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Boosting quantum efficiency using multi-stage parametric amplification
Optics Communications 269 (2007) 378–384 www.elsevier.com/locate/optcom
Boosting quantum efficiency using multi-stage parametric amplification J.W. Haus *, A. Pandey, P.E. Powers Electro-Optics Program, University of Dayton, Dayton, OH 45469-0245, United States Received 6 June 2006; received in revised form 8 August 2006; accepted 8 August 2006
Abstract We develop a numerical simulation to demonstrate increased quantum efficiency that can be achieved by using a second stage, phase matched crystal to convert signal energy to the idler wavelength. A pair of ZnGeP2 crystals with walkoff and pump absorption were simulated leading to a tripling of the idler output energy. The output beam characteristics are close to a Gaussian beam with an M2 around 1.1. 2006 Elsevier B.V. All rights reserved. PACS: 42.65.Yj MSC: 190.2620; 190.4410; 190.4420
1. Introduction A number of schemes have been devised to improve the quantum efficiency in a parametric down-conversion process from the pump to the idler. In scaling up the energy of OPOs, the problems of controlling back-conversion, attaining simple tuning, and a good transverse mode profile must be addressed. One approach to mitigate the effects of back conversion is to use a tandem optical parametric oscillator scheme [1]. This was proposed using one or two crystals in a single cavity [1,2]. Another coupled tandem OPO operated with higher efficiency using a ZnGeP2 (ZGP) crystal inside a cavity that was pumped by a KTiPO4 crystal inside a separate cavity [3,4]. Recently a OPO was used to seed two tandem optical parametric amplifiers (OPAs) with high efficiency energy conversion [5]. Typically such systems are numerically modeled one stage at a time, using programs such as SNLO [6]. However one must guess at parameters such as the beam profile *
Corresponding author. Tel.: +1 937 229 2394; fax: +1 937 229 2097. E-mail addresses: [email protected], jwhaus@udayton. edu (J.W. Haus). 0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.08.049
when trying to simulate the second stage interaction. For example, if significant walkoff is present in the first stage, then the output may have an elongation in the walkoff direction which makes the beam profile non-Gaussian. Our approach is to directly model both stages in a way that lets us take the calculated output of the first stage and propagate it to the second stage where it then serves as the input. Ultimately the conversion efficiency from the pump to the idler in a single phase matched interaction is dictated by the relative photon energies of the pump, signal, and idler. This is especially important when trying to generate wavelengths in the mid IR (3–12 lm) when starting with a near-IR pump. An example of one application where this is important is light detection and ranging (LIDAR) where a source with a strong coherent wave can be used for remote sensing in the atmosphere [7]. In these systems increased quantum efficiency can mean a smaller overall system or less power consumption both of which are driving factors for fielded LIDAR systems. The quantum efficiency from the pump to idler can be significantly improved by adding a second crystal where the signal from the first crystal acts as a pump in the second crystal thus
J.W. Haus et al. / Optics Communications 269 (2007) 378–384
amplifying the first stage idler. Such a scheme has been demonstrated as an intracavity element [8,9] using bulk crystals. It has also been proposed as a cascaded quasiphase matched interaction [10,11]. In birefringent crystals the addition of walkoff complicates the efficient extraction of pump energy. Furthermore due to the high gain provided by the pump and signal the first stage pump is depleted, thus making the first stage crystal longer would not gain much additional conversion efficiency. Here we use our model to demonstrate using optical parametric amplification that by adding a second stage nonlinear crystal pump to extract energy from the signal that the efficiency of the conversion process can be dramatically improved. Since in the first crystal the signal was amplified along with the idler, the second stage extracts additional energy from the signal to transfer it to the longwavelength idler. The case we consider here uses ZGP crystals, but the concept can be used in any situation where optimizing the conversion efficiency to the idler is needed. The results are not specific to either the materials or the wavelengths we used, as long as the crystal enables phase matching of the beams. Furthermore, two different crystals may be used to optimize the phase matching and walkoff for the two stages. In the case we present with ZGP, it is also important to consider the absorption of the pump. The two stage scheme presents a way to mitigate the pump absorption by using the first stage to generate an idler seed which is then amplified by a pump that is not absorbed in the second crystal, i.e. the signal from the first stage. In this paper we present results for two crystals using a single pass difference frequency (DFG) scheme. The schematic of the crystal geometry is shown in Fig. 1. The initial pump and signal are generated in a previous stage not shown in the diagram; this could be two laser sources or a previous optical parametric generation process. They are injected into the first crystal and a new idler frequency is generated. Between the first and second crystal a filter is used to eliminate the pump wave and a wave plate may be required to rotate the polarization of the signal or idler wave from the first crystal. The two waves are inserted into the second crystal, which is phase matched to extract energy from the first stage signal and boost the efficiency of the desired long wavelength output. Our model shows
WP ω1
ω1 ω2
Crystal 1
ω4
ω2
Crystal 2
ω3
ω2 ω3
F
Fig. 1. Diagram of the proposed experiment. The angular frequencies x1 and x2 represent the pump and signal waves for the first crystal. The waves are spectrally filtered (F) to remove the pump and perhaps a waveplate (WP) is used to rotate the relative polarizations of the signal and idler (x3). The waves are injected into the second crystal where the efficiency of the desired wave x3 is boosted.
