THE generation of coherent light sources at frequencies

Transcription

1 IEEE JOURNAL ON SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. XX, NO. Y, MAY Non-critical Phase Matched SHG of a DPSS Nd:YAG Laser in MgO:LiNbO 3 Carsten Langrock, David S. Hum, Eleni Diamanti, Mathieu Charbonneau-Lefort Abstract In this article, we discuss experimental results obtained from a recent experiment performed by a group of aspiring young scientists in the Ginzton Lab at Stanford University. A low-power diode-pumped cw solid-state laser (DPSS) operating at a wavelength of 1.6 µm was frequency doubled inside a MgO:LiNbO 3 crystal by means of temperature controlled non-critical phase matching. The characteristic dependence of the conversion efficiency on the phase mismatch could be shown to agree qualitatively with the underlying theory. Keywords SHG, DPSS, Nd:YAG, MgO:LiNbO 3, phase matching. I. Introduction THE generation of coherent light sources at frequencies covering an ever increasing portion of the electro-magnetic spectrum has been subject to research ever since the advent of the MASER. With the realization of the LASER, it became obvious that a multitude of materials with various atomic or molecular transitions could be used as gain media for the new light source. Only few materials, such as dyes and titanium-doped sapphire, exhibit a broad gain bandwidth, which allows spectral tuning of the laser. Due to the nature of stimulated emission, there is a restriction to how short a wavelength we can generate by means of a gain medium inside a resonant cavity. Even without this natural barrier, there are frequency bands that are either not covered by available laser materials or for which possible gain media cannot fulfill application required power scalability. One solution to the afore mentioned problems is frequency conversion of coherent laser radiation using nonlinear processes in liquids, gases, and crystals. In this article we will concentrate on second harmonic generation (SHG) in non-centrosymmetric piezoelectric crystals. In particular, we will discuss frequency doubling of 1.6 µm radiation produced by a DPSS cw Nd:YAG laser inside a MgO:LiNbO 3 crystal. This technology has been used in commercial products for many years to build DPSS lasers in the green (see for example Coherent s Verdi and Spectra-Physics Millennia). Recently, frequency conversion has experienced increased attention due to its applicability in the field of optical communication. Here, arbitrary and highly efficient shifting of WDM channels by means of passive optical components is of utmost importance and can eliminate the need to convert signals back and forth between the optical and electrical domain for signal processing. Even though this is not done by SHG, the underlying theory discussed in this article applies equally well to the nonlinear processes involved in these conversion schemes. In this article, we will establish a simple description of SHG and nonlinear processes in general based on Maxwell s equations using the so-called slowly varying envelope approximation to describe the evolution of the involved fields. We will present experimental results to exemplify the applicability of this theory to frequency doubling in nonlinear crystals. II. Introduction to second harmonic generation In our last article, we already mentioned that an electric field incident onto a medium will produce a frequency and orientation dependent polarization, which in turn will produce electromagnetic radiation. P (ω) = ɛ χ(ω)e(ω) (1) The electric susceptibility tensor χ(ω) will determine the strength and orientation of the generated polarization. Using a simple one dimensional harmonic oscillator model, we can get a rough estimate of the susceptibility for weak interactions in isotropic media. We will quickly go through some of the steps involved in this calculation. The displacement x(t) of the electron charge from the nucleus caused by a time dependent electric field E(t) can be modeled by m d2 x + kx = ee(t) dt2 where k represents the restoring force. Since the harmonic oscillator model is a linear system, an excitation at a frequency ω 1 will clearly cause a displacement which varies in time with the same frequency. We will therefore model the applied field E(t) and the displacement x(t) as follows. E(t) = Re{E(ω) exp(iωt)} x(t) = Re{x(ω) exp(iωt)} With these definitions, we finally arrive at the solution for x(ω). x(ω) = e m E(ω) ω 2 ω2 with ω 2 = k/m The displacement of the electron charge from the nucleus creates a polarization p(ω) = ex(ω). If we want to transform this microscopic polarization into a macroscopic one, we simply have to add up the resulting polarization inside a material with N oscillators per unit

