Abstract

We report nonlinear Cherenkov radiations (NCRs) in a Ti in-diffused LiNbO3 planar waveguide. The radiations were modulated exploiting different polarizations and orders of the guided modes, the fundamental wavelengths and the working temperatures. Some characteristics related to NCRs, such as radiation angles and relative intensities were investigated in detail. The experimental results matched well with theoretical calculations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

When the velocity of a charged particle exceeds that of light in medium, coherent light would be emitted at a specific angle, which is called Cherenkov radiation in conventional particle physics [1]. Similar effect was applied to nonlinear optics as the nonlinear Cherenkov radiation (NCR), which requires that the phase velocity of the nonlinear polarization wave (NPW) exceeds that of the harmonic wave. NCR corresponds to an automatically accomplished phase-matching diagram, which means the Cherenkov phase matching angle will change accordingly to the varying of the parameters such as the input wavelength or the working temperature. NCR can be observed in both bulk ferroelectric crystals where the radiation is enhanced by the domain walls, and in nonlinear waveguides where the value of the momentum uncertainty in the thin film is more than the transverse momentum mismatch due to uncertainty principle [2, 3]. In both material platforms, several approaches can be exploited to modulate the behavior of NCRs. Forward and backward reciprocal vectors parallel to the direction of the NPW can deaccelerate or accelerate the phase velocity of the NPWs in periodically-poled nonlinear waveguides [4] or two-dimensional nonlinear photonic crystals [5, 6], thus changing of the radiation angles as well as the nonlinear conversion efficiencies. Besides, when the interacting waves are set to be at different polarization states, i.e., using the birefringence of ferroelectric crystals, the phase velocity of the NPW can be modulated as well. By this means, Cherenkov type second harmonic generation (SHG), was experimentally demonstrated in an anomalous dispersive ultra-thin MgO doped LiNbO3 without phase velocity threshold [7]. Following the same idea, Cherenkov type different frequency generation (DFG), which cannot automatically be accomplished in normal dispersion materials, was realized in a KTP crystal [8]. In addition to utilizing the birefringence properties, the mode dispersion in waveguides can be engineered by choosing proper waveguide parameters, thus provide more flexibility in modulating NCRs. In this paper, we will report our results on NCR modulation in a Ti in-diffused LiNbO3 planar waveguide by mode dispersion.

To fabricate optical waveguides in LiNbO3, several methods, such as proton exchange, metal in-diffusion, femtosecond laser writing, ion implantation, optical grade dicing, and smart cut could be used [9, 10]. Here we chose Ti in-diffused LiNbO3 waveguide to investigate NCRs, because both the ordinary and extraordinary refractive indices are increased after Ti in-diffusion, thus both TE and TM modes are supported in these waveguides. Moreover, the fabrication procedure is relatively simple.

For Cherenkov type SHG in waveguides, it is a guided-to-radiation nonlinear process, i.e., the fundamental wave (FW) is in the guided mode while the second harmonic (SH) wave is radiated into the substrate. In a z-cut, x-propagating Ti in-diffused LiNbO3 waveguide, considering the non-zero elements of the second order nonlinear susceptibility of LiNbO3, the y and z-components of the second order nonlinear polarization P2ω are given by [11]:

Py=2ε0(d22Ey2+2d31EyEz)Pz=2ε0(d31Ey2+d33Ez2)
Ey corresponds to the electric field of the TE mode and Ez corresponds to that of the TM mode. Equation (1) contains four types of nonlinear processes, where the FWs take different waveguide modes and the radiated SH waves are of different polarizations. For simplicity, we adopt the notations which describe the nonlinear frequency conversions involving different polarizations in bulk crystals for the Cherenkov type SHGs: oo-o, eo-o, ee-e, and oo-e. It should be noted that ‘e’ is not exact for the SH wave, because there is a small angle, typically ten or more degrees, between the radiated SH wave and the x-axis of the crystal. The NCR’s phase-matching condition can be written as:
βi(ω)+βj(ω)=k(2ω)cos(θ)
Whereβi,j(ω)=2πNeffi,j(ω)/λ(ω), k(2ω)=2πn(2ω)/λ(2ω). Neffi,j denotes the effective refractive index of two guided modes at the FW, n is the refractive index of the SH wave in the substrate, and θ is the internal Cherenkov angle. Equation (2) is satisfied only when the wave-vector mismatch Δk=βi(ω)+βj(ω)k(2ω) is less than zero. The cladding layer of Ti in-diffused LiNbO3 waveguide is the air, while the refractive index of the guided layer is around 2.2, so Cherenkov SH waves can only radiate into the substrate, instead of into the air, due to the phase-matching requirement.