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that even a one pass DFG can greatly improve the conversion efficiency by a factor of three over a single crystal in which the pump has been depleted. We use ZGP as an example of how the efficiency can be boosted using a second ZGP crystal in this paper. We choose parameter values that are close to those reported in a recent experiment by Haidar et al. [12]. Our results for a single crystal were compared to the results of SNLO [6], which has been thoroughly experimentally validated. 1.1. Modeling Our numerical simulations optimize the conversion efficiency of a dual stage parametric amplifier. We extend the numerical approach of SNLO [6] to couple the output from one crystal into a second one; in general, the input to the second crystal is not a Gaussian profile beam. The process can be continued with optical elements, such as lenses, between the stages. In our simulations we use the refractive indices of ZGP based on Sellmeier equations in Refs. [13– 15]. We only consider amplification of a collinear phase matched signal-idler interaction. However, the model accounts for focusing and diffraction effects both on the propagating beams and on the phase matching. The three interacting waves are all single frequencies. Scaling the output to kilowatt powers is desirable for remote sensing applications [16] but additional non-collinear interactions and spontaneous processes such as spontaneous parametric fluorescence can be important [17,18]. In this paper we use three coupled equations to describe the nonlinear frequency conversion process for the signal field, Es, idler field, Ei and pump field, Ep. To be concrete we follow the paper of Haidar, Miyamoto and Ito [12] and use parameters that are close to those reported in their experiment. For definiteness we consider pumping the first ZGP crystal with a pump wavelength kp = 1.9 lm and a signal wavelength of ks = 2.4 lm. This is a logical choice based on the convenience of using a single KTP OPO to supply both wavelengths. The corresponding idler wavelength is ki = 9.12 lm. The wave propagates along the z axis with diffraction described through the transverse Laplacian 2 2 operator (r2? ¼ oxo 2 þ oyo 2 Þ. oEs xs 2 oEs xs vð2Þ ¼i r ? E s þ qs þi Ep Ei eiDkz ; oz 2ns c oy ns c oEi xi 2 oEi xi vð2Þ ¼i r ? E i þ qi þi Ep Es eiDkz ; oz 2ni c oy ni c oEp xp 2 oEp xp vð2Þ ¼i r? E p þ q p þi Es Ei eiDkz : oz 2np c oy np c
ð1Þ ð2Þ ð3Þ
The subscripts s, i and p denote the signal, idler and pump, respectively. The additional parameters are: the angular frequency, xa, the walkoff angles are denoted by qa, and indices of refraction by na, where a = s, i or p. The longitudinal wave vector mismatch is Dk = kp ks ki, kp, ks and ki are longitudinal wave vectors of pump, signal and idler. The nonlinear coefficient v(2) depends on the orientation of
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the crystal with respect to the propagation direction. The nonlinear coefficient is d36 = 75 ± 8 pm/V (v(2) = 2d) for frequency doubling at a pump wavelength of 9.6 lm [19]; however, using Miller’s rule the average value for the present cases is shifted close to 82 pm/V [20]. The effective nonlinearity for the experimental setup depends on the phase matching directions. The second stage uses the same set of equations with appropriate changes to the parameters to account for the different phase matching. In particular, in the second stage the pump is now the 2.4 lm output from the first stage, the idler is still 9.12 lm, but the signal is 3.26 lm. So it is seen that the 2.4 lm beam that is amplified in the first stage can now be used to generate even more 9.12 lm idler. Since the phase matching condition for this second stage is different we used the appropriate indices and walkoff angles. We calculated the effective nonlinearity using d36 = 82 pm/V as was done with the first stage. The equations are solved by a split step operator technique with two transverse coordinates. The symmetry of the spatial solutions is broken by the walkoff of the beams. A Gaussian temporal pulse shape was used with a pulse width of 10 ns full-width at half maximum. For the short crystal lengths used in this paper the effects of group velocity mismatch can be neglected and so we are able to consider the time domain as a series of cw power levels. We use the symmetry of the pulse shape and run our simulation over half the pulse width, which is divided into 20 time segments and hence 20 input quasi-cw power levels. In the transverse spatial dimension we typically used a 256 · 256 grid, but results were also compared with a 512 · 512 grid. The results for a single crystal are compared with the program SNLO and we find excellent agreement in comparing our outputs with SNLO [6].
Fig. 2. The peak fluence versus position z in the crystal.
wave energy relative to the pump wave energy, is 10.5% without pump absorption; the idler pulse energy is 0.1265 mJ. In Fig. 3 the pump and idler beam profiles at
2. Results As an initial step we calculated the output of the single ZGP crystal without absorption. The input parameters for this crystal are a Gaussian beam diameter of 1 mm FWHM and 10 ns pulse width. The powers for the input to the ZGP crystal were estimated from Ref. [12] as the KTP OPO is one source for the pump and signal inputs injected into the ZGP crystal. The pump pulse energy was 1.2 mJ, and the signal input was 0.8 mJ. The pump wave is polarized along the ordinary wave axis. The signal and idler are extra-ordinary propagating waves with walkoff angle values: qs1 = 12.01 · 103 rad and qi1 = 12.25 · 103 rad. The effective nonlinearity for the first crystal is deff = 80.5 pm/V. At phase matching the indices in the first crystal are: np1 = 3.152, ns1 = 3.162, and ni1 = 3.112. A step size of 50 lm was used for the propagation step. The peak fluence inside the first ZGP crystal is shown in Fig. 2. The logarithmic scale flattens the pump and signal depletion and gain effects, but the idler experiences a high gain. At the output of the crystal the pump energy is depleted to 0.59 mJ, and the signal energy is increased to 1.3 mJ. The conversion efficiency, as defined by the ratio of the idler
Fig. 3. (Top) The output pump fluence; (bottom) the output idler fluence. The signal profile is similar to the idler.
