable diode laser; PS, power splitter, PC1, PC2, polarization controller; PPLN1, Ridge-type Zn:PPLN waveguide; PPLN2, Ridge-type MgO:PPLN waveguide; Lens1, Lens2, 10× objective lens; Filter1, Filter2, infrared cut-off filter; BS1, BS2, beam splitter; CCD, charge-coupled device camera.

Fig. 5
Fig. 5

Results of the combination between HG 10 and HG 01 modes with different phase delays and amplitudes. The color and gray images indicate theoretical calculations and experimental results, respectively. a, Amplitude of HG 10 is larger than that of HG 01 . b, Amplitude of HG 10 is smaller than that of HG 01 . c, Amplitude of HG 10 and HG 01 is equal. The temperatures which are written below pictures indicate the deviations from the temperature at π/2 phase shift between two modes (HG 10 , HG 01 ).

Equations (2)

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k S H 2 k f K Q P M = 0
k S F , m k f , m 1 k f , m 2 K Q P M = 0

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