The refractive index profile of a Ti in-diffused LiNbO3 planar waveguide is determined by the thickness of the Ti film, the diffusion temperature and the diffusion time. After the waveguides have been fabricated at certain process parameters, NCRs could be modulated by changing the working temperature, the input wavelength and exploiting different modes involved in the nonlinear interactions.

The waveguide used in our experiment was fabricated by Ti in-diffusion. A sample with dimensions 20 × 10 × 0.5mm3 along X, Y, Z axes was prepared from a congruent LiNbO3 wafer. After cleaning, an 80-nm thick Ti layer was deposited on the –z face of the sample by electron-beam evaporation. Then Ti in-diffusion was carried out at 1050 °C for 9 hours in air ambient. The end facets of the waveguide were polished for optical measurements. Subsequently, the effective refractive index was characterized through a prism coupler at 632.8 nm. With the inverse WKB analysis [12], the diffusion depth was fitted to be 4.03 μm, thus we could obtain the refractive index profiles at different wavelengths according to [13–15]. We calculated the mode dispersion by the WKB analysis [16], finding four guided modes (TM0, TM1, TE0 and TE1) were supported at 1064 nm, while three modes (TM0, TM1 and TE0) were supported at the longer wavelength of 1342nm in this waveguide. Figure 1(a) shows the temperature dependent dispersion relations of the FW at 1064 nm and the SH wave at 532 nm. It is evident that for ee-e, oo-o, and eo-o processes, Eq. (2) is automatically completed since the relation n(2ω)>Neffi,j(ω) is always satisfied. While for the oo-e process, as shown in Fig. 1(b), NCR is phase-matched at the temperatures above 70°C when the FW is in the TE0 mode, while the critical phase-matching temperature goes down to 20°C when the FW is in the TE1 mode.

 

Fig. 1 (a) Temperature dependent dispersion relations of the FW at 1064nm and SH wave at 532 nm; (b) Zoom in view of the dispersion relations when the FW is in the TE0 and TE1 guided mode, while SH is e-polarized.

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Similar calculations were carried out at 1342 nm and the results were plotted in Fig. 2. One can see that three types of NCRs, which are ee-e, oo-o, and eo-o, can meet the NCR phase-matching requirements. However, for the oo-e process, NCR cannot occur at the temperatures below 200°C where n(2ω)<Neffi,j(ω).

 

Fig. 2 Temperature dependent dispersion relations of the FW at 1342nm and SH wave at 671 nm.

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2. Experimental setup

The schematic experimental setup is shown in Fig. 3. The fundamental source is a Q-switched Nd:YVO4 laser operating at 1064nm/1342nm with a pulse width of 20ns/30ns and a repetition rate of 10 kHz. The polarization of the near infrared beam was controlled by a half wave plate and was focused into the planar waveguide through a cylindrical lens with a 5-mm focal length. The waveguide was put in an oven for accurate temperature control. The generated NCRs were displayed on a screen which was placed 85 mm away from the end facet of the sample. Normally, one can observe two groups of SH spots symmetrically displayed on both sides of the horizontal line of the waveguide, and it can be explained as the following. Because the internal radiation angles are normally less than 20°, so part of the SH waves will experience multiple total internal reflections in the LiNbO3 substrate and exit from the end facet of the waveguide, see the inset in Fig. 3. The lower spots on the screen came from the SH waves directly hitting the end facet, as well as the ones last reflected at the upper surface of the substrate before hitting the end facet of the waveguide. While the upper spots came from the SH waves last reflected from the lower surface of the substrate.