J.W. Haus et al. / Optics Communications 269 (2007) 378–384
the output of the first stage ZGP crystal are plotted. The asymmetric form of the pump depletion is a result of the signal and idler walkoff. The maximum M2 for the pump parallel and perpendicular to the walkoff direction is 1.72, where M2 is measured using the second moment technique as defined by Siegman [21]. The maximum idler M2 in both directions is around 1.01. The peak of the pump fluence remains nearly constant in Fig. 2 despite significant depletion because of the walkoff for the signal and idler beams. The pump energy could be extracted more efficiently by compensating for the walkoff when the signal is injected into the crystal or by cutting the crystal into several lengths that walk the signal and idler beams in opposite directions and extract the energy at the peak of the fluence. Such schemes are commonly used for walkoff compensation, but only by using multiple phase matching schemes, such as our two stage approach, can the ultimate quantum efficiency be increased. Given the present technology for ZGP crystals there is appreciable linear absorption below 2 lm due to a Zn defect and the 1.9 lm pump will normally experience absorption. To determine the effect on the pump absorption we include this effect in our simulations as well. The absorption coefficient is estimated to be 94.2 m1 [6]. The top graph in Fig. 4 plots the peak fluence for the pump, signal, and idler. The pump value changes appreciably, an indication that the beam’s overall intensity has been affected; the output idler energy is 0.0672 mJ, which reduces the efficiency to 5.61% at the output of the first crystal. This is close to the output efficiency reported by Haidar et al. [12] for the 8 lm idler output. The M2 values for the output as a function of time are plotted on the bottom in Fig. 4. Similar to the case without absorption the signal and idler M2 square values remain around 1.01. The 2.4 lm and 9.12 lm beams are injected into a second ZGP crystal, which is now phase matched for a 2.4 lm pump to generate a 9.12 lm and an additional 3.26 lm signal. The 2.4 lm beam is polarized as an ordinary ray, while the 9.12 lm and 3.26 lm beams are extraordinary rays with walkoff parameters: qs2 = 12.61 · 103 rad and qi2 = 12.12 · 103 rad, respectively. The effective nonlinearity for the second crystal is 81.2 pm/V and the refractive indices are: np2 = 3.138, ns2 = 3.11 and ni2 = 3.148, at wavelengths 2.4 lm, 9.12 lm and 3.26 lm, respectively. In the second crystal we can access a larger nonlinearity for the crystal. In Fig. 5 the peak fluence inside the second crystal (top) and M2 (bottom) after the second crystal are displayed. This includes the effects of absorption in the first crystal. The first crystal has a length of 10 mm in all cases below. The 2.4 lm pump wavelength in the second crystal is not absorbed; we found that a crystal length of 20 mm is optimal for the best conversion efficiency. The output energy at the 9.12 lm wavelength from the second crystal is increased to 0.286 mJ so that the overall conversion efficiency from 1.9 lm to 9.12 lm is 17.32%; a factor of three increase in the overall efficiency compared to a single crystal. For a
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Fig. 4. (Top) Peak pump fluence with pump absorption versus crystal position. (Bottom) M2 values in the walkoff direction for the output of the first crystal versus time.
10 mm (second) crystal length the conversion efficiency is 10.5%. When pump absorption in the first crystal is eliminated, the two stage conversion efficiency is 21.06% for a 10 mm crystal and 23.84% for the 20 mm crystal. The bottom graph in Fig. 5 displays the M2 in the walkoff direction for the 9.12 lm beam, which has a maximum value around 1.1 near the pulse peak. The 2.4 lm output from the second crystal has a similar asymmetric profile shape as shown by the contour diagram in Fig. 6. The walkoff direction is clearly evident in the profile. The 9.12 lm output contour is symmetric with the center moved away from the origin. The beam retains its Gaussian characteristics. Again, as in the first stage crystal, the efficiency can be improved by segmenting the crystal to walk the beams over the maximum fluence at the wavelength 2.4 lm or by injecting the input beams at an angle that eliminates the walkoff. The conversion efficiency boost we find here is not limited to our choice of signal energy. Fig. 7 reports our findings for the output energy at 9.12 lm for different values of the input signal energy at 2.4 lm and fixed pump energy of 1.2 mJ. For signal energies above our previously chosen
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Fig. 6. (Top) Fluence profile of the 2.4 lm wave after the second crystal. (Bottom) Fluence profile of the 9.12 lm wave after the second crystal.
Fig. 5. (Top) Peak fluence in the second crystal versus position. (Bottom) M2 values in the walkoff direction for the output of the second crystal versus time.
value of 0.8 mJ we find a rapid saturation of the output because the stronger signal saturates the pump. Making the crystal longer in this case will not improve the conversion efficiency. For smaller signal energies there is always a boost in the efficiency even down to values of the energy that are smaller than 0.1 mJ. In a recent paper Vodopyanov et al. generated IR radiation using ZGP with 100 ns pulses and a cavity configuration [22]. Their beam width is almost the same as used in this paper and their highest pump energies have peak intensities which are comparable to our calculations reported here. For our parameters we expect 22% conversion, while they obtain 12% conversion efficiency with the OPO geometry. The two crystal boost in efficiency we propose here can be a significant improvement using a simple geometry. The enhancement we observe at 9.12 lm wavelength is not limited to this specific case. In Fig. 8 the results over the wavelength band from 5.4 lm to 9.12 lm is plotted.
Fig. 7. Output pulse energy versus input signal energy. The input pump energy is fixed at 1.2 mJ. The output wavelength is 9.12 lm. Four scenarios are analyzed: single crystal and double crystal systems each with (a 5 0) and without (a = 0) pump absorption in the first crystal.
