Abstract

We report a method for continuous tuning of ultraviolet (UV) radiation of second harmonic generation in a dispersive beta barium borate (BBO) whispering gallery mode resonator. The doubly resonant enhancement in a high quality factor resonator leads to high conversion efficiency but the resonator dispersion severely limits practical tuning range. By simultaneously varying the temperature of the resonator and the mechanical stress on the disk, we were able to experimentally demonstrate a continuous tuning range of 70 GHz of 317 nm laser light at 0.74%/mW conversion efficiency. The achieved tuning range is at least 35 times wider than that by either mechanical or temperature tuning alone.

© 2014 Optical Society of America

1. Introduction

Nonlinear frequency conversion such as second harmonic generation (SHG) has been widely used for producing coherent laser light that is otherwise difficult to generate directly. Efficient nonlinear conversion process requires high intensity of the fundamental laser. Optical resonators are then often used to enhance the nonlinear conversion efficiency in the configuration of singly or doubly resonant condition in which both pump and its harmonic field in the resonator are coherently buildup [1]. Monolithic dielectric resonators made of nonlinear crystals are of particular interest because of the wide wavelength range of the total internal reflection. Indeed, second harmonic generation in a doubly resonant monolithic ring resonator with four reflecting surfaces was first reported in 1993 [2]. More recently, whispering gallery mode (WGM) resonators, which are capable of ultrahigh quality factors (Q) and small mode volumes [3], have provided new opportunities in the nonlinear optics field. Efficient second and high-order harmonic generations have been demonstrated in WGM resonators made of mostly z-cut lithium niobate using either quasi-phase matching (QPM) [46] or non-critical phase matching (NCPM) [7]. Second harmonic generation is also observed in a GaAs microdisk using 4-bar QPM [8]. In a recent work [9], we demonstrated a novel wide-range cyclic phase matching method in WGM resonators made of xy-cut birefringent crystals, in which efficient second harmonic conversion covering the entire birefringent phase matching range becomes possible in a single monolithic resonator. This phase matching method works for non-ferroelectric birefringent crystals such as BBO, extending the application of WGM resonator enhanced SHG well into the UV regime.

High nonlinear optical conversion efficiencies in WGM resonators rely in part on the high Q factors, which correspond to narrow linewidths of the optical modes. In the double resonance-enhanced scheme, the mode frequency overlap between the pump and its second harmonic becomes a difficult condition to fulfill, especially with narrow linewidths. For the same reason, the SHG tuning range is very limited because the frequencies of the pump and second harmonics do not tune synchronously due to wavelength dispersion, making the use of double resonance-enhanced nonlinear conversion impractical for most spectroscopic applications. In this letter, we demonstrate that one can overcome this tuning limitation by using a combination of thermal and mechanical-stress tuning. We used an xy-cut BBO WGM resonator with a diameter of 2.0 mm and an intrinsic Q factor of 2.2 × 107 at the 634 nm wavelength. The SHG conversion efficiency into 317 nm UV of about 0.7%/mW was obtained using the cyclic phase matching [9]. The UV radiation tuning ranges were first investigated by mechanical compression and the temperature variation separately. We then combined these two tuning methods and achieved a 70 GHz continuous tuning range.

2. Experimental setup

The xy-cut WGM resonator is made of a z-cut BBO crystal substrate using mechanical polishing [10]. A disk preform is first cut out from the crystal substrate so that the optic axis is lying in the disk plane. The preform is then mounted on a rotation system so that the edge of the disk can be shaped into a wedge. This is then followed by progressive diamond polishing of disk edge to optical smoothness. The resulting high Q disk resonator is shown in the inset of Fig. 1. The resonator has a diameter of 2.0 mm and is mounted on a tuning fixture as described below.

 figure: Fig. 1

Fig. 1 Schematic of the experimental setup. LP, linear polarizer; HWP, half-wave plate; DM, dichroic mirror; BP, bandpass filter; T, temperature; U, voltage on the piezo. Inset: a photo of a BBO resonator.

