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Modelling a singly resonant, intracavity ring optical parametric oscillator
Optics Communications 216 (2003) 191–197 www.elsevier.com/locate/optcom
Modelling a singly resonant, intracavity ring optical parametric oscillator Preben Buchhave, Peter Tidemand-Lichtenberg *, Wei Hou, Ulrik L. Andersen, Haim Abitan The Optics Group, Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denmark Received 7 June 2002; received in revised form 16 September 2002; accepted 28 November 2002
Abstract We study theoretically and experimentally the dynamics of a single-frequency, unidirectional ring laser with an intracavity nonlinear singly resonant OPO-crystal in a coupled resonator. We find for a range of operating conditions good agreement between model results and measurements of the laser and OPO power output and of the temporal development of complex dynamic phenomena such as pulse shape, pulse duration, oscillatory transients and Q-switched operation of the laser. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The interaction of several nonlinear elements within a laser cavity leads to complex dynamic phenomena such as oscillatory transients at the onset of pumping, Q-switching when the cavity losses are modulated and self-oscillation and chaotic behaviour for certain phase angles of the circulating fields. A detailed understanding of the dynamics of the interaction between exited state population and the various circulating optical fields and an ability to predict these phenomena are essential for explaining the rich variety of experimental observations that can be made on such *
Corresponding author. Fax: +45-45-88-16-11. E-mail addresses: [email protected] (P. Buchhave), [email protected] (P. Tidemand-Lichtenberg).
systems. Theoretical tools are also essential in order to be able to design wave conversion devices for the generation of new coherent light wavelengths. The paper by Oshman and Harris [1] presents one of the first thorough analyses of intracavity OPO, but they simplified the coupled nonlinear equations by assuming all fields to be resonant and using a mean field approximation. Koch et al. [2] analysed two cascaded elements in a ring cavity, but did not include the interaction with the laser. They were then able to use the Jacobi elliptical solutions to the nonlinear interaction between the fields. Debuisschert et al. [3] performed a thorough numerical and analytical investigation of the intracavity OPO in the pulsed regime. The difference between their model and ours is that we perform a simulation of the time development of the
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(02)02299-X
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population inversion in the laser crystal and the circulating fields using the basic differential equations keeping all the nonlinear terms, while Debuisschert et al. simplify the equations by assuming single resonance (nondepleted signal resonant), using mean field approximation for both laser and OPO and assuming optimum phase adjustment (D/ ¼ p=2). In our model we can accept arbitrary phase relations between the fields and can handle the full range of interaction between the fields in a nonlinear crystal including depletion and nonlinear phase development. Debuisschert et al. derive coupled equations for the circulating laser power and signal power and an exponential growth for the idler power. Our work simulates the time development of the amplitude and phase of the circulating fields allowing phase mismatch, detunings and depletion, permitting us to investigate the effect of varying phase in the optical path, which can lead to strong modulation, Q-switching and instability (some of these effects are or will be described in other papers). Turnbull et al. [4] extend the analysis of Debuisschert et al. to include multimode pump lasers in continuous wave (CW)operation. The basic approximations are the same as those of Debuisschert et al. In an earlier paper [5] we used a different approach to the solution of the full nonlinear set of equations, but for the steady-state case and using the coupled power equations at optimal phase adjustment. Experimental results relating to the pulsed intracavity singly resonant OPO have been published in Turnbull et al. [6] and in a recent paper by Dubois et al. [7]. We have developed a new computerized model for vð2Þ nonlinear processes in both intracavity and external cavity systems. The model is based on the interaction between the laser population inversion and the circulating intracavity laser field including both amplitude and phase. The circulating laser field is coupled to fields circulating in coupled resonators. Both passive losses and losses resulting from nonlinear interactions in optically nonlinear crystals are accounted for. Seen from the laser, these coupled resonators act as losses, and since the time constants of the resonators are much smaller than that of the laser cavity (a factor of 60 in our experiment) they can be considered instan-
taneously adjusted to the laser field (adiabatic approximation). By including the phases of the circulating fields in the model it is possible to predict phenomena such as mode pulling, mode hopping, field modulations and Q-switching due to path length changes (piezo-mirrors) in both transient and CW-operation. In this publication, the model is used to explain experimental results for a singly resonant, unidirectional ring OPO placed in the circulating field of a unidirectional diode-pumped bow-tie ring laser. This system has been operated successfully both CW and Q-switched. To reiterate: our model is not a numerical solution to a simplified set of equations for the power in the coupled waves, but rather a simulation of the time development of the interacting population inversion in the laser and the circulating fields using the full set of nonlinear equations. This allows us to treat phenomena such as phase modulation in the optical path, phase mismatch, arbitrary phase relations between the fields at the entrance to the nonlinear elements and the effects of coupled cavities with phase modulation.