 

Fig. 3 Schematic experimental setup. The inset on the upper left illustrates the origination of the two SH spots on the screen.

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3. Results and discussions

Firstly, the FW was chosen to be at 1064 nm with 45-degree polarization and the temperature of the waveguide was set to 50°C. Four groups of NCRs should be observed in total, which corresponded to the four processes utilizing different nonlinear coefficients described in Eq. (1). As shown in left part of Fig. 4, group ①, ② and ④ can be clearly observed and each group of radiations included two or more Cherenkov radiation spots, which were attributed to Cherenkov SHG or sum frequency generation (SFG) between different guided modes. Group ① and ② were Cherenkov type frequency up-conversions via the oo-e and ee-e process, where the FWs in each group were in the waveguide modes with same polarization but different orders. As for group ④, the spots therein were eo-o type Cherenkov SFGs between different order TE and TM modes. The radiations in Group ③ through the oo-o process were much weaker than the radiations in group ②, which can be seen from the relative output intensities in Table 1. In experiment, to obtain a suitable contrast ratio between group ③ and ② which were close to each other, we set the polarization of the FW to be mainly at the horizontal, thus group ③ could be visualized as shown in the middle part of Fig. 4.

 

Fig. 4 Cherenkov SHGs and SFGs at the fundamental wavelength of 1064nm. The left part illustrates the radiation patterns on the screen. The middle part is a zoom in view of the radiations in group ①, ② and ③. The right part gives the phase-matching types and the involved guide modes for the four groups of radiations.

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Tables Icon

Table 1. Calculated relative output intensities of NCRs

Figure 5 shows the behavior of the oo-e type NCR at different temperatures. There was only one pair of spots induced by Cherenkov SHG of the FW in the TE1 mode at 25°C, between which was a bright spot generated from the collinear guided-to-guided SHG process. When the temperature was increased to 50°C, a new pair of spots appeared, which were generated through Cherenkov SFG of the TE1 and TE0 modes. Meanwhile, the collinear SHG spot weakened obviously because the working temperature was far from the collinear phase-matching temperature at about 20°C. When the temperature was further increased to 75°C, the phase-matching condition for Cherenkov SHG of the FW in the TE0 mode was satisfied, which was consistent with our predictions in Fig. 1(b).

 

Fig. 5 oo-e type NCRs at different temperatures. The guide modes involved are indicated beside the radiation spots.

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The measured external radiation angles varying with the working temperature of the waveguide was given in Fig. 6. With temperature increasing, the radiation angles of the oo-e process became larger, while for ee-e, eo-o, and oo-o processes the radiation angles remained almost the same because temperature has little influence on the wave-vector mismatches for the three processes.

 

Fig. 6 Experimental (the circle) and theoretical (the line) external radiation angles varying with temperatures for different types of NCRs.

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In order to investigate the behavior of the NCRs in the Ti in-diffused waveguide at different wavelengths, we changed the incident light to a nanosecond pulsed 1342 nm laser. Then three kinds of radiation spots was observed, as shown in Fig. 7, when the input beam was 45° polarized. Group ①, ② and ③ were obtained through eo-o, oo-o and ee-e type NCRs respectively. The number of radiation spots in each group was determined by the guided mode supported at 1342 nm, where were TE0, TM0 and TM1 as mentioned previously. The external NCR angles of the ee-e, oo-o and eo-o processes, where the FW only took the lowest order guided modes, were measured to be 26.3°, 29.7° and 39.5°, which matched well with the calculated values 26.0°, 29.7° and 39.0°, respectively. In experiment, oo-e type NCR could not be observed in the temperature range from 25 °C to 200 °C, which was consistent with the theoretical predictions in Fig. 2.

 

Fig. 7 Three groups of NCRs at the fundamental wavelength of 1342nm were shown in the left part. Right part is the zoom in view of the radiation spots in group ② and ③. The guide modes involved are indicated beside the radiation spots.