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The overall sketch of the experimental setup is also shown in Fig. 1. The fundamental pump laser is a tunable external cavity diode laser at 634 nm. A linear polarizer and a half-wave plate are used to produce an s-polarized pump laser beam. A convex lens then focuses the beam at the sapphire prism to evanescently excite the TE polarized WGMs in the BBO resonator. An output lens is used to re-collimate the reflected beam. A UV reflective mirror serves as a dichroic filter. The transmitted pump is detected using a silicon detector (PD1). An additional UV bandpass filter is inserted before a UV-sensitive GaP detector (PD2) to block any remaining pump signal. With this arrangement, both pump and second harmonic signals are monitored simultaneously and displayed on a digital oscilloscope. For mechanical tuning, the bottom steel post is attached to a piezo actuator to introduce compression strain to the resonator. The entire resonator setup including the mount is actively temperature controlled for thermal tuning.

3. Experimental results

WGMs are excited and detected by tuning the pump laser frequencies across them. A typical mode spectrum gives multiple Lorentzian dips in the detected signal reflected from the prism, as shown in the upper curve of Fig. 2. The unloaded (intrinsic) mode linewidth is usually measured by using a larger coupling gap thus operating in the under-coupled regime. The corresponding Q factor is then determined by Q = f/∆f, where f is the resonant frequency and ∆f is the linewidth. The measured unloaded linewidth of the mode in the center of the plot is 22 MHz corresponding to a Q of 2.2 × 107 at 634 nm. This Q factor is typically limited by the surface scattering loss. Unlike angle-cut BBO resonators [11], the xy-cut geometry shows obvious uneven stiffness around the disk edge and requires more attention in polishing to achieve good surface optical quality.

 figure: Fig. 2

Fig. 2 Reflected pump and second harmonic output power as pump is detuned across several WGMs. Inset: UV beam fluorescence spots on a white paper with and without the green fluorescent ink.

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The second harmonic signal appears when the doubly resonant condition is fulfilled as shown in Fig. 2. A peak power of 7.4 μW at the second harmonic wavelength was obtained. The dashed arrow shows the in-coupled power of 1.0 mW at pump. The corresponding conversion efficiency (PSH/Ppump2) is about 0.7%/mW, which is several order of magnitude larger than that of a small single-pass doubling setup [12] and is comparable to Fabry-Perot cavity enhanced SHG setup [13,14]. It should be noted that, when the second harmonic output power is optimized, the loaded linewidth at the pump is 140 MHz which is about 7 times larger than the intrinsic linewidth, showing that the pump mode is over-coupled while optimizing coupling of the shorter-wavelength second harmonics [15,16]. However, no serious attempt was made to optimize the overall coupling of the pump. The inset in Fig. 2 gives the photos of the UV fluorescent spots on a white paper screen with and without green fluorescent ink painting. One can also clearly see the spectral dispersion of the pump and second harmonic signals.

The observed high conversion efficiency relies in part on the double resonance condition. In other words, the second harmonic field also builds up coherently in the resonator. The double-resonance enhanced second harmonic generation requires a good mode overlap between the pump and SH in both spatial and spectral domains. The latter one is depicted in Fig. 3(a). Considering the surface Rayleigh scattering-limited Q factors, the resonance profile at SH is expected to be wider than that at pump. The output SH signal in PD2 is then a convolution of both mode shapes and the nonlinear conversion process.

 figure: Fig. 3

Fig. 3 (a) Schematic of doubly resonant condition: left, optimized; right, not optimized, where red curves indicate the pump while magenta curves the second harmonics which have wider width due to larger Rayleigh scattering loss at shorter wavelength; (b) Tuning spectra with increasing voltage applied on the piezo as labeled on the curves and with fixed temperature.

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In order to maintain the double-resonance condition while tuning the second harmonic frequency, the modes at the pump and SH will have to be tuned synchronously. Consider linearized frequency tuning of WGMs as expressed approximately in the following [17]:

1fdfdX=1ndndX1rdrdX
where X can be either temperature T or voltage U applied on the piezo, n and r represent the index of refraction and the radius of the resonator respectively. Tuning the modes at the pump and SH together means that the right part values need to be the same at both wavelengths. In other words, the mode frequency shifting rate at SH should be equal to the twice of that at pump.