2. Theory Since we are working experimentally with single-frequency, unidirectional ring laser systems our calculations are based on a single-frequency travelling wave ring laser model. We have further simplified the calculations by assuming plane wave interactions; however, extension to weakly focused Gaussian beams can easily be carried out by the introduction of suitable Boyd–Kleinmann factors [1,8,9]. In the present case, the Rayleigh range of the beams focused into the active components are long enough to allow us to use the plane wave model as a good approximation to the experiment. In the following, all lengths are given as optical path lengths, i.e., the physical length times the index for the particular wave in the material. All wavenumbers are given as the vacuum wavenumber for the particular wave. The laser is described by three coupled rate equations, one for the population inversion, N , one for the amplitude of the circulating field, jaj, and one for the phase of the
P. Buchhave et al. / Optics Communications 216 (2003) 191–197
circulating field, u [10]. The rate equation for the population inversion of a simplified four-level laser is dN N 2 ¼ rp rðmlas Þjaj N ; dt s2
ð1Þ
where rp is the pumping rate (the number of excitations per unit volume per unit time), s2 is the excited state lifetime (due primarily to spontaneous emission from the upper laser level) and rðmlas Þ is the stimulated emission cross section (gain per unit length per incident photon per excited state atom) at the laser frequency, mlas rðmlas Þ includes the Lorentzian gain profile for the laser transition. The electric field in the cavity, Eðz; tÞ, may be written in terms of the complex amplitude, Aðz; tÞ, or the normalized complex amplitude, aðz; tÞ: Eðz; tÞ ¼ Aðz; tÞeiðklas z2pmlas tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2ghmlas aðz; tÞeiðklas z2pmlas tÞ ;
ð2Þ
where klas is the wave vector modulus for the laser field and hmlas is the energy of the laser photon. The impedance ofpthe medium, g, is given by ffiffiffiffiffiffiffiffiffiffiffi g ¼ g0 =n with g0 ¼ l0 =e0 , where g0 is the impedance of the vacuum, e0 is the permittivity of the vacuum and n is the index of refraction of the medium. The absolute square of the normalized complex amplitude, aðz; tÞ, equals the photon flux density, / 2
jaj ¼ / ¼
I ; hmlas
ð3Þ
where I is the circulating intensity (circulating power per unit area) of the laser field. The circulating laser power, Plas , is then given by Plas ¼ Alas hmlas /, where Alas is the cross sectional area of the laser beam in the laser medium. The rate equation for the normalized complex amplitude is djaj 1 llas 1 ¼ c0 rðmlas ÞN jaj; ð4Þ dt 2 le sc where llas is the optical length of the laser crystal, le is the optical cavity length, c0 is the speed of light in vacuo and sc is the cavity lifetime given by 1 1 c0 ¼ Llas ¼ ðLi þ Lout þ Lnl Þ; sc Dt le
ð5Þ
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where Dt ¼ le =c0 is the cavity round trip time and Llas the total power loss for one round trip of the laser field through the cavity. Llas is the sum of internal passive power losses, Li , output coupling, Lout ¼ Tout , where Tout is the power transmission of the output coupler, and nonlinear conversion power losses, Lnl . The rate equation for the phase is [10] du llas ¼ 2pðmc mlas Þ þ 2pðm0 mlas Þs2 c0 rðmas ÞN ; dt le ð6Þ where mc is the cavity resonance frequency and m0 is the laser gain centre frequency. Solution of these equations will give information on the dynamics (and steady-state values) of important laser parameters such as excited state population, circulating laser field amplitude and phase and laser frequency detuning (mode pulling) in the presence of coupled resonators containing nonlinear elements. The conversion between circulating single-frequency fields (e.g., circulating laser field acting as a pump and OPO signal and idler fields) interacting in the nonlinear crystals is governed by the three coupled mode equations (see e.g. [11]): das ðzÞ ¼ igap ðzÞai ðzÞeiDkz ; dz dai ðzÞ ¼ igap ðzÞas ðzÞeiDkz ; dz dap ðzÞ ¼ igas ðzÞai ðzÞeiDkz ; dz
ð7Þ ð8Þ ð9Þ
where j s; i; p represent signal, idler and pump, respectively, Dk ks þ ki kp and g is a gain factor containing i.a. the effective nonlinear coefficient, deff [As/V2 ] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : ð10Þ g ¼ 8p2 hms mi mp g3 deff The classic solution for the fields after travelling the length lOPO through the OPO-crystal is given in [12] in terms of normalized field amplitudes, 2 jaj ðlOPO Þj and an equation for the total phase mismatch h ¼ Dkz þ D/ with D/ /s þ /i /p . This solution includes arbitrary phases between the waves at the entrance to the crystal and a possible wave vector mismatch while travelling through the
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crystal. We have implemented this solution in a mathematics program on a PC (Mathematica) [13], which involved the solution of a third-order equation and looking up the Jacobi sine-functions. However, it turns out that it is faster to solve the three coupled differential equations directly with the build-in differential equation solvers (Dsolve[]). Consequently, the following simulation results have been obtained by including the Dsolve[] call directly in the program loop to obtain values for the complex amplitudes of the circulating fields at the exit of the crystal as a function of the complex amplitudes of the fields at the entrance to the crystal. This solution also includes both arbitrary amplitudes and phases of the fields at the entrance to the crystal and a possible phase mismatch on the way through the crystal. The method covers all standard and advanced forms of conversion such as second harmonic generation (SHG), second harmonic generation with nonzero input, optical parametric amplification (OPA) and sum–difference generation, depending upon the phase relationship of the waves at the entrance to the crystal and the phase matching condition.