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We also investigated the efficiencies of the NCRs. By applying the nonlinear coupled-mode theory [17], the output power of the NCRs can be written as:

PSHPFW2[ω2d(2)βSHS]21tanθ
where PFW is the power of the FW coupled into the waveguide, βSH is the propagation constant of the SH wave, d(2) denotes the second order nonlinear coefficient. Sis the normalized overlap integral between the FW and SH electromagnetic field. Assuming PFW to be the same, we can calculate the relative radiation intensities for different processes at 50 °C. As shown in Table 1, oo-e type NCR at 1064 nm has the most intense SH output, which has a small radiation angle and a large overlap integral due to small phase mismatch, though the nonlinear coefficient for this process is not the biggest. The output of the two ee-e processes were less intense, which utilizes the largest nonlinear coefficient. As for oo-o and eo-o processes, the small nonlinear coefficients, small overlap integrals and large radiation angles lead to far less SH powers than the former two processes. Calculations were in good accordance with the experimental observations.

4. Conclusions

Ti in-diffused LiNbO3 is a promising platform for integrated nonlinear photonics, and in this paper, a comprehensive theoretical and experimental study on NCRs in a Ti in-diffused LiNbO3 planar waveguide was presented. The mode dispersions were numerically calculated to find out the phase-matching requirements for NCRs. In experiment, we observed rich phenomena of NCRs which were realized through Cherenkov type second harmonic generation of the same guided mode as well as sum frequency generation between guided modes with different polarizations and different orders. The dependence of the radiations on the wavelength of the input FW and the working temperature were also studied and experimentally confirmed. The relative output intensity of the radiations were mainly determined by the second-order nonlinear coefficient, the overlapping integral and the radiation angle, and the numerical results were consistent with experimental observations. The phase-matching conditions required in this work are for Cherenkov type frequency up-conversions. When the requirements are not satisfied, for example, below some critical temperatures in Fig. 1(b), Cherenkov type frequency down-conversions could be realized. Thus, Cherenkov type different frequency generations (DFG) and spontaneous parametric down conversions (SPDC) based on mode dispersion tailoring might be further research topics in Ti in-diffused LiNbO3 waveguides, which would provide compact tunable laser light sources and entangled photon sources respectively.

Funding

National Key R&D Program of China (2017YFA0303700); National Natural Science Foundation of China (NSFC) (11674171, 91636106, 11627810, 11621091); Jiangsu Science Foundation (BK20151374); Fundamental Research Funds for the Central Universities (020414380064); PAPD of Jiangsu Higher Education Institutions.

References and links

1. P. A. Cherenkov, “Visible emission of clean liquids by action of radiation,” Dokl. Akad. Nauk SSSR 2, 451 (1934).

2. S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009). [CrossRef]  

3. V. Roppo, K. Kalinowski, Y. Sheng, W. Krolikowski, C. Cojocaru, and J. Trull, “Unified approach to Cerenkov second harmonic generation,” Opt. Express 21(22), 25715–25726 (2013). [CrossRef]   [PubMed]  

4. Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov Radiation in Nonlinear Photonic Crystal Waveguides,” Phys. Rev. Lett. 100(16), 163904 (2008). [CrossRef]   [PubMed]  

5. C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N. Zhu, “Experimental realization of Cerenkov up-conversions in a 2D nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101 (2012). [CrossRef]  

6. C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N. Zhu, “Čerenkov difference frequency generation in a two-dimensional nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113 (2012). [CrossRef]  

7. H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108(22), 223901 (2012). [CrossRef]   [PubMed]  

8. R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016). [CrossRef]  

9. M. Bazzana and C. Sada, “Optical waveguides in lithium niobate: Recent developments and applications,” Appl. Phys. Rev. 2(4), 040603 (2015). [CrossRef]  

10. J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015). [CrossRef]   [PubMed]  

11. A. Yariv, Quantum Electronics (Wiley, 1975), Chap. 16.

12. A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48(5), 313–316 (1984). [CrossRef]  

13. P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas, and S. K. Lahiri, “Simulation of refractive index profiles for titanium indiffused lithium niobate channel waveguides,” Fiber Integr. Opt. 15(2), 135–147 (1996). [CrossRef]  