For the mechanical tuning, it is well known that the mechanical strain will not only cause the change of the physical dimension but also induce anisotropy, which in turn causes different mode shifting rates for TE and TM modes [18,19]. Figure 3(b) shows the observed pump and SH signals with increasing the piezo voltage U. As one can see, the optimum mode overlap is achieved at U = 7 V, at which the SH peak becomes narrower and centered with the mode at pump. The narrower SH signal profile is due to the nonlinear response to the pump resonance condition. Moreover, by observing the SH peak position relative to the pump dip at U = 4 V and 7 V, one can conclude that the SH mode shifts slower than twice of the speed at the pump mode. Figure 3(b) also shows a 3 dB UV tuning range of about 1 GHz at the pump or 2 GHz at SH, corresponding to about 6 V in the voltage on piezo. In addition, this measurement also provides a quick way to determine the spectral overlap condition for any doubly resonant SHG in a WGM resonator.

With the temperature tuning, the mode shift rate is determined by both thermal expansion and thermorefractive coefficient according to Eq. (1). It is known that BBO crystals have negative thermal optical coefficients: dno/dT = - 16.6 × 10−6 /°C, dne/dT = - 9.3 × 10−6 /°C and positive thermal expansion coefficients [20]. For a TM polarized mode whose refractive index oscillates from the ordinary to extraordinary one [9], its temperature dependent tuning rate should then be larger than TE polarized mode at the same wavelength as opposed to the mechanical dependent tuning. Measurements at different temperatures similar to that in Fig. 3 of the mechanical tuning were carried out. The frequency tuning rate at the pump mode is about −5 GHz/°C, which is consistent with the expected value of −4.7 GHz/°C using the two known thermal expansion coefficients and dno/dT. The measured 3 dB UV tuning range is 0.2 °C in temperature and about 2 GHz in frequency.

The non-synchronized frequency tuning of the doubly resonant modes by either mechanical or temperature tuning alone results in very limited tuning ranges of these devices. As a result, it is then difficult if not impossible to set the SH at a specific frequency. Because of the opposite relative tuning rates of the two tuning methods, however, they can be combined to yield broadband tuning. As described above, if we increase the temperature by 0.2 °C and raise the voltage on the piezo by about 6 V at the same time, the corresponding pump and SH frequency shift concurrently by equal amount, resulting in nearly unchanged SH peak power as shown in Fig. 4. By changing the temperature for a range of 3.4 °C and the voltage range of about 100 V, we observed a continuous tuning range of the 317 nm signal over 70 GHz, which is more than 35 times better than each tuning method alone. In fact, this range is only experimentally limited by the piezo displacement and the corresponding mechanical contact in the current implementation setup.

 figure: Fig. 4

Fig. 4 Tuning spectra with simultaneously increasing both the temperature T and piezo voltage U. The temperature step is 0.2 °C and piezo voltage step is about 6 V.

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4. Conclusion

In conclusion, we have demonstrated a method for continuous tuning double-resonance condition for SHG in high-Q dielectric resonators. The combined temperature and mechanical tuning allows a widely tunable UV light source using cyclic phase matching in a xy-cut BBO whispering gallery mode resonator. Similar approach should apply to other doubly resonant systems. The tuning method paves the way for more practical use of double-resonant enhancement in dielectric resonators. An example of such use would be the new nonlinear crystals KBBF [21]. With the cyclic phase matching and continuous tuning method described here, it may be now possible to extend tunable continuous SHG into the vacuum UV wavelength regime at low pump powers.

Acknowledgments

This work was performed at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. We acknowledge discussions with Ivan S Grudinin and Lukas M Baumgartel. G. Lin acknowledges support from the NASA Postdoctoral Program, administered by Oak Ridge Associated Universities (ORAU).