2 12 IEEE JOURNAL ON SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. XX, NO. Y, MAY 22 volume. The macroscopic polarization density is then given by P (ω) = e2 N m E(ω) ω 2 (2) ω2 If we compare Eqn. 2 and Eqn. 1, we can easily identify the term that represents the linear susceptibility in this simple model. Knowing the susceptibility, we can also calculate the frequency dependent refractive index n(ω), which is defined by n 2 (ω) = 1 + χ(ω). Even though this model seems to be fairly crude, numerical values obtained by this formula agree well with experimental results. The model can be extended easily to include multiple resonances of varying strengths. So far, we have only talked about linear systems. It should be clear that second harmonic generation is inherently a nonlinear process, which cannot be described by the harmonic oscillator model. We will have to introduce anharmonicity into the model. This can be done by adding a nonlinear restoring force. Starting from the harmonic model mentioned above, the second simplest model can be written as follows. m d2 x dt 2 + kx αx2 = ee(t) It should be clear that a restoring force quadratic in the displacement can only be caused by a cubic contribution in the potential, since F = V. This in turn can only happen if the material under study is noncentrosymmetric, which is the case for LiNbO 3. Even without explicitly solving this equation, it should not be surprising that the displacement will include a term with a 2ωt time dependence. This will lead to a nonlinear polarization at 2ω, which will produce electromagnetic radiation at the same frequency. Using the anharmonic oscillator model, we have shown that Eqn. 1 is only a linear approximation to reality. A more complete version of this relation can be written as P = ɛ { χ (1) E + χ (2) :E E + χ (3) ::E :E E +...} In this article, we will only examine the effects of the χ (2) term pertaining to second harmonic generation. Sum and difference frequency generation are also described by this second order susceptibility. To accurately model the effect of this nonlinear polarization, we have to go back to Maxwell s equations. In a uniform and source-free medium, these can be written as E = µ H t H = D t + J + P t = E = B where D represents the real portion of the dielectric response, J contains the contribution of the in-phase conductivity, and P stands for the nonlinear polarization. The scalar wave equation can then be written in the following form; { 2 Ê x Ê y Ê z 2 = µ 2ˆD } t 2 + 2ˆP t 2 + Ĵ (3) t We will model all field quantities like with ˆQ(x, y, z, t) = Re{Q(x, y, z, t) exp(ω t k z)} k 2 = ω2 ɛ(ω ) c 2 = ω2 n2 (ω ) c 2 We will now introduce the slowly varying envelope approximation. This approximation is based on the fact that the envelope quantities Q(x, y, z, t) will vary slowly in time compared to ω and slowly in space compared to k. We can therefore reduces the order of the resulting differential equation by neglecting second derivatives with respect to time and the spatial z component. Relating the current density to the electric field by Ĵ = σê and introducing the following abbreviations α µ σω µ and η 2k ɛ ɛ(ω ) 377 n Ω we arrive at the following slowly varying envelope equation; E z + j ( 2 ) E 2k x E y 2 + αe = jk 2 E jk 2ɛ ɛ(ω ) D k ω ɛ ɛ(ω ) D t jω η 2 P Up to this point, we haven t really talked about dispersion. To simplify the above equation, we will make the dispersion-free approximation that the susceptibility is constant over the frequency range of interest. We can therefore approximate Eqn. 1 by P (t) = ɛ χ(ω)e(t) And similarly all higher order contributions to the polarization. With this approximation, the slowly varying envelope equation becomes E z + j 2k ( 2 E x E y 2 ) +αe+ 1 E c/n t = jω η 2 P (4) A more complete version of Eqn. 4 would include dispersion and Poynting vector walk-off. Since we will not need these terms in our simplified discussion of second harmonic generation, the formula presented here will suffice. We will further compact this expression by neglecting diffraction, loss, and transient effects. After dropping the appropriate terms in Eqn. 4, we arrive at the most basic expression possible that will us allow to discuss nonlinear effects; E z = jω η 2 P (5)