J.W. Haus et al. / Optics Communications 269 (2007) 378–384
Fig. 8. Pulse energy versus output wavelength over a wide band in the mid IR. Four scenarios are analyzed: single crystal and double crystal systems each with (a 5 0) and without (a = 0) pump absorption in the first crystal.
Four cases are considered for each wavelength. The ZGP parameter values for each set of wavelengths in the first and second stages are listed in the Table 1. The wavelengths for the pump and signal in the first stage were chosen based on down-conversion of a YAG laser wavelength to convenient IR wavelengths that can be injected into the ZGP crystals. In each case the same input pump and signal pulse energies and pulse widths are used. In the first stage pump absorption results in a reduction of the output signal and idler energy. 2.1. Conclusions When the long-wavelength idler is the desired output, a two stage parametric amplifier system can be used to significantly increase the quantum efficiency of the down-conver-
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sion process over a single crystal amplifier. The signal and idler are amplified together in the first stage, but the second stage boosts the desired idler output by extracting energy from the signal and transferring it to the idler. This is especially useful when the injected signal has an energy that is comparable to the pump energy. Further work can be done to optimize the performance since there are many parameters, for instance the beam diameter can be changed and the beams can be focused in the medium. Furthermore, the first crystal length can be optimized and it can be cut into in two parts to walk the beams back through the pump beam and the beams can be asymmetrically displaced from their mutual centers. The input pump and/or signal powers can be adjusted to convert more energy. Although the efficiency boosting strategy is advantageous when a strong injected signal is already present, a much weaker signal can give a significant increase in the efficiency too. We conclude that a second stage is warranted to boost the long-wavelength output over a wide range of input signal energies. Our technique could be especially useful for shorter pulse operation where a cavity cannot oscillate and linewidths for the pump and signal may be broader. Our technique does not demand special alignment or design of the experimental setup. To illustrate the effect we used ZGP as a nonlinear medium the specific wavelengths we chose are typical and the two stages can be designed for a range of frequencies. The two stage process boosted the quantum efficiency by over a factor of three more than a single stage can achieve. We report this for a case with pump absorption in the first stage, which further restricts the length of the first crystal. The output results with and without pump absorption after the first stage differed by a factor of two; however, after the second stage the outputs were largely identical. Our model shows a large improvement in the quantum efficiency making it practical to consider such a scheme experimentally.
Table 1 Values used in the simulation across the mid IR regime Wavelength (lm)
Walkoff (mRad)
kp = 1.76 ks = 2.61 ki = 5.404
qs = 9.48 qi = 9.37
kp = 1.8 ks = 2.57 ki = 6.0078
qs = 10.5 qi = 10.45
kp = 1.85 ks = 2.51 ki = 7.0356
qs = 11.38 qi = 11.43
kp = 1.89 ks = 2.46 ki = 8.1568
qs = 11.83 qi = 12.0
kp = 1.9 ks = 2.4 ki = 9.12
qs = 12.01 qi = 12.25
ap (m1) deff (pm/V)
Wavelength (lm)
Walkoff (mRad)
deff (pm/V)
27.18 64.8
kp = 2.61 ks = 5.404 ki = 5.048
qs = 12.22 qi = 12.19
81.0
40.64 71.4
kp = 2.57 ks = 6.0078 ki = 4.4913
qs = 12.28 qi = 12.15
81.1
62.5 77.0
kp = 2.51 ks = 7.0356 ki = 3.9021
qs = 12.4 qi = 12.13
81.2
87.94 79.6
kp = 2.46 ks = 8.1568 ki = 3.5223
qs = 12.53 qi = 12.13
81.2
94.16 80.5
kp = 2.4 ks = 9.12 ki = 3.2757
qs = 12.61 qi = 12.13
81.2
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Acknowledgement The authors would like to acknowledge informative conversations with Loren Eyres and James Morehead at JDSU Commercial Lasers in Milpitas, CA. References [1] G.T. Moore, K. Koch, IEEE J. Quant. Electron. 31 (1995) 761. [2] G.T. Moore, K. Koch, M. Dearborn, M. Vaidyanathan, IEEE J. Quant. Electron. 34 (1998) 803. [3] P.B. Phua, K.S. Lai, R.F. Wu, T.C. Chong, Opt. Lett. 23 (1998) 1562. [4] P.B. Phua, K.S. Lai, R.F. Wu, T.C. Chong, Appl. Opt. 38 (1999) 563. [5] G. Arisholm, Ø. Nordseth, G. Rustad, Opt. Exp. 12 (2004) 4189. [6] SNLO is a program by A.V. Smith, it is available on the web at:. [7] F.K. Tittel, D. Richterand, A. Fried, Top. Appl. Phys. 89 (2003) 445. [8] M.E. Dearborn, K. Koch, G.T. Moore, J.C. Diels, Opt. Lett. 23 (1998) 759.