14. P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Analysis of titanium concentration and refractive index profiles of Ti: LiNbO3 channel waveguide,” J. Opt. 39(4), 175–180 (2010). [CrossRef]  

15. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984). [CrossRef]  

16. A. Gedeon, “Comparison between rigorous theory and WKB-analysis of modes in graded-index waveguides,” Opt. Commun. 12(3), 329–332 (1974). [CrossRef]  

17. H. Tamada, “Coupled-mode analysis of second harmonic generation in the form of Cerenkov radiation from a planar optical waveguide,” IEEE J. Quantum Electron. 27(3), 502–508 (1991). [CrossRef]  

References

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  1. P. A. Cherenkov, “Visible emission of clean liquids by action of radiation,” Dokl. Akad. Nauk SSSR 2, 451 (1934).
  2. S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
    [Crossref]
  3. V. Roppo, K. Kalinowski, Y. Sheng, W. Krolikowski, C. Cojocaru, and J. Trull, “Unified approach to Cerenkov second harmonic generation,” Opt. Express 21(22), 25715–25726 (2013).
    [Crossref] [PubMed]
  4. Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov Radiation in Nonlinear Photonic Crystal Waveguides,” Phys. Rev. Lett. 100(16), 163904 (2008).
    [Crossref] [PubMed]
  5. C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N. Zhu, “Experimental realization of Cerenkov up-conversions in a 2D nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101 (2012).
    [Crossref]
  6. C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N. Zhu, “Čerenkov difference frequency generation in a two-dimensional nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113 (2012).
    [Crossref]
  7. H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
    [Crossref] [PubMed]
  8. R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
    [Crossref]
  9. M. Bazzana and C. Sada, “Optical waveguides in lithium niobate: Recent developments and applications,” Appl. Phys. Rev. 2(4), 040603 (2015).
    [Crossref]
  10. J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
    [Crossref] [PubMed]
  11. A. Yariv, Quantum Electronics (Wiley, 1975), Chap. 16.
  12. A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48(5), 313–316 (1984).
    [Crossref]
  13. P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas, and S. K. Lahiri, “Simulation of refractive index profiles for titanium indiffused lithium niobate channel waveguides,” Fiber Integr. Opt. 15(2), 135–147 (1996).
    [Crossref]
  14. P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Analysis of titanium concentration and refractive index profiles of Ti: LiNbO3 channel waveguide,” J. Opt. 39(4), 175–180 (2010).
    [Crossref]
  15. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
    [Crossref]
  16. A. Gedeon, “Comparison between rigorous theory and WKB-analysis of modes in graded-index waveguides,” Opt. Commun. 12(3), 329–332 (1974).
    [Crossref]
  17. H. Tamada, “Coupled-mode analysis of second harmonic generation in the form of Cerenkov radiation from a planar optical waveguide,” IEEE J. Quantum Electron. 27(3), 502–508 (1991).
    [Crossref]

2016 (1)

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

2015 (2)

M. Bazzana and C. Sada, “Optical waveguides in lithium niobate: Recent developments and applications,” Appl. Phys. Rev. 2(4), 040603 (2015).
[Crossref]

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

2013 (1)

2012 (3)

C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N. Zhu, “Experimental realization of Cerenkov up-conversions in a 2D nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101 (2012).
[Crossref]

C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N. Zhu, “Čerenkov difference frequency generation in a two-dimensional nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113 (2012).
[Crossref]

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
[Crossref] [PubMed]

2010 (1)

P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Analysis of titanium concentration and refractive index profiles of Ti: LiNbO3 channel waveguide,” J. Opt. 39(4), 175–180 (2010).
[Crossref]

2009 (1)

S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
[Crossref]

2008 (1)

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov Radiation in Nonlinear Photonic Crystal Waveguides,” Phys. Rev. Lett. 100(16), 163904 (2008).
[Crossref] [PubMed]

1996 (1)

P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas, and S. K. Lahiri, “Simulation of refractive index profiles for titanium indiffused lithium niobate channel waveguides,” Fiber Integr. Opt. 15(2), 135–147 (1996).
[Crossref]