3 SHG OF A DPSS IN MGO:LINBO 3 13 Having gone through the derivation of the slowly varying envelope equation, let us now look at second harmonic generation. In this χ (2) process, the electric fields and polarizations involved will be modeled the following way. E 1 (z, t) = Ẽ1 2 exp i(ω 1t k 1 z) + c.c. E 2 (z, t) = Ẽ2 2 exp i(ω 2t k 2 z) + c.c. P 1 (z, t) = P 1 2 exp i(ω 1t k 1 z) + c.c. P 2 (z, t) = P 2 2 exp i(ω 2t k 2 z) + c.c. To calculate the nonlinear polarization caused by the fundamental electric field E 1, we simply evaluate the afore mentioned relation P (z, t) = ɛ χ (2) E 2 (z, t) ɛ 2d E 2 (z, t) (6) where we have introduced the nonlinear coefficient d by definition. This leads immediately to P (z, t) = 2ɛ d Ẽ2 1 4 exp i(2ω 1t 2k 1 z) + c.c. with = 2ɛ d Ẽ2 1 4 exp i kz exp i(ω 2t k 2 z) + c.c. ω 2 = 2ω 1 and k k 2 2k 1 Comparing this last expression with our definition of P 2 (z, t), we can easily identify the envelope quantity P 2 as P 2 = ɛ d Ẽ2 1 exp i kz (7) We can now directly insert Eqn. 7 into Eqn. 5. This leads to the following differential equation for the generating process; Ẽ2 z = iη 2ω 1 ɛ dẽ2 1 exp i kz (8) After creating an electric field at ω 2 = 2ω 1, we have to examine what kind of polarization will get created by a combination of the fundamental and second harmonic fields. Once again, we will use Eqn. 6 find the polarization; P (z, t) = 2ɛ d Ẽ2Ẽ 1 2 exp i kz exp i(ω 1 t k 1 z) + c.c. Comparing this with our definition of P 1 (z, t), we can identify the envelope quantity P 1 as P 1 = 2dɛ Ẽ 2 Ẽ1 exp i kz (9) Combining Eqn. 9 and Eqn. 5, we arrive at the differential equation describing the reverse generation process; Ẽ1 z = iη 1ω 1 ɛ dẽ2ẽ 1 exp i kz (1) The coupled equations (8) and (1) describe second harmonic generation in the absence of loss and diffraction. They can be solved exactly, but we will only consider the special case where the fundamental is not affected too much by the SHG process. The negligence of pump depletion is equivalent to only considering small conversion efficiency. In most practical cases, this turns out to be a good approximation. We are therefore left with a single differential equation, Eqn. 8, in which Ẽ1 is independent of z. Simple integration over the crystal length L, leads to the following expression for the power density of the second harmonic signal at the exit face of the nonlinear crystal; Ẽ2 2 ( ) kl = η 2 (ω 1 ɛ 2η dẽ2 1 L)2 sinc 2 (11) 2 2 We can clearly see that the phase mismatch k will determine the generated second harmonic power. The optimum conversion could be achieved by eliminating the phase mismatch. How this can be done will be explained next. Let us take a second look at the definition of the phase mismatch. k = k 2 2k 1 = 2πn(2ω 1) λ 2ω1 2 2πn(ω 1) λ ω1 = 4π λ ω1 (n(2ω 1 ) n(ω 1 )) This relation between the phase mismatch and the refractive indices at the fundamental and second harmonic implies that we have to make n(2ω 1 ) = n(ω 1 ) to achieve perfect phase matching. In general, this is not possible, of course, and depends on the material properties of the nonlinear medium. In a uniaxial crystal, there are two axis with different refractive indices, one of which is called the ordinary axis, with the other named extraordinary (or optical) axis. In the case of LiNbO 3, it turns out that n e (2ω 1 ) n o (ω 1 ) for a fundamental wavelength around 1 µm. This, and the fact that the nonlinear coefficient responsible for second harmonic generation in this geometry is non-zero, can be regarded as good luck. Even though the involved d 31 coefficient is not the largest nonlinear coefficient in LiNbO 3, the ability to achieve perfect phase matching allows efficient second harmonic generation in a bulk crystal. We are required to propagate perpendicular to the optical axis with the fundamental being polarized along the ordinary axis to take advantage of the d 31 coefficient and phase matching. Since the refractive indices n e and n o are temperature dependent, we have a convenient way to exactly match them for a relatively wide range of fundamental wavelengths by heating the crystal to the required temperature. This method of phase matching is called non-critical phase matching, since it does not rely on angle tuning; a method that was explained in our previous article on amplitude and phase modulators. It has the advantage of exhibiting an increased range of angular acceptance and