[9] I.B. Zotova, Y.J. Ding, X.D. Mu, J.B. Khurgin, Opt. Lett. 28 (2003) 552. [10] K.A. Tillman, D.T. Reid, D. Artigas, T.Y. Jiang, J. Opt. Soc. Am. B 21 (2004) 1551. [11] D. Artigas, D.T. Reid, Opt. Commun. 210 (2002) 113. [12] S. Haidar, K. Miyamoto, H. Ito, J. Phys. D: Appl. Phys. 37 (2004) 3347. [13] D.E. Zelmon, E.A. Hanning, P.G. Schunemann, J. Opt. Soc. Am. B 18 (2001) 1307. [14] D.W. Fischer, M.C. Ohmer, J. Appl. Phys. 81 (1997) 425. [15] C. Ghosh, Appl. Opt. 37 (1998) 1205. [16] R.D. Peterson, K.L. Schepler, J.L. Brown, P.G. Schunemann, J. Opt. Soc. Am. B 12 (1995) 2142. [17] Y. Guan, J.W. Haus, P.E. Powers, J. Opt. Soc. Am. B 21 (2004) 1225. [18] Y. Guan, J.W. Haus, P.E. Powers, Phys. Rev. A 71 (2005) 023809. [19] P.D. Mason, D.J. Jackson, E.K. Gorton, Opt. Commun. 110 (1994) 163. [20] Arlee Smith, Private communication: We thank Arlee Smith for clarifying this point. [21] A.E. Siegman, Proc. Soc. Photo-Opt. Instrum. Eng 1224 (1990) 2. [22] K.L. Vodopyanov, F. Ganikhanov, J.P. Maffetone, I. Zwieback, W. Ruderman, Opt. Lett. 25 (2000) 841.
Boosting quantum efficiency using multi-stage parametric amplification J.W. Haus *, A. Pandey, P.E. Powers Electro-Optics Program, University of Dayton, Dayton, OH 45469-0245, United States Received 6 June 2006; received in revised form 8 August 2006; accepted 8 August 2006
Abstract We develop a numerical simulation to demonstrate increased quantum efficiency that can be achieved by using a second stage, phase matched crystal to convert signal energy to the idler wavelength. A pair of ZnGeP2 crystals with walkoff and pump absorption were simulated leading to a tripling of the idler output energy. The output beam characteristics are close to a Gaussian beam with an M2 around 1.1. 2006 Elsevier B.V. All rights reserved. PACS: 42.65.Yj MSC: 190.2620; 190.4410; 190.4420
1. Introduction A number of schemes have been devised to improve the quantum efficiency in a parametric down-conversion process from the pump to the idler. In scaling up the energy of OPOs, the problems of controlling back-conversion, attaining simple tuning, and a good transverse mode profile must be addressed. One approach to mitigate the effects of back conversion is to use a tandem optical parametric oscillator scheme [1]. This was proposed using one or two crystals in a single cavity [1,2]. Another coupled tandem OPO operated with higher efficiency using a ZnGeP2 (ZGP) crystal inside a cavity that was pumped by a KTiPO4 crystal inside a separate cavity [3,4]. Recently a OPO was used to seed two tandem optical parametric amplifiers (OPAs) with high efficiency energy conversion [5]. Typically such systems are numerically modeled one stage at a time, using programs such as SNLO [6]. However one must guess at parameters such as the beam profile *
Corresponding author. Tel.: +1 937 229 2394; fax: +1 937 229 2097. E-mail addresses: [email protected], jwhaus@udayton. edu (J.W. Haus). 0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.08.049
when trying to simulate the second stage interaction. For example, if significant walkoff is present in the first stage, then the output may have an elongation in the walkoff direction which makes the beam profile non-Gaussian. Our approach is to directly model both stages in a way that lets us take the calculated output of the first stage and propagate it to the second stage where it then serves as the input. Ultimately the conversion efficiency from the pump to the idler in a single phase matched interaction is dictated by the relative photon energies of the pump, signal, and idler. This is especially important when trying to generate wavelengths in the mid IR (3–12 lm) when starting with a near-IR pump. An example of one application where this is important is light detection and ranging (LIDAR) where a source with a strong coherent wave can be used for remote sensing in the atmosphere [7]. In these systems increased quantum efficiency can mean a smaller overall system or less power consumption both of which are driving factors for fielded LIDAR systems. The quantum efficiency from the pump to idler can be significantly improved by adding a second crystal where the signal from the first crystal acts as a pump in the second crystal thus
J.W. Haus et al. / Optics Communications 269 (2007) 378–384
amplifying the first stage idler. Such a scheme has been demonstrated as an intracavity element [8,9] using bulk crystals. It has also been proposed as a cascaded quasiphase matched interaction [10,11]. In birefringent crystals the addition of walkoff complicates the efficient extraction of pump energy. Furthermore due to the high gain provided by the pump and signal the first stage pump is depleted, thus making the first stage crystal longer would not gain much additional conversion efficiency. Here we use our model to demonstrate using optical parametric amplification that by adding a second stage nonlinear crystal pump to extract energy from the signal that the efficiency of the conversion process can be dramatically improved. Since in the first crystal the signal was amplified along with the idler, the second stage extracts additional energy from the signal to transfer it to the longwavelength idler. The case we consider here uses ZGP crystals, but the concept can be used in any situation where optimizing the conversion efficiency to the idler is needed. The results are not specific to either the materials or the wavelengths we used, as long as the crystal enables phase matching of the beams. Furthermore, two different crystals may be used to optimize the phase matching and walkoff for the two stages. In the case we present with ZGP, it is also important to consider the absorption of the pump. The two stage scheme presents a way to mitigate the pump absorption by using the first stage to generate an idler seed which is then amplified by a pump that is not absorbed in the second crystal, i.e. the signal from the first stage. In this paper we present results for two crystals using a single pass difference frequency (DFG) scheme. The schematic of the crystal geometry is shown in Fig. 1. The initial pump and signal are generated in a previous stage not shown in the diagram; this could be two laser sources or a previous optical parametric generation process. They are injected into the first crystal and a new idler frequency is generated. Between the first and second crystal a filter is used to eliminate the pump wave and a wave plate may be required to rotate the polarization of the signal or idler wave from the first crystal. The two waves are inserted into the second crystal, which is phase matched to extract energy from the first stage signal and boost the efficiency of the desired long wavelength output. Our model shows
WP ω1
ω1 ω2
Crystal 1
ω4
ω2
Crystal 2
ω3
ω2 ω3
F
Fig. 1. Diagram of the proposed experiment. The angular frequencies x1 and x2 represent the pump and signal waves for the first crystal. The waves are spectrally filtered (F) to remove the pump and perhaps a waveplate (WP) is used to rotate the relative polarizations of the signal and idler (x3). The waves are injected into the second crystal where the efficiency of the desired wave x3 is boosted.