3. Computer simulations The computations are carried out as an incremental time-step simulation. For each time step, Dt, the circulating fields are assumed known. Based on the known amplitudes and phases, the passive losses and the nonlinear conversion losses are found. The passive losses were determined by comparing model results to the measurements. The total passive loss used for the laser cavity was 2%. The losses for the OPO were assumed dominated by the mirror leakage as given in Section 4. The conversion loss is found by calculating the change in the complex amplitude of the pump, Dap ¼ a14 a13 , signal, Das ¼ a16 a15 and idler, Dai ¼ a18 a17 (see Fig. 1) by solving the differential equations for the nonlinear interaction. The loss coefficients, Li , Lout and Lnl ¼ ðja14 j ja13 jÞ=ja13 j are used to compute the laser cavity loss rate, 1=sc , according to Eq. (5). The three differential laser rate equations are then used as difference equations to find new values of population inversion and laser
Fig. 1. Illustrating the calculation of new values of the circulating fields after a time increment. a1 is input to the nonlinear crystal from the laser, a7 is the return to the laser, a13 , a15 , a17 are pump, signal and idler input to nonlinear crystal and a14 , a16 , a18 are the corresponding outputs from the nonlinear crystal.
field. With the new laser field, a01 (including a possible feedback of the fundamental from the coupled cavity resulting from less than perfect mirror coatings), and the values of the output from the nonlinear crystal, we calculate the new inputs to the nonlinear crystal. The formulas for the new input fields to the nonlinear crystal are: a013 ¼
a01 t1;p eikp z1;j þ Dap rtot;p eikp ðztot;p lOPO Þ ; 1 rtot;p eikp ztot;p
ð11Þ
a015 ¼
Das rtot;s eiks ðztot;s lOPO Þ ; 1 rtot;s eiks ztot;s
ð12Þ
a017 ¼
Dai rtot;i eiki ðztot;i lOPO Þ ; 1 rtot;i eiki ztot;i
ð13Þ
where t1;j is the amplitude transmittance of the input mirror to the coupled cavity, z1;j is the optical distance from mirror M1 to the crystal, rtot;j is the total amplitude reflectivity of the feedback loop and ztot;j is the total optical path length of the loop. These formulas assume a smaller time constant for the coupled cavity than for the laser cavity (adiabatic approximation for the coupled cavities – a condition which is readily fulfilled for the relatively small cavity lengths and low cavity Q-values used in the experiments). As shown in Eq. (5), sc includes loss contributions from passive losses in the resonators, output coupling and nonlinear conversion. Thus, for each time increment, the passive losses and the output coupling must be recalculated and the nonlinear loss computed by solving the differential equations. If the generated field is further used as a pump in a second coupled resonator, the output fields of the first nonlinear converter must be input to the second nonlinear converter, and all the
P. Buchhave et al. / Optics Communications 216 (2003) 191–197
Fig. 2. Unidirectional ring laser coupled to a triangular ring OPO-cavity.
differential equations must be solved together (see [5] for a solution to the steady-state problem of a cascaded SHG and OPO).
195
The method as described above includes all the nonlinearities of the problem without any linearization of either laser rate equations or nonlinear conversions. However, the finite time step of course represents an approximation, which has to be reduced to acceptable levels by using sufficiently small values of Dt. How small the value must be depends on the dynamics of the problem. High frequency relaxation oscillations of course require small values of Dt, while quasi-steady-state problems can be solved with larger values of Dt. The value must be chosen by estimation or trial-and-error in each case. It is characteristic of this method that it simulates the actual physical processes very closely. In a sense it is just as difficult to make the program work as it is to run a real physical setup. If the initial fields and the physical constants are not given physically realistic values, the program will not run, but crash after a few iterations.
Fig. 3. (a) Model results for population inversion, N, amplitude of circulating laser field, ja1 j, and circulating OPO signal field, jas j, as a function of time at a pump rate, rp ¼ 8 W. (b) Model results for population inversion, N, laser amplitude, ja1 j, and OPO signal, jas j, as a function of time at a pump rate, rp ¼ 10 W.
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We now report the use of this method on the problem of an intracavity, singly resonant OPO as described above, and using the experimental parameters described in the following section.