4 14 IEEE JOURNAL ON SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. XX, NO. Y, MAY 22 Parameter n o n e a a a a b b b TABLE I Temperature dependent Sellmeier coefficients for the ordinary and extraordinary refractive index of congruent LiNbO 3. Second Harmonic Power (µw) Experiment Theory zero Poynting vector walk-off compared to critical phase matching. In the case of LiNbO 3, the wavelength and temperature dependence of the refractive index is given by the following empirically determined Sellmeier equation; n 2 = a 1 + a 2 + b 1 F λ 2 (a 3 + b 2 F ) 2 + b 3F + a 4 λ 2 F = (T T )(T + T + 546) with T = 24.5 C The coefficients for congruent LiNbO 3 are listed in Table I. Even though we used MgO:LiNbO 3 in our experiment, these values are a good starting point to determine the temperature dependent phase mismatch. To calculate the refractive index, we have to insert the wavelength expressed in µm and the temperature in C. To calculate the expected conversion efficiency, we can have to evaluate the following expression (applicable only in the low conversion efficiency limit for η < 25%); η = P 2ω = 8π2 d 2 eff P ω L 2 ( ) kl P ω n 2ω n ω ɛ λ 2 sinc 2 (12) ω A eff 2 with d eff effective nonlinear coefficient L interaction length (i.e. length of crystal) A eff effective area of gaussian beam at fundamental frequency This concludes our theoretical treatment of second harmonic generation in a nonlinear crystal. In the next section, we will present experimental data pertaining to SHG in a 6 mm long MgO:LiNbO 3 crystal. III. Experimental Results In our experimental setup, we used a cw Nd:YAG laser operating at 1.6 µm as the source for SHG of 532 nm green light. The output power of the laser was measured to be approximately 3 mw with a beam diameter of 1.6 mm at the focusing optic. Various focal lengths focusing optics were used to maximize the conversion efficiency. The tuning curve shown in Fig. 1 was taken using Temperature ( C) Fig. 1. Temperature tuning curve of a 6 mm long MgO:LiNbO 3 crystal. The theoretical plot has been calculated as explained in the text. The temperature offset of almost 7 C between the experimental data and the theoretical plot has been removed for display purposes. a 88 mm focal length lens, producing a 37 µm spot size inside the crystal. With Eqn. 12, we calculated the second harmonic power assuming d eff = d pm/v. The temperature dependence of the phase mismatch was determined by the above mentioned Sellmeier equation and the coefficients shown in Table I. It has to be mentioned that we calculated a zero mismatch temperature of C, which differs from the experimental results by about 78 C. This serious discrepancy between theory and experiment was caused by a combination of the following factors. First of all, the thermistor connected to the crystal oven did not measure the temperature of the crystal accurately, since it was attached to the outside of the oven. This alone can only account for an error of a few degrees Celsius, though. In a later experiment we discovered that the thermistor did not even measure the temperature of the casing correctly. Using a bi-metal surface thermometer, we could see temperature differences of up to 4 C. Clearly, the themistor did not make good contact with the oven. The Sellmeier coefficients used to calculate the phase mismatch only apply to congruent LiNbO 3 and not MgO:LiNbO 3. Even though these two materials are very similar, a lower phase matching temperature is conceivable. Besides this temperature shift, the measured temperature tuning curve shown in Fig. 1 agrees very well with the theoretical calculation based on the non-depleted pump approximation. A conversion efficiency of.1% is certainly too low to be of any practical significance. To test the theoretical prediction that we can only generate second harmonic radiation efficiently when the fundamental is polarized perpendicularly to the optical axis, we used a half-waveplate to rotate the incident 1.6 µm beam. Since the waveplate rotates the polarization by