References and links

1. A. Ashkin, G. Boyd, and J. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966). [CrossRef]  

2. K. Fiedler, S. Schiller, R. Paschotta, P. Kürz, and J. Mlynek, “Highly efficient frequency doubling with a doubly resonant monolithic total-internal-reflection ring resonator,” Opt. Lett. 18(21), 1786–1788 (1993). [CrossRef]   [PubMed]  

3. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef]   [PubMed]  

4. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004). [CrossRef]   [PubMed]  

5. K. Sasagawa and M. Tsuchiya, “Highly efficient third harmonic generation in a periodically poled MgO: LiNbO3 disk resonator,” Appl. Phys. Express 2(12), 122401 (2009). [CrossRef]  

6. J. Moore, M. Tomes, T. Carmon, and M. Jarrahi, “Continuous-wave ultraviolet emission through fourth-harmonic generation in a whispering-gallery resonator,” Opt. Express 19(24), 24139–24146 (2011). [CrossRef]   [PubMed]  

7. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010). [CrossRef]   [PubMed]  

8. P. S. Kuo and S. S. Glenn, “Second-harmonic generation using 4-bar quasi-phasematching in a GaAs microdisk cavity,” arXiv:1210.1984 (2012).

9. G. Lin, J. Fürst, D. V. Strekalov, and N. Yu, “Wide-range cyclic phase matching and second harmonic generation in whispering gallery resonators,” Appl. Phys. Lett. 103(18), 181107 (2013). [CrossRef]  

10. A. Savchenkov, V. Ilchenko, A. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70(5), 051804 (2004). [CrossRef]  

11. G. Lin, J. Fürst, D. V. Strekalov, I. S. Grudinin, and N. Yu, “High-Q UV whispering gallery mode resonators made of angle-cut BBO crystals,” Opt. Express 20(19), 21372–21378 (2012). [CrossRef]   [PubMed]  

12. K. Hara, K. Matsumoto, T. Onda, W. Nagashima, and I. Shoji, “Efficient ultraviolet second-harmonic generation from a walk-off-compensating β-BaB2O4 device with a new structure fabricated by room-temperature bonding,” Appl. Phys. Express 5(5), 052201 (2012). [CrossRef]  

13. S. J. Rehse and S. A. Lee, “Generation of 125 mW frequency stabilized continuous-wave tunable laser light at 295 nm by frequency doubling in a BBO crystal,” Opt. Commun. 213(4-6), 347–350 (2002). [CrossRef]  

14. Y. Kaneda, J. M. Yarborough, L. Li, N. Peyghambarian, L. Fan, C. Hessenius, M. Fallahi, J. Hader, J. V. Moloney, Y. Honda, M. Nishioka, Y. Shimizu, K. Miyazono, H. Shimatani, M. Yoshimura, Y. Mori, Y. Kitaoka, and T. Sasaki, “Continuous-wave all-solid-state 244 nm deep-ultraviolet laser source by fourth-harmonic generation of an optically pumped semiconductor laser using CsLiB6O10 in an external resonator,” Opt. Lett. 33(15), 1705–1707 (2008). [CrossRef]   [PubMed]  

15. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, V. S., “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17(6), 1051–1057 (2000). [CrossRef]  

16. I. S. Grudinin, G. Lin, and N. Yu, “Polarization conversion loss in birefringent crystalline resonators,” Opt. Lett. 38(14), 2410–2412 (2013). [CrossRef]   [PubMed]  

17. D. V. Strekalov, R. J. Thompson, L. M. Baumgartel, I. S. Grudinin, and N. Yu, “Temperature measurement and stabilization in a birefringent whispering gallery mode resonator,” Opt. Express 19(15), 14495–14501 (2011). [CrossRef]   [PubMed]  