4. Experiment The experimental setup is shown in Fig. 2. A unidirectional ring laser, a Nd:YVO4 bow-tie laser defined by the four mirrors, M1(ROC ¼ 1, R ¼ 99.9% at 1064 nm), M2 (ROC ¼ 1, R ¼ 99.9% at 1064 nm), M3 (ROC ¼ )100 mm, R ¼ 99.9% at 1064 nm) and M4 (ROC ¼ )150 mm, R ¼ 99.9% at 1064 nm), is coupled to a triangular ring resonator, defined by the three mirrors M5 (ROC ¼ )75 mm, R ¼ 99.9% at 2400–2700 nm), M6 (ROC ¼ )75 mm, R ¼ 99.9% at 2400–2700 nm) and M7 (ROC ¼ 1, R ¼ 99.9% at 2400–2700 nm), at the two dichroic beamsplitters, BS1 (ROC ¼ 1, R ¼ 98.5% at 1064 nm, T ¼ 99:4% at 1400–10,000 nm at 54°) and BS2 (ROC ¼ 1, R ¼ 98.5% at 1064 nm, T ¼ 99.4% at 1400–10,000 nm at 54°). The distances d1 , d2 , d3 and d4 are, respectively, 154, 208, 176 and 207 mm. The laser is pumped by a fibre coupled semiconductor diode laser. The OPOcrystal is a 2 cm long, 0.5 mm thick PPLN designed for a signal wavelength of 1.82 lm and an idler wavelength of 2.56 lm (broad band AR). The pump beam waist in the PPLN crystal was 70 lm with a Rayleigh range of 15 mm. The PPLN was temperature stabilized at 150 °C in an electrically controlled oven. The PPLN losses as determined from reflections were about 0.5%. The optical rectification discriminates the two directions of circulation of the laser field by means of the reciprocal polarization rotation in the half wave plate, k=2, and the nonreciprocal polarization rotation in the Faraday crystal, FR, (TGG-crystal) in combination with the polarization discrimination by the beamsplitters. The laser may be Q-switched by insertion of a mechanical chopper or a Cr:YAG saturable absorber.
laser field amplitude, jaj, and normalized OPO signal amplitude, jas j. The calculation simulates Qswitched operation by using a high initial value for the population inversion. The reflectivities of the OPO feedback mirrors were chosen to simulate a singly resonant (signal resonant) OPO, and the feedback for the fundamental and the idler were chosen to be negligible. The laser field shows the initial rise in the circulating laser field due to the Q-switching. After reaching threshold for the OPO, the laser field is strongly depleted due to the onset of the OPO-process. The laser field drops below threshold for the OPO due to the high conversion losses, and the OPO ceases operation. The laser field then builds up again, and with a sufficiently high residual population inversion the OPO may start again as shown in Fig. 3(b). It is noticeable that the
5. Results Fig. 3(a) shows results of calculations of laser parameters: population inversion, N , normalized
Fig. 4. Experimental results for circulating laser power, Plas , at two different pump rates, (a) Pp ¼ 8 W and (b) Pp ¼ 10 W.
P. Buchhave et al. / Optics Communications 216 (2003) 191–197
laser field is very quickly depleted when the OPO exceeds threshold whereas the excited state population remains relatively high. Using even higher laser pump rates it is possible to produce multiple OPO-pulses before the excited state population is reduced to a level below the value needed for the circulating laser field to reach the threshold for the OPO. Fig. 4 shows experimental results obtained with a mechanical Q-switch (a rotating chopper). The figure shows a signal proportional to the circulating laser power at two different pump levels. The laser field was detected by measuring the power leakage through one of the mirrors (M3 in Fig. 2). The peak circulating intensity was determined to be 1:1 105 W=cm2 .
6. Conclusion A computer simulation of the time evolution of the circulating fields in a single-frequency, unidirectional ring laser with an intracavity nonlinear OPO-crystal is developed. The model is based on a computer simulation that keeps all the nonlinearities of the laser and the nonlinear crystal at play in the system. The model is tested on a singly resonant, intracavity OPO and compared to a diodepumped bow-tie unidirectional Nd:YVO4 laser with an intracavity singly resonant PPLN OPO.
197
Very good agreement is found, both for values of laser and OPO thresholds and for the dynamic properties of the system such as OPO-rise time and the time separation between OPO-pulses at high pump levels.