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that even a one pass DFG can greatly improve the conversion efficiency by a factor of three over a single crystal in which the pump has been depleted. We use ZGP as an example of how the efficiency can be boosted using a second ZGP crystal in this paper. We choose parameter values that are close to those reported in a recent experiment by Haidar et al. [12]. Our results for a single crystal were compared to the results of SNLO [6], which has been thoroughly experimentally validated. 1.1. Modeling Our numerical simulations optimize the conversion efficiency of a dual stage parametric amplifier. We extend the numerical approach of SNLO [6] to couple the output from one crystal into a second one; in general, the input to the second crystal is not a Gaussian profile beam. The process can be continued with optical elements, such as lenses, between the stages. In our simulations we use the refractive indices of ZGP based on Sellmeier equations in Refs. [13– 15]. We only consider amplification of a collinear phase matched signal-idler interaction. However, the model accounts for focusing and diffraction effects both on the propagating beams and on the phase matching. The three interacting waves are all single frequencies. Scaling the output to kilowatt powers is desirable for remote sensing applications [16] but additional non-collinear interactions and spontaneous processes such as spontaneous parametric fluorescence can be important [17,18]. In this paper we use three coupled equations to describe the nonlinear frequency conversion process for the signal field, Es, idler field, Ei and pump field, Ep. To be concrete we follow the paper of Haidar, Miyamoto and Ito [12] and use parameters that are close to those reported in their experiment. For definiteness we consider pumping the first ZGP crystal with a pump wavelength kp = 1.9 lm and a signal wavelength of ks = 2.4 lm. This is a logical choice based on the convenience of using a single KTP OPO to supply both wavelengths. The corresponding idler wavelength is ki = 9.12 lm. The wave propagates along the z axis with diffraction described through the transverse Laplacian 2 2 operator (r2? ¼ oxo 2 þ oyo 2 Þ. oEs xs 2 oEs xs vð2Þ ¼i r ? E s þ qs þi Ep Ei eiDkz ; oz 2ns c oy ns c oEi xi 2 oEi xi vð2Þ ¼i r ? E i þ qi þi Ep Es eiDkz ; oz 2ni c oy ni c oEp xp 2 oEp xp vð2Þ ¼i r? E p þ q p þi Es Ei eiDkz : oz 2np c oy np c
ð1Þ ð2Þ ð3Þ
The subscripts s, i and p denote the signal, idler and pump, respectively. The additional parameters are: the angular frequency, xa, the walkoff angles are denoted by qa, and indices of refraction by na, where a = s, i or p. The longitudinal wave vector mismatch is Dk = kp ks ki, kp, ks and ki are longitudinal wave vectors of pump, signal and idler. The nonlinear coefficient v(2) depends on the orientation of
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the crystal with respect to the propagation direction. The nonlinear coefficient is d36 = 75 ± 8 pm/V (v(2) = 2d) for frequency doubling at a pump wavelength of 9.6 lm [19]; however, using Miller’s rule the average value for the present cases is shifted close to 82 pm/V [20]. The effective nonlinearity for the experimental setup depends on the phase matching directions. The second stage uses the same set of equations with appropriate changes to the parameters to account for the different phase matching. In particular, in the second stage the pump is now the 2.4 lm output from the first stage, the idler is still 9.12 lm, but the signal is 3.26 lm. So it is seen that the 2.4 lm beam that is amplified in the first stage can now be used to generate even more 9.12 lm idler. Since the phase matching condition for this second stage is different we used the appropriate indices and walkoff angles. We calculated the effective nonlinearity using d36 = 82 pm/V as was done with the first stage. The equations are solved by a split step operator technique with two transverse coordinates. The symmetry of the spatial solutions is broken by the walkoff of the beams. A Gaussian temporal pulse shape was used with a pulse width of 10 ns full-width at half maximum. For the short crystal lengths used in this paper the effects of group velocity mismatch can be neglected and so we are able to consider the time domain as a series of cw power levels. We use the symmetry of the pulse shape and run our simulation over half the pulse width, which is divided into 20 time segments and hence 20 input quasi-cw power levels. In the transverse spatial dimension we typically used a 256 · 256 grid, but results were also compared with a 512 · 512 grid. The results for a single crystal are compared with the program SNLO and we find excellent agreement in comparing our outputs with SNLO [6].