18. V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998). [CrossRef]  

19. W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 14 (2001). [CrossRef]  

20. D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987). [CrossRef]  

21. C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996). [CrossRef]  

References

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  1. A. Ashkin, G. Boyd, and J. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
    [Crossref]
  2. K. Fiedler, S. Schiller, R. Paschotta, P. Kürz, and J. Mlynek, “Highly efficient frequency doubling with a doubly resonant monolithic total-internal-reflection ring resonator,” Opt. Lett. 18(21), 1786–1788 (1993).
    [Crossref] [PubMed]
  3. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
    [Crossref] [PubMed]
  4. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
    [Crossref] [PubMed]
  5. K. Sasagawa and M. Tsuchiya, “Highly efficient third harmonic generation in a periodically poled MgO: LiNbO3 disk resonator,” Appl. Phys. Express 2(12), 122401 (2009).
    [Crossref]
  6. J. Moore, M. Tomes, T. Carmon, and M. Jarrahi, “Continuous-wave ultraviolet emission through fourth-harmonic generation in a whispering-gallery resonator,” Opt. Express 19(24), 24139–24146 (2011).
    [Crossref] [PubMed]
  7. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
    [Crossref] [PubMed]
  8. P. S. Kuo and S. S. Glenn, “Second-harmonic generation using 4-bar quasi-phasematching in a GaAs microdisk cavity,” arXiv:1210.1984 (2012).
  9. G. Lin, J. Fürst, D. V. Strekalov, and N. Yu, “Wide-range cyclic phase matching and second harmonic generation in whispering gallery resonators,” Appl. Phys. Lett. 103(18), 181107 (2013).
    [Crossref]
  10. A. Savchenkov, V. Ilchenko, A. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70(5), 051804 (2004).
    [Crossref]
  11. G. Lin, J. Fürst, D. V. Strekalov, I. S. Grudinin, and N. Yu, “High-Q UV whispering gallery mode resonators made of angle-cut BBO crystals,” Opt. Express 20(19), 21372–21378 (2012).
    [Crossref] [PubMed]
  12. K. Hara, K. Matsumoto, T. Onda, W. Nagashima, and I. Shoji, “Efficient ultraviolet second-harmonic generation from a walk-off-compensating β-BaB2O4 device with a new structure fabricated by room-temperature bonding,” Appl. Phys. Express 5(5), 052201 (2012).
    [Crossref]
  13. S. J. Rehse and S. A. Lee, “Generation of 125 mW frequency stabilized continuous-wave tunable laser light at 295 nm by frequency doubling in a BBO crystal,” Opt. Commun. 213(4-6), 347–350 (2002).
    [Crossref]
  14. Y. Kaneda, J. M. Yarborough, L. Li, N. Peyghambarian, L. Fan, C. Hessenius, M. Fallahi, J. Hader, J. V. Moloney, Y. Honda, M. Nishioka, Y. Shimizu, K. Miyazono, H. Shimatani, M. Yoshimura, Y. Mori, Y. Kitaoka, and T. Sasaki, “Continuous-wave all-solid-state 244 nm deep-ultraviolet laser source by fourth-harmonic generation of an optically pumped semiconductor laser using CsLiB6O10 in an external resonator,” Opt. Lett. 33(15), 1705–1707 (2008).
    [Crossref] [PubMed]
  15. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, V. S., “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17(6), 1051–1057 (2000).
    [Crossref]
  16. I. S. Grudinin, G. Lin, and N. Yu, “Polarization conversion loss in birefringent crystalline resonators,” Opt. Lett. 38(14), 2410–2412 (2013).
    [Crossref] [PubMed]
  17. D. V. Strekalov, R. J. Thompson, L. M. Baumgartel, I. S. Grudinin, and N. Yu, “Temperature measurement and stabilization in a birefringent whispering gallery mode resonator,” Opt. Express 19(15), 14495–14501 (2011).
    [Crossref] [PubMed]
  18. V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998).
    [Crossref]
  19. W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 14 (2001).
    [Crossref]
  20. D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
    [Crossref]
  21. C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
    [Crossref]

2013 (2)

G. Lin, J. Fürst, D. V. Strekalov, and N. Yu, “Wide-range cyclic phase matching and second harmonic generation in whispering gallery resonators,” Appl. Phys. Lett. 103(18), 181107 (2013).
[Crossref]

I. S. Grudinin, G. Lin, and N. Yu, “Polarization conversion loss in birefringent crystalline resonators,” Opt. Lett. 38(14), 2410–2412 (2013).
[Crossref] [PubMed]

2012 (2)