References [1] M.K. Oshman, S.E. Harris, IEEE QE-4 (1968) 491. [2] K. Koch, G.T. Moore, E.C. Cheung, J. Opt. Soc. Am. B 12 (1995) 2268. [3] T. Debuisschert, J. Raffy, J.-P. Pocholle, M. Pauchon, J. Opt. Soc. Am. B 13 (1996) 1569. [4] G.A. Turnbull, M.H. Dunn, M. Ebrahimzadeh, Appl. Phys. B 66 (1998) 701. [5] P. Buchhave, H. Abitan, P. Tidemand-Lichtenberg, Opt. Commun. 200 (2001) 359. [6] G.A. Turnbull, D.J.M. Stothard, M. Ebrahimzadeh, M.H. Dunn, IEEE QE-35 (1999) 1666. [7] A. Dubois, S. Victori, T. Lepine, P. Georges, A. Brun, Appl. Phys. B 67 (1998) 181. [8] G.D. Boyd, D.A. Kleinman, J. Appl. Phys. 39 (1968) 3597. [9] P. Tidemand-Lichtenberg, P. Buchhave, Opt. Commun. 180 (2000) 159. [10] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, 1997. [11] B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, Wiley, New York, 1991. [12] J.A. Armstrong, N. Bloembergen, J. Ducuing, P.S. Pershan, Phys. Rev. 127 (1962) 1918. [13] H. Abitan, Ph.D. Thesis, Technical University of Denmark, 2001.
Modelling a singly resonant, intracavity ring optical parametric oscillator Preben Buchhave, Peter Tidemand-Lichtenberg *, Wei Hou, Ulrik L. Andersen, Haim Abitan The Optics Group, Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denmark Received 7 June 2002; received in revised form 16 September 2002; accepted 28 November 2002
Abstract We study theoretically and experimentally the dynamics of a single-frequency, unidirectional ring laser with an intracavity nonlinear singly resonant OPO-crystal in a coupled resonator. We find for a range of operating conditions good agreement between model results and measurements of the laser and OPO power output and of the temporal development of complex dynamic phenomena such as pulse shape, pulse duration, oscillatory transients and Q-switched operation of the laser. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The interaction of several nonlinear elements within a laser cavity leads to complex dynamic phenomena such as oscillatory transients at the onset of pumping, Q-switching when the cavity losses are modulated and self-oscillation and chaotic behaviour for certain phase angles of the circulating fields. A detailed understanding of the dynamics of the interaction between exited state population and the various circulating optical fields and an ability to predict these phenomena are essential for explaining the rich variety of experimental observations that can be made on such *
Corresponding author. Fax: +45-45-88-16-11. E-mail addresses: [email protected] (P. Buchhave), [email protected] (P. Tidemand-Lichtenberg).
systems. Theoretical tools are also essential in order to be able to design wave conversion devices for the generation of new coherent light wavelengths. The paper by Oshman and Harris [1] presents one of the first thorough analyses of intracavity OPO, but they simplified the coupled nonlinear equations by assuming all fields to be resonant and using a mean field approximation. Koch et al. [2] analysed two cascaded elements in a ring cavity, but did not include the interaction with the laser. They were then able to use the Jacobi elliptical solutions to the nonlinear interaction between the fields. Debuisschert et al. [3] performed a thorough numerical and analytical investigation of the intracavity OPO in the pulsed regime. The difference between their model and ours is that we perform a simulation of the time development of the
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(02)02299-X
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population inversion in the laser crystal and the circulating fields using the basic differential equations keeping all the nonlinear terms, while Debuisschert et al. simplify the equations by assuming single resonance (nondepleted signal resonant), using mean field approximation for both laser and OPO and assuming optimum phase adjustment (D/ ¼ p=2). In our model we can accept arbitrary phase relations between the fields and can handle the full range of interaction between the fields in a nonlinear crystal including depletion and nonlinear phase development. Debuisschert et al. derive coupled equations for the circulating laser power and signal power and an exponential growth for the idler power. Our work simulates the time development of the amplitude and phase of the circulating fields allowing phase mismatch, detunings and depletion, permitting us to investigate the effect of varying phase in the optical path, which can lead to strong modulation, Q-switching and instability (some of these effects are or will be described in other papers). Turnbull et al. [4] extend the analysis of Debuisschert et al. to include multimode pump lasers in continuous wave (CW)operation. The basic approximations are the same as those of Debuisschert et al. In an earlier paper [5] we used a different approach to the solution of the full nonlinear set of equations, but for the steady-state case and using the coupled power equations at optimal phase adjustment. Experimental results relating to the pulsed intracavity singly resonant OPO have been published in Turnbull et al. [6] and in a recent paper by Dubois et al. [7]. We have developed a new computerized model for vð2Þ nonlinear processes in both intracavity and external cavity systems. The model is based on the interaction between the laser population inversion and the circulating intracavity laser field including both amplitude and phase. The circulating laser field is coupled to fields circulating in coupled resonators. Both passive losses and losses resulting from nonlinear interactions in optically nonlinear crystals are accounted for. Seen from the laser, these coupled resonators act as losses, and since the time constants of the resonators are much smaller than that of the laser cavity (a factor of 60 in our experiment) they can be considered instan-
taneously adjusted to the laser field (adiabatic approximation). By including the phases of the circulating fields in the model it is possible to predict phenomena such as mode pulling, mode hopping, field modulations and Q-switching due to path length changes (piezo-mirrors) in both transient and CW-operation. In this publication, the model is used to explain experimental results for a singly resonant, unidirectional ring OPO placed in the circulating field of a unidirectional diode-pumped bow-tie ring laser. This system has been operated successfully both CW and Q-switched. To reiterate: our model is not a numerical solution to a simplified set of equations for the power in the coupled waves, but rather a simulation of the time development of the interacting population inversion in the laser and the circulating fields using the full set of nonlinear equations. This allows us to treat phenomena such as phase modulation in the optical path, phase mismatch, arbitrary phase relations between the fields at the entrance to the nonlinear elements and the effects of coupled cavities with phase modulation.