1991 (1)

H. Tamada, “Coupled-mode analysis of second harmonic generation in the form of Cerenkov radiation from a planar optical waveguide,” IEEE J. Quantum Electron. 27(3), 502–508 (1991).
[Crossref]

1984 (2)

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[Crossref]

A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48(5), 313–316 (1984).
[Crossref]

1974 (1)

A. Gedeon, “Comparison between rigorous theory and WKB-analysis of modes in graded-index waveguides,” Opt. Commun. 12(3), 329–332 (1974).
[Crossref]

1934 (1)

P. A. Cherenkov, “Visible emission of clean liquids by action of radiation,” Dokl. Akad. Nauk SSSR 2, 451 (1934).

An, N.

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
[Crossref] [PubMed]

Arie, A.

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
[Crossref]

Bazzana, M.

M. Bazzana and C. Sada, “Optical waveguides in lithium niobate: Recent developments and applications,” Appl. Phys. Rev. 2(4), 040603 (2015).
[Crossref]

Biswas, J. C.

P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Analysis of titanium concentration and refractive index profiles of Ti: LiNbO3 channel waveguide,” J. Opt. 39(4), 175–180 (2010).
[Crossref]

P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas, and S. K. Lahiri, “Simulation of refractive index profiles for titanium indiffused lithium niobate channel waveguides,” Fiber Integr. Opt. 15(2), 135–147 (1996).
[Crossref]

Chen, C. D.

C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N. Zhu, “Experimental realization of Cerenkov up-conversions in a 2D nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101 (2012).
[Crossref]

C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N. Zhu, “Čerenkov difference frequency generation in a two-dimensional nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113 (2012).
[Crossref]

Chen, X.

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
[Crossref] [PubMed]

Cheng, Y.

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

Cherenkov, P. A.

P. A. Cherenkov, “Visible emission of clean liquids by action of radiation,” Dokl. Akad. Nauk SSSR 2, 451 (1934).

Cojocaru, C.

Datt, S.

P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas, and S. K. Lahiri, “Simulation of refractive index profiles for titanium indiffused lithium niobate channel waveguides,” Fiber Integr. Opt. 15(2), 135–147 (1996).
[Crossref]

Deng, X.

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
[Crossref] [PubMed]

Du, L.

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

Edwards, G. J.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[Crossref]

Fang, W.

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

Fang, Z.

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

Ganguly, P.

P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Analysis of titanium concentration and refractive index profiles of Ti: LiNbO3 channel waveguide,” J. Opt. 39(4), 175–180 (2010).
[Crossref]

P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas, and S. K. Lahiri, “Simulation of refractive index profiles for titanium indiffused lithium niobate channel waveguides,” Fiber Integr. Opt. 15(2), 135–147 (1996).
[Crossref]

Gao, Z. D.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov Radiation in Nonlinear Photonic Crystal Waveguides,” Phys. Rev. Lett. 100(16), 163904 (2008).
[Crossref] [PubMed]

Gedeon, A.

A. Gedeon, “Comparison between rigorous theory and WKB-analysis of modes in graded-index waveguides,” Opt. Commun. 12(3), 329–332 (1974).
[Crossref]

Hu, X. P.

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N. Zhu, “Čerenkov difference frequency generation in a two-dimensional nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113 (2012).
[Crossref]

C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N. Zhu, “Experimental realization of Cerenkov up-conversions in a 2D nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101 (2012).
[Crossref]

Kalinowski, K.

Kaul, A. N.

A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48(5), 313–316 (1984).
[Crossref]

Kivshar, Y.

S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
[Crossref]

Koynov, K.

S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
[Crossref]

Krolikowski, W.

V. Roppo, K. Kalinowski, Y. Sheng, W. Krolikowski, C. Cojocaru, and J. Trull, “Unified approach to Cerenkov second harmonic generation,” Opt. Express 21(22), 25715–25726 (2013).
[Crossref] [PubMed]

S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
[Crossref]

Lahiri, S. K.