G. Lin, J. Fürst, D. V. Strekalov, I. S. Grudinin, and N. Yu, “High-Q UV whispering gallery mode resonators made of angle-cut BBO crystals,” Opt. Express 20(19), 21372–21378 (2012).
[Crossref] [PubMed]

K. Hara, K. Matsumoto, T. Onda, W. Nagashima, and I. Shoji, “Efficient ultraviolet second-harmonic generation from a walk-off-compensating β-BaB2O4 device with a new structure fabricated by room-temperature bonding,” Appl. Phys. Express 5(5), 052201 (2012).
[Crossref]

2011 (2)

2010 (1)

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref] [PubMed]

2009 (1)

K. Sasagawa and M. Tsuchiya, “Highly efficient third harmonic generation in a periodically poled MgO: LiNbO3 disk resonator,” Appl. Phys. Express 2(12), 122401 (2009).
[Crossref]

2008 (1)

2004 (2)

A. Savchenkov, V. Ilchenko, A. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70(5), 051804 (2004).
[Crossref]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[Crossref] [PubMed]

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

2002 (1)

S. J. Rehse and S. A. Lee, “Generation of 125 mW frequency stabilized continuous-wave tunable laser light at 295 nm by frequency doubling in a BBO crystal,” Opt. Commun. 213(4-6), 347–350 (2002).
[Crossref]

2001 (1)

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 14 (2001).
[Crossref]

2000 (1)

1998 (1)

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998).
[Crossref]

1996 (1)

C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
[Crossref]

1993 (1)

1987 (1)

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[Crossref]

1966 (1)

A. Ashkin, G. Boyd, and J. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
[Crossref]

Andersen, U. L.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref] [PubMed]

Ashkin, A.

A. Ashkin, G. Boyd, and J. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
[Crossref]

Baumgartel, L. M.

Boyd, G.

A. Ashkin, G. Boyd, and J. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
[Crossref]

Carmon, T.

Chen, C. T.

C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
[Crossref]

Davis, L.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[Crossref]

Deng, D.

C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
[Crossref]

Dziedzic, J.

A. Ashkin, G. Boyd, and J. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
[Crossref]

Eimerl, D.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[Crossref]

Elser, D.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref] [PubMed]

Fallahi, M.

Fan, L.

Fiedler, K.

Fürst, J.

G. Lin, J. Fürst, D. V. Strekalov, and N. Yu, “Wide-range cyclic phase matching and second harmonic generation in whispering gallery resonators,” Appl. Phys. Lett. 103(18), 181107 (2013).
[Crossref]

G. Lin, J. Fürst, D. V. Strekalov, I. S. Grudinin, and N. Yu, “High-Q UV whispering gallery mode resonators made of angle-cut BBO crystals,” Opt. Express 20(19), 21372–21378 (2012).
[Crossref] [PubMed]

Fürst, J. U.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref] [PubMed]

Gorodetsky, M. L.

Graham, E. K.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[Crossref]

Grudinin, I. S.

Hader, J.

Hara, K.

K. Hara, K. Matsumoto, T. Onda, W. Nagashima, and I. Shoji, “Efficient ultraviolet second-harmonic generation from a walk-off-compensating β-BaB2O4 device with a new structure fabricated by room-temperature bonding,” Appl. Phys. Express 5(5), 052201 (2012).
[Crossref]

Hare, J.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 14 (2001).
[Crossref]

Haroche, S.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998).
[Crossref]

Hessenius, C.

Honda, Y.

Ilchenko, V.

A. Savchenkov, V. Ilchenko, A. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70(5), 051804 (2004).
[Crossref]

Ilchenko, V. S.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[Crossref] [PubMed]

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 14 (2001).
[Crossref]

M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, V. S., “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17(6), 1051–1057 (2000).
[Crossref]

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998).
[Crossref]

Jarrahi, M.

Kaneda, Y.

Kitaoka, Y.

Kürz, P.

Lassen, M.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref] [PubMed]

Lee, S. A.