2. Theory Since we are working experimentally with single-frequency, unidirectional ring laser systems our calculations are based on a single-frequency travelling wave ring laser model. We have further simplified the calculations by assuming plane wave interactions; however, extension to weakly focused Gaussian beams can easily be carried out by the introduction of suitable Boyd–Kleinmann factors [1,8,9]. In the present case, the Rayleigh range of the beams focused into the active components are long enough to allow us to use the plane wave model as a good approximation to the experiment. In the following, all lengths are given as optical path lengths, i.e., the physical length times the index for the particular wave in the material. All wavenumbers are given as the vacuum wavenumber for the particular wave. The laser is described by three coupled rate equations, one for the population inversion, N , one for the amplitude of the circulating field, jaj, and one for the phase of the
P. Buchhave et al. / Optics Communications 216 (2003) 191–197
circulating field, u [10]. The rate equation for the population inversion of a simplified four-level laser is dN N 2 ¼ rp rðmlas Þjaj N ; dt s2
ð1Þ
where rp is the pumping rate (the number of excitations per unit volume per unit time), s2 is the excited state lifetime (due primarily to spontaneous emission from the upper laser level) and rðmlas Þ is the stimulated emission cross section (gain per unit length per incident photon per excited state atom) at the laser frequency, mlas rðmlas Þ includes the Lorentzian gain profile for the laser transition. The electric field in the cavity, Eðz; tÞ, may be written in terms of the complex amplitude, Aðz; tÞ, or the normalized complex amplitude, aðz; tÞ: Eðz; tÞ ¼ Aðz; tÞeiðklas z2pmlas tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2ghmlas aðz; tÞeiðklas z2pmlas tÞ ;
ð2Þ
where klas is the wave vector modulus for the laser field and hmlas is the energy of the laser photon. The impedance ofpthe medium, g, is given by ffiffiffiffiffiffiffiffiffiffiffi g ¼ g0 =n with g0 ¼ l0 =e0 , where g0 is the impedance of the vacuum, e0 is the permittivity of the vacuum and n is the index of refraction of the medium. The absolute square of the normalized complex amplitude, aðz; tÞ, equals the photon flux density, / 2
jaj ¼ / ¼
I ; hmlas
ð3Þ
where I is the circulating intensity (circulating power per unit area) of the laser field. The circulating laser power, Plas , is then given by Plas ¼ Alas hmlas /, where Alas is the cross sectional area of the laser beam in the laser medium. The rate equation for the normalized complex amplitude is djaj 1 llas 1 ¼ c0 rðmlas ÞN jaj; ð4Þ dt 2 le sc where llas is the optical length of the laser crystal, le is the optical cavity length, c0 is the speed of light in vacuo and sc is the cavity lifetime given by 1 1 c0 ¼ Llas ¼ ðLi þ Lout þ Lnl Þ; sc Dt le
ð5Þ
193
where Dt ¼ le =c0 is the cavity round trip time and Llas the total power loss for one round trip of the laser field through the cavity. Llas is the sum of internal passive power losses, Li , output coupling, Lout ¼ Tout , where Tout is the power transmission of the output coupler, and nonlinear conversion power losses, Lnl . The rate equation for the phase is [10] du llas ¼ 2pðmc mlas Þ þ 2pðm0 mlas Þs2 c0 rðmas ÞN ; dt le ð6Þ where mc is the cavity resonance frequency and m0 is the laser gain centre frequency. Solution of these equations will give information on the dynamics (and steady-state values) of important laser parameters such as excited state population, circulating laser field amplitude and phase and laser frequency detuning (mode pulling) in the presence of coupled resonators containing nonlinear elements. The conversion between circulating single-frequency fields (e.g., circulating laser field acting as a pump and OPO signal and idler fields) interacting in the nonlinear crystals is governed by the three coupled mode equations (see e.g. [11]): das ðzÞ ¼ igap ðzÞai ðzÞeiDkz ; dz dai ðzÞ ¼ igap ðzÞas ðzÞeiDkz ; dz dap ðzÞ ¼ igas ðzÞai ðzÞeiDkz ; dz
ð7Þ ð8Þ ð9Þ
where j s; i; p represent signal, idler and pump, respectively, Dk ks þ ki kp and g is a gain factor containing i.a. the effective nonlinear coefficient, deff [As/V2 ] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : ð10Þ g ¼ 8p2 hms mi mp g3 deff The classic solution for the fields after travelling the length lOPO through the OPO-crystal is given in [12] in terms of normalized field amplitudes, 2 jaj ðlOPO Þj and an equation for the total phase mismatch h ¼ Dkz þ D/ with D/ /s þ /i /p . This solution includes arbitrary phases between the waves at the entrance to the crystal and a possible wave vector mismatch while travelling through the
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crystal. We have implemented this solution in a mathematics program on a PC (Mathematica) [13], which involved the solution of a third-order equation and looking up the Jacobi sine-functions. However, it turns out that it is faster to solve the three coupled differential equations directly with the build-in differential equation solvers (Dsolve[]). Consequently, the following simulation results have been obtained by including the Dsolve[] call directly in the program loop to obtain values for the complex amplitudes of the circulating fields at the exit of the crystal as a function of the complex amplitudes of the fields at the entrance to the crystal. This solution also includes both arbitrary amplitudes and phases of the fields at the entrance to the crystal and a possible phase mismatch on the way through the crystal. The method covers all standard and advanced forms of conversion such as second harmonic generation (SHG), second harmonic generation with nonzero input, optical parametric amplification (OPA) and sum–difference generation, depending upon the phase relationship of the waves at the entrance to the crystal and the phase matching condition.