P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Analysis of titanium concentration and refractive index profiles of Ti: LiNbO3 channel waveguide,” J. Opt. 39(4), 175–180 (2010).
[Crossref]

P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas, and S. K. Lahiri, “Simulation of refractive index profiles for titanium indiffused lithium niobate channel waveguides,” Fiber Integr. Opt. 15(2), 135–147 (1996).
[Crossref]

Lawrence, M.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[Crossref]

Lin, J.

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

Ming, N. B.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov Radiation in Nonlinear Photonic Crystal Waveguides,” Phys. Rev. Lett. 100(16), 163904 (2008).
[Crossref] [PubMed]

Neshev, D.

S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
[Crossref]

Ni, R.

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

Qi, Z.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov Radiation in Nonlinear Photonic Crystal Waveguides,” Phys. Rev. Lett. 100(16), 163904 (2008).
[Crossref] [PubMed]

Qiao, L.

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

Ren, H.

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
[Crossref] [PubMed]

Roppo, V.

Sada, C.

M. Bazzana and C. Sada, “Optical waveguides in lithium niobate: Recent developments and applications,” Appl. Phys. Rev. 2(4), 040603 (2015).
[Crossref]

Saltiel, S.

S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
[Crossref]

Sen, D. C.

P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas, and S. K. Lahiri, “Simulation of refractive index profiles for titanium indiffused lithium niobate channel waveguides,” Fiber Integr. Opt. 15(2), 135–147 (1996).
[Crossref]

Sheng, Y.

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

V. Roppo, K. Kalinowski, Y. Sheng, W. Krolikowski, C. Cojocaru, and J. Trull, “Unified approach to Cerenkov second harmonic generation,” Opt. Express 21(22), 25715–25726 (2013).
[Crossref] [PubMed]

S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
[Crossref]

Song, J.

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

Tamada, H.

H. Tamada, “Coupled-mode analysis of second harmonic generation in the form of Cerenkov radiation from a planar optical waveguide,” IEEE J. Quantum Electron. 27(3), 502–508 (1991).
[Crossref]

Thyagarajan, K.

A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48(5), 313–316 (1984).
[Crossref]

Trull, J.

Voloch-Bloch, N.

S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
[Crossref]

Wang, M.

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

Wang, N.

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

Wu, Y.

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

Xu, P.

C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N. Zhu, “Čerenkov difference frequency generation in a two-dimensional nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113 (2012).
[Crossref]

C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N. Zhu, “Experimental realization of Cerenkov up-conversions in a 2D nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101 (2012).
[Crossref]

Xu, Y.

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

Xu, Y. L.

C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N. Zhu, “Čerenkov difference frequency generation in a two-dimensional nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113 (2012).
[Crossref]

Zhang, Y.

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N. Zhu, “Experimental realization of Cerenkov up-conversions in a 2D nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101 (2012).
[Crossref]

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov Radiation in Nonlinear Photonic Crystal Waveguides,” Phys. Rev. Lett. 100(16), 163904 (2008).
[Crossref] [PubMed]

Zhao, G.

C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N. Zhu, “Experimental realization of Cerenkov up-conversions in a 2D nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101 (2012).
[Crossref]

C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N. Zhu, “Čerenkov difference frequency generation in a two-dimensional nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113 (2012).
[Crossref]

Zheng, Y.

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
[Crossref] [PubMed]

Zhu, S. N.

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N. Zhu, “Čerenkov difference frequency generation in a two-dimensional nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113 (2012).
[Crossref]

C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N. Zhu, “Experimental realization of Cerenkov up-conversions in a 2D nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101 (2012).
[Crossref]

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov Radiation in Nonlinear Photonic Crystal Waveguides,” Phys. Rev. Lett. 100(16), 163904 (2008).
[Crossref] [PubMed]

Zou, J.