Fig. 2. The peak fluence versus position z in the crystal.
wave energy relative to the pump wave energy, is 10.5% without pump absorption; the idler pulse energy is 0.1265 mJ. In Fig. 3 the pump and idler beam profiles at
2. Results As an initial step we calculated the output of the single ZGP crystal without absorption. The input parameters for this crystal are a Gaussian beam diameter of 1 mm FWHM and 10 ns pulse width. The powers for the input to the ZGP crystal were estimated from Ref. [12] as the KTP OPO is one source for the pump and signal inputs injected into the ZGP crystal. The pump pulse energy was 1.2 mJ, and the signal input was 0.8 mJ. The pump wave is polarized along the ordinary wave axis. The signal and idler are extra-ordinary propagating waves with walkoff angle values: qs1 = 12.01 · 103 rad and qi1 = 12.25 · 103 rad. The effective nonlinearity for the first crystal is deff = 80.5 pm/V. At phase matching the indices in the first crystal are: np1 = 3.152, ns1 = 3.162, and ni1 = 3.112. A step size of 50 lm was used for the propagation step. The peak fluence inside the first ZGP crystal is shown in Fig. 2. The logarithmic scale flattens the pump and signal depletion and gain effects, but the idler experiences a high gain. At the output of the crystal the pump energy is depleted to 0.59 mJ, and the signal energy is increased to 1.3 mJ. The conversion efficiency, as defined by the ratio of the idler
Fig. 3. (Top) The output pump fluence; (bottom) the output idler fluence. The signal profile is similar to the idler.
J.W. Haus et al. / Optics Communications 269 (2007) 378–384
the output of the first stage ZGP crystal are plotted. The asymmetric form of the pump depletion is a result of the signal and idler walkoff. The maximum M2 for the pump parallel and perpendicular to the walkoff direction is 1.72, where M2 is measured using the second moment technique as defined by Siegman [21]. The maximum idler M2 in both directions is around 1.01. The peak of the pump fluence remains nearly constant in Fig. 2 despite significant depletion because of the walkoff for the signal and idler beams. The pump energy could be extracted more efficiently by compensating for the walkoff when the signal is injected into the crystal or by cutting the crystal into several lengths that walk the signal and idler beams in opposite directions and extract the energy at the peak of the fluence. Such schemes are commonly used for walkoff compensation, but only by using multiple phase matching schemes, such as our two stage approach, can the ultimate quantum efficiency be increased. Given the present technology for ZGP crystals there is appreciable linear absorption below 2 lm due to a Zn defect and the 1.9 lm pump will normally experience absorption. To determine the effect on the pump absorption we include this effect in our simulations as well. The absorption coefficient is estimated to be 94.2 m1 [6]. The top graph in Fig. 4 plots the peak fluence for the pump, signal, and idler. The pump value changes appreciably, an indication that the beam’s overall intensity has been affected; the output idler energy is 0.0672 mJ, which reduces the efficiency to 5.61% at the output of the first crystal. This is close to the output efficiency reported by Haidar et al. [12] for the 8 lm idler output. The M2 values for the output as a function of time are plotted on the bottom in Fig. 4. Similar to the case without absorption the signal and idler M2 square values remain around 1.01. The 2.4 lm and 9.12 lm beams are injected into a second ZGP crystal, which is now phase matched for a 2.4 lm pump to generate a 9.12 lm and an additional 3.26 lm signal. The 2.4 lm beam is polarized as an ordinary ray, while the 9.12 lm and 3.26 lm beams are extraordinary rays with walkoff parameters: qs2 = 12.61 · 103 rad and qi2 = 12.12 · 103 rad, respectively. The effective nonlinearity for the second crystal is 81.2 pm/V and the refractive indices are: np2 = 3.138, ns2 = 3.11 and ni2 = 3.148, at wavelengths 2.4 lm, 9.12 lm and 3.26 lm, respectively. In the second crystal we can access a larger nonlinearity for the crystal. In Fig. 5 the peak fluence inside the second crystal (top) and M2 (bottom) after the second crystal are displayed. This includes the effects of absorption in the first crystal. The first crystal has a length of 10 mm in all cases below. The 2.4 lm pump wavelength in the second crystal is not absorbed; we found that a crystal length of 20 mm is optimal for the best conversion efficiency. The output energy at the 9.12 lm wavelength from the second crystal is increased to 0.286 mJ so that the overall conversion efficiency from 1.9 lm to 9.12 lm is 17.32%; a factor of three increase in the overall efficiency compared to a single crystal. For a
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Fig. 4. (Top) Peak pump fluence with pump absorption versus crystal position. (Bottom) M2 values in the walkoff direction for the output of the first crystal versus time.
10 mm (second) crystal length the conversion efficiency is 10.5%. When pump absorption in the first crystal is eliminated, the two stage conversion efficiency is 21.06% for a 10 mm crystal and 23.84% for the 20 mm crystal. The bottom graph in Fig. 5 displays the M2 in the walkoff direction for the 9.12 lm beam, which has a maximum value around 1.1 near the pulse peak. The 2.4 lm output from the second crystal has a similar asymmetric profile shape as shown by the contour diagram in Fig. 6. The walkoff direction is clearly evident in the profile. The 9.12 lm output contour is symmetric with the center moved away from the origin. The beam retains its Gaussian characteristics. Again, as in the first stage crystal, the efficiency can be improved by segmenting the crystal to walk the beams over the maximum fluence at the wavelength 2.4 lm or by injecting the input beams at an angle that eliminates the walkoff. The conversion efficiency boost we find here is not limited to our choice of signal energy. Fig. 7 reports our findings for the output energy at 9.12 lm for different values of the input signal energy at 2.4 lm and fixed pump energy of 1.2 mJ. For signal energies above our previously chosen
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Fig. 6. (Top) Fluence profile of the 2.4 lm wave after the second crystal. (Bottom) Fluence profile of the 9.12 lm wave after the second crystal.