3. Computer simulations The computations are carried out as an incremental time-step simulation. For each time step, Dt, the circulating fields are assumed known. Based on the known amplitudes and phases, the passive losses and the nonlinear conversion losses are found. The passive losses were determined by comparing model results to the measurements. The total passive loss used for the laser cavity was 2%. The losses for the OPO were assumed dominated by the mirror leakage as given in Section 4. The conversion loss is found by calculating the change in the complex amplitude of the pump, Dap ¼ a14 a13 , signal, Das ¼ a16 a15 and idler, Dai ¼ a18 a17 (see Fig. 1) by solving the differential equations for the nonlinear interaction. The loss coefficients, Li , Lout and Lnl ¼ ðja14 j ja13 jÞ=ja13 j are used to compute the laser cavity loss rate, 1=sc , according to Eq. (5). The three differential laser rate equations are then used as difference equations to find new values of population inversion and laser
Fig. 1. Illustrating the calculation of new values of the circulating fields after a time increment. a1 is input to the nonlinear crystal from the laser, a7 is the return to the laser, a13 , a15 , a17 are pump, signal and idler input to nonlinear crystal and a14 , a16 , a18 are the corresponding outputs from the nonlinear crystal.
field. With the new laser field, a01 (including a possible feedback of the fundamental from the coupled cavity resulting from less than perfect mirror coatings), and the values of the output from the nonlinear crystal, we calculate the new inputs to the nonlinear crystal. The formulas for the new input fields to the nonlinear crystal are: a013 ¼
a01 t1;p eikp z1;j þ Dap rtot;p eikp ðztot;p lOPO Þ ; 1 rtot;p eikp ztot;p
ð11Þ
a015 ¼
Das rtot;s eiks ðztot;s lOPO Þ ; 1 rtot;s eiks ztot;s
ð12Þ
a017 ¼
Dai rtot;i eiki ðztot;i lOPO Þ ; 1 rtot;i eiki ztot;i
ð13Þ
where t1;j is the amplitude transmittance of the input mirror to the coupled cavity, z1;j is the optical distance from mirror M1 to the crystal, rtot;j is the total amplitude reflectivity of the feedback loop and ztot;j is the total optical path length of the loop. These formulas assume a smaller time constant for the coupled cavity than for the laser cavity (adiabatic approximation for the coupled cavities – a condition which is readily fulfilled for the relatively small cavity lengths and low cavity Q-values used in the experiments). As shown in Eq. (5), sc includes loss contributions from passive losses in the resonators, output coupling and nonlinear conversion. Thus, for each time increment, the passive losses and the output coupling must be recalculated and the nonlinear loss computed by solving the differential equations. If the generated field is further used as a pump in a second coupled resonator, the output fields of the first nonlinear converter must be input to the second nonlinear converter, and all the
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Fig. 2. Unidirectional ring laser coupled to a triangular ring OPO-cavity.
differential equations must be solved together (see [5] for a solution to the steady-state problem of a cascaded SHG and OPO).
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The method as described above includes all the nonlinearities of the problem without any linearization of either laser rate equations or nonlinear conversions. However, the finite time step of course represents an approximation, which has to be reduced to acceptable levels by using sufficiently small values of Dt. How small the value must be depends on the dynamics of the problem. High frequency relaxation oscillations of course require small values of Dt, while quasi-steady-state problems can be solved with larger values of Dt. The value must be chosen by estimation or trial-and-error in each case. It is characteristic of this method that it simulates the actual physical processes very closely. In a sense it is just as difficult to make the program work as it is to run a real physical setup. If the initial fields and the physical constants are not given physically realistic values, the program will not run, but crash after a few iterations.
