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Theoretical investigation of impedance matching in the process of sum-frequency generation in an external resonator

Yan Xiao-Juan Ma Wei-Guang Tan Wei

Theoretical investigation of impedance matching in the process of sum-frequency generation in an external resonator

Yan Xiao-Juan, Ma Wei-Guang, Tan Wei
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  • The sum-frequency conversion efficiency is directly proportional to the product of two fundamental laser powers. Therefore, sum-frequency conversion efficiency is rather low when the fundamental beams pass through a nonlinear crystal only once. External resonant technique as an effective means of improving the powers of the fundamental light has been widely applied to the field of nonlinear frequency conversion. This technique can greatly improve the sum-frequency conversion efficiency and is particularly suitable for the situation in which the input power of the fundamental frequency lasers bas been limited. The implementation of high efficient sum-frequency generation in an external resonator requires that the fundamental frequency laser should be efficiently coupled to the external cavity. Therefore, the system needs to achieve impedance matching. In the part of theoretical analysis, first, we derive the enhancement factor when travelingwave cavity is resonant, and then, establish the theoretical models of doubly resonant and singly resonant sum-frequency generation in an external resonator respectively. The variation of enhancement factors as functions of reflectivity of the input couplers and power of the input fundamental light for doubly resonant and singly resonant sum-frequency systems is derived from Boyd-Kleinman theory in detail based on the theoretical models described in the text. The expressions of enhancement factors reflect the nonlinear correlation characteristics of two fundamental light beams in the process of sum-frequency generation. In the part of numerical simulation, firstly, we draw contour plots of output power as functions of reflectivity of the input couplers at two input frequencies in the doubly resonant sum-frequency system by theoretical simulation, and achieve an optimum reflectivity of the input couplers under the condition of different powers of input fundamental light. Secondly, we draw the contour plots of output power as functions of the reflectivity of the input coupler at the resonant frequency, and the input power of non-resonant frequency light in the singly resonant sum-frequency system by theoretical simulation, and achieve an optimum reflectivity of the input coupler at the resonant frequency. These optimum values enable the system to achieve impedance matching; consequently, the sum-frequency conversion efficiency is improved. Finally, this paper analyzes the influence of input power on the impedance matching, and shows that the optimal coupling mirror reflectivity of the resonant fundamental frequency will decrease with the increase of incident power of the other resonant or non-resonant fundamental frequency laser, otherwise, the resonant incident power of its own has less influence on the optimal coupling mirror reflectivity, whether the system undergoes doubly resonant or singly resonant sum-frequency. In addition, if the coupling mirror reflectivity exceeds the optimum value, the power of sum-frequency light will decrease rapidly, while if it is less than the optimum value, the power of sum-frequency light decreases relatively slowly. Therefore an input coupler that may yield over-coupling should be avoided. These results will have a certain guiding significance to related experiments.
      Corresponding author: Yan Xiao-Juan, yanxj@sxu.edu.cn
    • Funds: Project supported by the National Basic Research Program of China (Grant No. 2012CB921603), the National Natural Science Foundation of China (Grant Nos. 61178009, 61275213, 11304189), and the Natural Science Foundation of Shanxi Province, China (Grant No. 2015021105).
    [1]

    Franken P A, Hill A E, Peters C W, Weinreich G 1961 Phys. Rev. Lett. 7 118

    [2]

    Maker P D, Terhune R W, Nisenoff C M, Savage C 1962 Phys. Rev. Lett. 8 21

    [3]

    Giordmine J 1962 Phys. Rev. Lett. 8 19

    [4]

    Foltynowicz A, Ban T, Masłwski P, Adler F, Ye J 2011 Phys. Rev. Lett. 107 233002

    [5]

    Sugiyama K, Kawajiri S, Yabu N, Matsumoto K, Kitano M 2010 Appl. Opt. 49 5510

    [6]

    Hollemann G, Braun B, Dorsch F, Hennig P, Heistulf P, Kutschki U, Voelckel H 2000 Proc. SPIE 3954 140

    [7]

    Wang P Y, Xie S Y, Bo Y, Wang B S, Zuo J W, Wang Z C, Shen Y, Zhang F F, Wei K, Jin K, Xu Y T, Xu J L, Peng Q J, Zhang J Y, Lei W Q, Cui D F, Zhang Y D, Xu Z Y 2014 Chin. Phys. B 23 094208

    [8]