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

Appl. Phys. Lett. (2)

R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y. Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency generation exploiting birefringence of KTP,” Appl. Phys. Lett. 108(3), 031104 (2016).
[Crossref]

C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N. Zhu, “Čerenkov difference frequency generation in a two-dimensional nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113 (2012).
[Crossref]

Appl. Phys. Rev. (1)

M. Bazzana and C. Sada, “Optical waveguides in lithium niobate: Recent developments and applications,” Appl. Phys. Rev. 2(4), 040603 (2015).
[Crossref]

Dokl. Akad. Nauk SSSR (1)

P. A. Cherenkov, “Visible emission of clean liquids by action of radiation,” Dokl. Akad. Nauk SSSR 2, 451 (1934).

Fiber Integr. Opt. (1)

P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas, and S. K. Lahiri, “Simulation of refractive index profiles for titanium indiffused lithium niobate channel waveguides,” Fiber Integr. Opt. 15(2), 135–147 (1996).
[Crossref]

IEEE J. Quantum Electron. (2)

H. Tamada, “Coupled-mode analysis of second harmonic generation in the form of Cerenkov radiation from a planar optical waveguide,” IEEE J. Quantum Electron. 27(3), 502–508 (1991).
[Crossref]

S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, “Cerenkovtype second-harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11), 1465–1472 (2009).
[Crossref]

J. Opt. (1)

P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Analysis of titanium concentration and refractive index profiles of Ti: LiNbO3 channel waveguide,” J. Opt. 39(4), 175–180 (2010).
[Crossref]

J. Phys. D Appl. Phys. (1)

C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N. Zhu, “Experimental realization of Cerenkov up-conversions in a 2D nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101 (2012).
[Crossref]

Opt. Commun. (2)

A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48(5), 313–316 (1984).
[Crossref]

A. Gedeon, “Comparison between rigorous theory and WKB-analysis of modes in graded-index waveguides,” Opt. Commun. 12(3), 329–332 (1974).
[Crossref]

Opt. Express (1)

Opt. Quantum Electron. (1)

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[Crossref]

Phys. Rev. Lett. (2)

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
[Crossref] [PubMed]

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov Radiation in Nonlinear Photonic Crystal Waveguides,” Phys. Rev. Lett. 100(16), 163904 (2008).
[Crossref] [PubMed]

Sci. Rep. (1)

J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao, W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate microresonators using femtosecond laser micromachining,” Sci. Rep. 5(1), 8072 (2015).
[Crossref] [PubMed]

Other (1)

A. Yariv, Quantum Electronics (Wiley, 1975), Chap. 16.

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Figures (7)

Fig. 1
Fig. 1 (a) Temperature dependent dispersion relations of the FW at 1064nm and SH wave at 532 nm; (b) Zoom in view of the dispersion relations when the FW is in the TE0 and TE1 guided mode, while SH is e-polarized.
Fig. 2
Fig. 2 Temperature dependent dispersion relations of the FW at 1342nm and SH wave at 671 nm.
Fig. 3
Fig. 3 Schematic experimental setup. The inset on the upper left illustrates the origination of the two SH spots on the screen.
Fig. 4
Fig. 4 Cherenkov SHGs and SFGs at the fundamental wavelength of 1064nm. The left part illustrates the radiation patterns on the screen. The middle part is a zoom in view of the radiations in group ①, ② and ③. The right part gives the phase-matching types and the involved guide modes for the four groups of radiations.
Fig. 5
Fig. 5 oo-e type NCRs at different temperatures. The guide modes involved are indicated beside the radiation spots.
Fig. 6
Fig. 6 Experimental (the circle) and theoretical (the line) external radiation angles varying with temperatures for different types of NCRs.
Fig. 7
Fig. 7 Three groups of NCRs at the fundamental wavelength of 1342nm were shown in the left part. Right part is the zoom in view of the radiation spots in group ② and ③. The guide modes involved are indicated beside the radiation spots.

Tables (1)

Tables Icon

Table 1 Calculated relative output intensities of NCRs

Equations (3)

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

P y = 2 ε 0 ( d 22 E y 2 + 2 d 31 E y E z ) P z = 2 ε 0 ( d 31 E y 2 + d 33 E z 2 )
β i ( ω ) + β j ( ω ) = k ( 2 ω ) cos ( θ )
P S H P F W 2 [ ω 2 d ( 2 ) β S H S ] 2 1 tan θ

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