Fig. 3. (a) Model results for population inversion, N, amplitude of circulating laser field, ja1 j, and circulating OPO signal field, jas j, as a function of time at a pump rate, rp ¼ 8 W. (b) Model results for population inversion, N, laser amplitude, ja1 j, and OPO signal, jas j, as a function of time at a pump rate, rp ¼ 10 W.
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We now report the use of this method on the problem of an intracavity, singly resonant OPO as described above, and using the experimental parameters described in the following section.
4. Experiment The experimental setup is shown in Fig. 2. A unidirectional ring laser, a Nd:YVO4 bow-tie laser defined by the four mirrors, M1(ROC ¼ 1, R ¼ 99.9% at 1064 nm), M2 (ROC ¼ 1, R ¼ 99.9% at 1064 nm), M3 (ROC ¼ )100 mm, R ¼ 99.9% at 1064 nm) and M4 (ROC ¼ )150 mm, R ¼ 99.9% at 1064 nm), is coupled to a triangular ring resonator, defined by the three mirrors M5 (ROC ¼ )75 mm, R ¼ 99.9% at 2400–2700 nm), M6 (ROC ¼ )75 mm, R ¼ 99.9% at 2400–2700 nm) and M7 (ROC ¼ 1, R ¼ 99.9% at 2400–2700 nm), at the two dichroic beamsplitters, BS1 (ROC ¼ 1, R ¼ 98.5% at 1064 nm, T ¼ 99:4% at 1400–10,000 nm at 54°) and BS2 (ROC ¼ 1, R ¼ 98.5% at 1064 nm, T ¼ 99.4% at 1400–10,000 nm at 54°). The distances d1 , d2 , d3 and d4 are, respectively, 154, 208, 176 and 207 mm. The laser is pumped by a fibre coupled semiconductor diode laser. The OPOcrystal is a 2 cm long, 0.5 mm thick PPLN designed for a signal wavelength of 1.82 lm and an idler wavelength of 2.56 lm (broad band AR). The pump beam waist in the PPLN crystal was 70 lm with a Rayleigh range of 15 mm. The PPLN was temperature stabilized at 150 °C in an electrically controlled oven. The PPLN losses as determined from reflections were about 0.5%. The optical rectification discriminates the two directions of circulation of the laser field by means of the reciprocal polarization rotation in the half wave plate, k=2, and the nonreciprocal polarization rotation in the Faraday crystal, FR, (TGG-crystal) in combination with the polarization discrimination by the beamsplitters. The laser may be Q-switched by insertion of a mechanical chopper or a Cr:YAG saturable absorber.
laser field amplitude, jaj, and normalized OPO signal amplitude, jas j. The calculation simulates Qswitched operation by using a high initial value for the population inversion. The reflectivities of the OPO feedback mirrors were chosen to simulate a singly resonant (signal resonant) OPO, and the feedback for the fundamental and the idler were chosen to be negligible. The laser field shows the initial rise in the circulating laser field due to the Q-switching. After reaching threshold for the OPO, the laser field is strongly depleted due to the onset of the OPO-process. The laser field drops below threshold for the OPO due to the high conversion losses, and the OPO ceases operation. The laser field then builds up again, and with a sufficiently high residual population inversion the OPO may start again as shown in Fig. 3(b). It is noticeable that the
5. Results Fig. 3(a) shows results of calculations of laser parameters: population inversion, N , normalized
Fig. 4. Experimental results for circulating laser power, Plas , at two different pump rates, (a) Pp ¼ 8 W and (b) Pp ¼ 10 W.
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laser field is very quickly depleted when the OPO exceeds threshold whereas the excited state population remains relatively high. Using even higher laser pump rates it is possible to produce multiple OPO-pulses before the excited state population is reduced to a level below the value needed for the circulating laser field to reach the threshold for the OPO. Fig. 4 shows experimental results obtained with a mechanical Q-switch (a rotating chopper). The figure shows a signal proportional to the circulating laser power at two different pump levels. The laser field was detected by measuring the power leakage through one of the mirrors (M3 in Fig. 2). The peak circulating intensity was determined to be 1:1 105 W=cm2 .
6. Conclusion A computer simulation of the time evolution of the circulating fields in a single-frequency, unidirectional ring laser with an intracavity nonlinear OPO-crystal is developed. The model is based on a computer simulation that keeps all the nonlinearities of the laser and the nonlinear crystal at play in the system. The model is tested on a singly resonant, intracavity OPO and compared to a diodepumped bow-tie unidirectional Nd:YVO4 laser with an intracavity singly resonant PPLN OPO.
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Very good agreement is found, both for values of laser and OPO thresholds and for the dynamic properties of the system such as OPO-rise time and the time separation between OPO-pulses at high pump levels.
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