    Yan X J, Li Z X, Zhang Y Z, Tan W, Fu X F, Ma W G, Zhang L, Yin W B, Jia S T 2012 Acta Sin. Quantum Opt. 18 197 (in Chinese) [闫晓娟, 李志新, 张永智, 谭巍, 付小芳, 马维光, 张雷, 尹王保, 贾锁堂 2012 量子光学学报 18 197]

    [9]

    Boyd G D, Kleinman D A 1968 J. Appl. Phys. 39 3597

    [10]

    Wen X, Han Y H, Bai J D, He J, Wang Y H, Yang B D, Wang J M 2014 Opt. Express 22 32293

    [11]

    Yang W H, Wang Y J, Zheng Y H, Lu H D 2015 Opt. Express 23 19624

    [12]

    Yan X J, Li Z X, Zhang Y Z, Wang L, Hu Z Y, Ma W G, Zhang L, Yin W B, Jia S T 2011 Acta Phys. Sin. 60 104210 (in Chinese) [闫晓娟, 李志新, 张永志, 王乐, 胡志裕, 马维光, 张雷, 尹王保, 贾锁堂 2011 物理学报 60 104210]

    [13]

    Tan W, Fu X F, Li Z X, Zhao G, Yan X J, Ma W G, Dong L, Zhang L, Yin W B, Jia S T 2013 Acta Phys. Sin. 62 094211 (in Chinese) [谭巍, 付小芳, 李志新, 赵刚, 闫晓娟, 马维光, 董磊, 张雷, 尹王保, 贾锁堂 2013 物理学报 62 094211]

    [14]

    Bienfang J C, Denman C A, Grime B W, Hillman P D, Moore G T, Telle J M 2003 Opt. Lett. 28 2219

    [15]

    Kumagai H 2007 Opt. Lett. 32 62

    [16]

    Andersen M T, Schlosser P J, Hastie J E, Tidemand-Lichtenberg P, Dawson M D, Pedersen C 2009 Opt. Express 17 6010

    [17]

    Mimoun E, Sarlo L D, Zondy J J, Dalibard J, Gerbier F 2010 Appl. Phys. B 99 31

    [18]

    Samblowski A, Vollmer C E, Baune C, Fiurek J, Schnabel R 2014 Opt. Lett. 39 2979

  • [1]

    Franken P A, Hill A E, Peters C W, Weinreich G 1961 Phys. Rev. Lett. 7 118

    [2]

    Maker P D, Terhune R W, Nisenoff C M, Savage C 1962 Phys. Rev. Lett. 8 21

    [3]

    Giordmine J 1962 Phys. Rev. Lett. 8 19

    [4]

    Foltynowicz A, Ban T, Masłwski P, Adler F, Ye J 2011 Phys. Rev. Lett. 107 233002

    [5]

    Sugiyama K, Kawajiri S, Yabu N, Matsumoto K, Kitano M 2010 Appl. Opt. 49 5510

    [6]

    Hollemann G, Braun B, Dorsch F, Hennig P, Heistulf P, Kutschki U, Voelckel H 2000 Proc. SPIE 3954 140

    [7]

    Wang P Y, Xie S Y, Bo Y, Wang B S, Zuo J W, Wang Z C, Shen Y, Zhang F F, Wei K, Jin K, Xu Y T, Xu J L, Peng Q J, Zhang J Y, Lei W Q, Cui D F, Zhang Y D, Xu Z Y 2014 Chin. Phys. B 23 094208

    [8]

    Yan X J, Li Z X, Zhang Y Z, Tan W, Fu X F, Ma W G, Zhang L, Yin W B, Jia S T 2012 Acta Sin. Quantum Opt. 18 197 (in Chinese) [闫晓娟, 李志新, 张永智, 谭巍, 付小芳, 马维光, 张雷, 尹王保, 贾锁堂 2012 量子光学学报 18 197]

    [9]

    Boyd G D, Kleinman D A 1968 J. Appl. Phys. 39 3597

    [10]

    Wen X, Han Y H, Bai J D, He J, Wang Y H, Yang B D, Wang J M 2014 Opt. Express 22 32293

    [11]

    Yang W H, Wang Y J, Zheng Y H, Lu H D 2015 Opt. Express 23 19624

    [12]

    Yan X J, Li Z X, Zhang Y Z, Wang L, Hu Z Y, Ma W G, Zhang L, Yin W B, Jia S T 2011 Acta Phys. Sin. 60 104210 (in Chinese) [闫晓娟, 李志新, 张永志, 王乐, 胡志裕, 马维光, 张雷, 尹王保, 贾锁堂 2011 物理学报 60 104210]