S. J. Rehse and S. A. Lee, “Generation of 125 mW frequency stabilized continuous-wave tunable laser light at 295 nm by frequency doubling in a BBO crystal,” Opt. Commun. 213(4-6), 347–350 (2002).
[Crossref]

Lefevre-Seguin, V.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998).
[Crossref]

Lefèvre-Seguin, V.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 14 (2001).
[Crossref]

Leuchs, G.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref] [PubMed]

Li, L.

Lin, G.

Long, R.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 14 (2001).
[Crossref]

Maleki, L.

A. Savchenkov, V. Ilchenko, A. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70(5), 051804 (2004).
[Crossref]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[Crossref] [PubMed]

Marquardt, C.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref] [PubMed]

Matsko, A.

A. Savchenkov, V. Ilchenko, A. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70(5), 051804 (2004).
[Crossref]

Matsko, A. B.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[Crossref] [PubMed]

Matsumoto, K.

K. Hara, K. Matsumoto, T. Onda, W. Nagashima, and I. Shoji, “Efficient ultraviolet second-harmonic generation from a walk-off-compensating β-BaB2O4 device with a new structure fabricated by room-temperature bonding,” Appl. Phys. Express 5(5), 052201 (2012).
[Crossref]

Miyazono, K.

Mlynek, J.

Moloney, J. V.

Moore, J.

Mori, Y.

Nagashima, W.

K. Hara, K. Matsumoto, T. Onda, W. Nagashima, and I. Shoji, “Efficient ultraviolet second-harmonic generation from a walk-off-compensating β-BaB2O4 device with a new structure fabricated by room-temperature bonding,” Appl. Phys. Express 5(5), 052201 (2012).
[Crossref]

Nishioka, M.

Onda, T.

K. Hara, K. Matsumoto, T. Onda, W. Nagashima, and I. Shoji, “Efficient ultraviolet second-harmonic generation from a walk-off-compensating β-BaB2O4 device with a new structure fabricated by room-temperature bonding,” Appl. Phys. Express 5(5), 052201 (2012).
[Crossref]

Paschotta, R.

Peyghambarian, N.

Pryamikov, A. D.

Raimond, J. M.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998).
[Crossref]

Rehse, S. J.

S. J. Rehse and S. A. Lee, “Generation of 125 mW frequency stabilized continuous-wave tunable laser light at 295 nm by frequency doubling in a BBO crystal,” Opt. Commun. 213(4-6), 347–350 (2002).
[Crossref]

Sasagawa, K.

K. Sasagawa and M. Tsuchiya, “Highly efficient third harmonic generation in a periodically poled MgO: LiNbO3 disk resonator,” Appl. Phys. Express 2(12), 122401 (2009).
[Crossref]

Sasaki, T.

Savchenkov, A.

A. Savchenkov, V. Ilchenko, A. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70(5), 051804 (2004).
[Crossref]

Savchenkov, A. A.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[Crossref] [PubMed]

Schiller, S.

Shimatani, H.

Shimizu, Y.

Shoji, I.

K. Hara, K. Matsumoto, T. Onda, W. Nagashima, and I. Shoji, “Efficient ultraviolet second-harmonic generation from a walk-off-compensating β-BaB2O4 device with a new structure fabricated by room-temperature bonding,” Appl. Phys. Express 5(5), 052201 (2012).
[Crossref]

Strekalov, D. V.

G. Lin, J. Fürst, D. V. Strekalov, and N. Yu, “Wide-range cyclic phase matching and second harmonic generation in whispering gallery resonators,” Appl. Phys. Lett. 103(18), 181107 (2013).
[Crossref]

G. Lin, J. Fürst, D. V. Strekalov, I. S. Grudinin, and N. Yu, “High-Q UV whispering gallery mode resonators made of angle-cut BBO crystals,” Opt. Express 20(19), 21372–21378 (2012).
[Crossref] [PubMed]

D. V. Strekalov, R. J. Thompson, L. M. Baumgartel, I. S. Grudinin, and N. Yu, “Temperature measurement and stabilization in a birefringent whispering gallery mode resonator,” Opt. Express 19(15), 14495–14501 (2011).
[Crossref] [PubMed]

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref] [PubMed]

Tang, D.

C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
[Crossref]

Thompson, R. J.