Fig. 5. (Top) Peak fluence in the second crystal versus position. (Bottom) M2 values in the walkoff direction for the output of the second crystal versus time.
value of 0.8 mJ we find a rapid saturation of the output because the stronger signal saturates the pump. Making the crystal longer in this case will not improve the conversion efficiency. For smaller signal energies there is always a boost in the efficiency even down to values of the energy that are smaller than 0.1 mJ. In a recent paper Vodopyanov et al. generated IR radiation using ZGP with 100 ns pulses and a cavity configuration [22]. Their beam width is almost the same as used in this paper and their highest pump energies have peak intensities which are comparable to our calculations reported here. For our parameters we expect 22% conversion, while they obtain 12% conversion efficiency with the OPO geometry. The two crystal boost in efficiency we propose here can be a significant improvement using a simple geometry. The enhancement we observe at 9.12 lm wavelength is not limited to this specific case. In Fig. 8 the results over the wavelength band from 5.4 lm to 9.12 lm is plotted.
Fig. 7. Output pulse energy versus input signal energy. The input pump energy is fixed at 1.2 mJ. The output wavelength is 9.12 lm. Four scenarios are analyzed: single crystal and double crystal systems each with (a 5 0) and without (a = 0) pump absorption in the first crystal.
J.W. Haus et al. / Optics Communications 269 (2007) 378–384
Fig. 8. Pulse energy versus output wavelength over a wide band in the mid IR. Four scenarios are analyzed: single crystal and double crystal systems each with (a 5 0) and without (a = 0) pump absorption in the first crystal.
Four cases are considered for each wavelength. The ZGP parameter values for each set of wavelengths in the first and second stages are listed in the Table 1. The wavelengths for the pump and signal in the first stage were chosen based on down-conversion of a YAG laser wavelength to convenient IR wavelengths that can be injected into the ZGP crystals. In each case the same input pump and signal pulse energies and pulse widths are used. In the first stage pump absorption results in a reduction of the output signal and idler energy. 2.1. Conclusions When the long-wavelength idler is the desired output, a two stage parametric amplifier system can be used to significantly increase the quantum efficiency of the down-conver-
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sion process over a single crystal amplifier. The signal and idler are amplified together in the first stage, but the second stage boosts the desired idler output by extracting energy from the signal and transferring it to the idler. This is especially useful when the injected signal has an energy that is comparable to the pump energy. Further work can be done to optimize the performance since there are many parameters, for instance the beam diameter can be changed and the beams can be focused in the medium. Furthermore, the first crystal length can be optimized and it can be cut into in two parts to walk the beams back through the pump beam and the beams can be asymmetrically displaced from their mutual centers. The input pump and/or signal powers can be adjusted to convert more energy. Although the efficiency boosting strategy is advantageous when a strong injected signal is already present, a much weaker signal can give a significant increase in the efficiency too. We conclude that a second stage is warranted to boost the long-wavelength output over a wide range of input signal energies. Our technique could be especially useful for shorter pulse operation where a cavity cannot oscillate and linewidths for the pump and signal may be broader. Our technique does not demand special alignment or design of the experimental setup. To illustrate the effect we used ZGP as a nonlinear medium the specific wavelengths we chose are typical and the two stages can be designed for a range of frequencies. The two stage process boosted the quantum efficiency by over a factor of three more than a single stage can achieve. We report this for a case with pump absorption in the first stage, which further restricts the length of the first crystal. The output results with and without pump absorption after the first stage differed by a factor of two; however, after the second stage the outputs were largely identical. Our model shows a large improvement in the quantum efficiency making it practical to consider such a scheme experimentally.
Table 1 Values used in the simulation across the mid IR regime Wavelength (lm)
Walkoff (mRad)
kp = 1.76 ks = 2.61 ki = 5.404
qs = 9.48 qi = 9.37
kp = 1.8 ks = 2.57 ki = 6.0078
qs = 10.5 qi = 10.45
kp = 1.85 ks = 2.51 ki = 7.0356
qs = 11.38 qi = 11.43
kp = 1.89 ks = 2.46 ki = 8.1568
qs = 11.83 qi = 12.0
kp = 1.9 ks = 2.4 ki = 9.12
qs = 12.01 qi = 12.25
ap (m1) deff (pm/V)
Wavelength (lm)
Walkoff (mRad)
deff (pm/V)
27.18 64.8
kp = 2.61 ks = 5.404 ki = 5.048
qs = 12.22 qi = 12.19
81.0
40.64 71.4
kp = 2.57 ks = 6.0078 ki = 4.4913
qs = 12.28 qi = 12.15
81.1
62.5 77.0
kp = 2.51 ks = 7.0356 ki = 3.9021
qs = 12.4 qi = 12.13
81.2
87.94 79.6
kp = 2.46 ks = 8.1568 ki = 3.5223
qs = 12.53 qi = 12.13
81.2
94.16 80.5
kp = 2.4 ks = 9.12 ki = 3.2757
qs = 12.61 qi = 12.13
81.2
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Acknowledgement The authors would like to acknowledge informative conversations with Loren Eyres and James Morehead at JDSU Commercial Lasers in Milpitas, CA. References [1] G.T. Moore, K. Koch, IEEE J. Quant. Electron. 31 (1995) 761. [2] G.T. Moore, K. Koch, M. Dearborn, M. Vaidyanathan, IEEE J. Quant. Electron. 34 (1998) 803. [3] P.B. Phua, K.S. Lai, R.F. Wu, T.C. Chong, Opt. Lett. 23 (1998) 1562. [4] P.B. Phua, K.S. Lai, R.F. Wu, T.C. Chong, Appl. Opt. 38 (1999) 563. [5] G. Arisholm, Ø. Nordseth, G. Rustad, Opt. Exp. 12 (2004) 4189. [6] SNLO is a program by A.V. Smith, it is available on the web at:
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