    [13]

    Tan W, Fu X F, Li Z X, Zhao G, Yan X J, Ma W G, Dong L, Zhang L, Yin W B, Jia S T 2013 Acta Phys. Sin. 62 094211 (in Chinese) [谭巍, 付小芳, 李志新, 赵刚, 闫晓娟, 马维光, 董磊, 张雷, 尹王保, 贾锁堂 2013 物理学报 62 094211]

    [14]

    Bienfang J C, Denman C A, Grime B W, Hillman P D, Moore G T, Telle J M 2003 Opt. Lett. 28 2219

    [15]

    Kumagai H 2007 Opt. Lett. 32 62

    [16]

    Andersen M T, Schlosser P J, Hastie J E, Tidemand-Lichtenberg P, Dawson M D, Pedersen C 2009 Opt. Express 17 6010

    [17]

    Mimoun E, Sarlo L D, Zondy J J, Dalibard J, Gerbier F 2010 Appl. Phys. B 99 31

    [18]

    Samblowski A, Vollmer C E, Baune C, Fiurek J, Schnabel R 2014 Opt. Lett. 39 2979

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  • Received Date:  10 September 2015
  • Accepted Date:  23 October 2015
  • Published Online:  05 February 2016

Theoretical investigation of impedance matching in the process of sum-frequency generation in an external resonator

    Corresponding author: Yan Xiao-Juan, yanxj@sxu.edu.cn
  • 1. School of Physics Electronic Engineering, Shanxi University, Taiyuan 030006, China;
  • 2. Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006, China
Fund Project:  Project supported by the National Basic Research Program of China (Grant No. 2012CB921603), the National Natural Science Foundation of China (Grant Nos. 61178009, 61275213, 11304189), and the Natural Science Foundation of Shanxi Province, China (Grant No. 2015021105).

Abstract: The sum-frequency conversion efficiency is directly proportional to the product of two fundamental laser powers. Therefore, sum-frequency conversion efficiency is rather low when the fundamental beams pass through a nonlinear crystal only once. External resonant technique as an effective means of improving the powers of the fundamental light has been widely applied to the field of nonlinear frequency conversion. This technique can greatly improve the sum-frequency conversion efficiency and is particularly suitable for the situation in which the input power of the fundamental frequency lasers bas been limited. The implementation of high efficient sum-frequency generation in an external resonator requires that the fundamental frequency laser should be efficiently coupled to the external cavity. Therefore, the system needs to achieve impedance matching. In the part of theoretical analysis, first, we derive the enhancement factor when travelingwave cavity is resonant, and then, establish the theoretical models of doubly resonant and singly resonant sum-frequency generation in an external resonator respectively. The variation of enhancement factors as functions of reflectivity of the input couplers and power of the input fundamental light for doubly resonant and singly resonant sum-frequency systems is derived from Boyd-Kleinman theory in detail based on the theoretical models described in the text. The expressions of enhancement factors reflect the nonlinear correlation characteristics of two fundamental light beams in the process of sum-frequency generation. In the part of numerical simulation, firstly, we draw contour plots of output power as functions of reflectivity of the input couplers at two input frequencies in the doubly resonant sum-frequency system by theoretical simulation, and achieve an optimum reflectivity of the input couplers under the condition of different powers of input fundamental light. Secondly, we draw the contour plots of output power as functions of the reflectivity of the input coupler at the resonant frequency, and the input power of non-resonant frequency light in the singly resonant sum-frequency system by theoretical simulation, and achieve an optimum reflectivity of the input coupler at the resonant frequency. These optimum values enable the system to achieve impedance matching; consequently, the sum-frequency conversion efficiency is improved. Finally, this paper analyzes the influence of input power on the impedance matching, and shows that the optimal coupling mirror reflectivity of the resonant fundamental frequency will decrease with the increase of incident power of the other resonant or non-resonant fundamental frequency laser, otherwise, the resonant incident power of its own has less influence on the optimal coupling mirror reflectivity, whether the system undergoes doubly resonant or singly resonant sum-frequency. In addition, if the coupling mirror reflectivity exceeds the optimum value, the power of sum-frequency light will decrease rapidly, while if it is less than the optimum value, the power of sum-frequency light decreases relatively slowly. Therefore an input coupler that may yield over-coupling should be avoided. These results will have a certain guiding significance to related experiments.

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