Tomes, M.

Treussart, F.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998).
[Crossref]

Tsuchiya, M.

K. Sasagawa and M. Tsuchiya, “Highly efficient third harmonic generation in a periodically poled MgO: LiNbO3 disk resonator,” Appl. Phys. Express 2(12), 122401 (2009).
[Crossref]

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

Velichansky, V. L.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998).
[Crossref]

Velsko, S.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[Crossref]

Volikov, P. S.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998).
[Crossref]

von Klitzing, W.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 14 (2001).
[Crossref]

Wong, K. L.

C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
[Crossref]

Wu, B.

C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
[Crossref]

Xu, Z. Y.

C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
[Crossref]

Yarborough, J. M.

Ye, N.

C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
[Crossref]

Yoshimura, M.

Yu, N.

Zalkin, A.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[Crossref]

Zhang, J.

C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
[Crossref]

Appl. Phys. Express (2)

K. Sasagawa and M. Tsuchiya, “Highly efficient third harmonic generation in a periodically poled MgO: LiNbO3 disk resonator,” Appl. Phys. Express 2(12), 122401 (2009).
[Crossref]

K. Hara, K. Matsumoto, T. Onda, W. Nagashima, and I. Shoji, “Efficient ultraviolet second-harmonic generation from a walk-off-compensating β-BaB2O4 device with a new structure fabricated by room-temperature bonding,” Appl. Phys. Express 5(5), 052201 (2012).
[Crossref]

Appl. Phys. Lett. (2)

G. Lin, J. Fürst, D. V. Strekalov, and N. Yu, “Wide-range cyclic phase matching and second harmonic generation in whispering gallery resonators,” Appl. Phys. Lett. 103(18), 181107 (2013).
[Crossref]

C. T. Chen, Z. Y. Xu, D. Deng, J. Zhang, K. L. Wong, B. Wu, N. Ye, and D. Tang, “The vacuum ultraviolet phase‐matching characteristics of nonlinear optical KBe2BO3F2 crystal,” Appl. Phys. Lett. 68(21), 2930–2932 (1996).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Ashkin, G. Boyd, and J. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
[Crossref]

J. Appl. Phys. (1)

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[Crossref]

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

Nature (1)

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

New J. Phys. (1)

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 14 (2001).
[Crossref]

Opt. Commun. (2)

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998).
[Crossref]

S. J. Rehse and S. A. Lee, “Generation of 125 mW frequency stabilized continuous-wave tunable laser light at 295 nm by frequency doubling in a BBO crystal,” Opt. Commun. 213(4-6), 347–350 (2002).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (1)

A. Savchenkov, V. Ilchenko, A. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70(5), 051804 (2004).
[Crossref]

Phys. Rev. Lett. (2)

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[Crossref] [PubMed]

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref] [PubMed]

Other (1)

P. S. Kuo and S. S. Glenn, “Second-harmonic generation using 4-bar quasi-phasematching in a GaAs microdisk cavity,” arXiv:1210.1984 (2012).

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Figures (4)

Fig. 1
Fig. 1 Schematic of the experimental setup. LP, linear polarizer; HWP, half-wave plate; DM, dichroic mirror; BP, bandpass filter; T, temperature; U, voltage on the piezo. Inset: a photo of a BBO resonator.
Fig. 2
Fig. 2 Reflected pump and second harmonic output power as pump is detuned across several WGMs. Inset: UV beam fluorescence spots on a white paper with and without the green fluorescent ink.
Fig. 3
Fig. 3 (a) Schematic of doubly resonant condition: left, optimized; right, not optimized, where red curves indicate the pump while magenta curves the second harmonics which have wider width due to larger Rayleigh scattering loss at shorter wavelength; (b) Tuning spectra with increasing voltage applied on the piezo as labeled on the curves and with fixed temperature.
Fig. 4
Fig. 4 Tuning spectra with simultaneously increasing both the temperature T and piezo voltage U. The temperature step is 0.2 °C and piezo voltage step is about 6 V.

Equations (1)

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1 f df dX = 1 n dn dX 1 r dr dX

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