5 SHG OF A DPSS IN MGO:LINBO Experiment Theory 3 Second Harmonic Power (µw) 1 5 Maximum SHG Power (µw) Half Waveplate Angle (degree) Fig. 2. Generated second harmonic power as a function of the fundamental s polarization. The predicted dependence is clearly visible. 2θ, we expected to observe a {sin(2θ)} 2 dependence of the generated SHG power. Fig. 2 supports this theoretical prediction nicely. The reason for the SHG power not to go all the way to zero when the incident light was polarized parallel to the optical axis, was the power meter s background noise in the low µw. One could mistake these points for power created by using the much larger d 33 nonlinear coefficient. In bulk MgO:LiNbO 3 phase matching cannot be achieved in this geometry, though, since both fundamental and second harmonic are polarized along the extraordinary axis and will therefore experience different refractive indices. In this case, second harmonic generation would only occur over a coherence length L c = π/ k. At room temperature, this would amount to an effective interaction length of approximately 4 µm. Even though d eff = d 33 is seven times larger than d 31, a reduction of L by a factor of 1 will clearly dominate the conversion efficiency. This quick calculation shows the importance of phase matching. Recently, techniques have been developed to take advantage of the larger d 33 coefficient and yet achieve phase matching. This new technique is called quasi phase matching (QPM) and is based on the ability to effectively flip the sign of the nonlinear coefficient every coherence length. We will discuss this amazing method in more detail in a future article. After all these nearly perfect experimental results, let us present a measurement that did not quite work out as expected. It is immediately obvious that the choice of the focusing optic s focal length will influence the conversion efficiency, since it determines the spot size inside the crystal as well as the effective interaction length. Focusing tightly inside the crystal will increase the power density and therefore the conversion efficiency, but will also lower the effective interaction length due to a decreased Rayleigh range z R = πω 2 /λ. On the other hand, focusing Fig. 3. length Focal Length (mm) Generated second harmonic power as a function of focal more loosely will increase the interaction length, but will decrease the power density. The existence of an optimum spot size as a function of crystal length is therefore immediately believable. As a starting point, choosing the spot size in such a way as to match twice the Rayleigh range to the crystal length, L = 2z R, is certainly a good idea. An exact calculation shows that the optimum conversion efficiency is achieved at L = 5.7z R. A 2% increase in conversion efficiency can be achieved following the calculation as compared to confocal focusing. In our experiment, we tried to optimize the conversion efficiency for four different lenses. The result is shown in Fig. 3. For a crystal length of 6 mm, confocal focussing should have been achieved using a 75 mm focal length lens. For maximum conversion, an even shorter focal length lens of about 45 mm should have been used. The general trend shown in Fig. 3 agrees with the theoretical prediction, but the maximum conversion efficiency does not occur at the predicted point for which we do not have a satisfactory explanation at this point. IV. Conclusions In this article, we have reviewed the theory of second harmonic generation in nonlinear media as well as the concept of phase matching. We have shown the importance of phase matching using a numerical example and experimental data pertaining to SHG in MgO:LiNbO 3. Excellent agreement between theory and experiment could be achieved for most of the measurements, which confirms the applicability of the underlying theoretical considerations. In a future article, we will revisit second harmonic generation in a different nonlinear material as well as introduce the concept of quasi phase matching, which helped to increase the conversion efficiency tremendously in recent waveguide based devices. Increasing the fundamental s power density by using a Q-switched Nd:YAG should

6 16 IEEE JOURNAL ON SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. XX, NO. Y, MAY 22 lead to an increase in conversion efficiency. We will check this intuitively apparent statement in a future experiment using KD P. Acknowledgments The authors would like to acknowledge the help of Karel Urbanek who showed us that nonlinear optics really does work when you are patient. Carsten Langrock (S 2) received the Diploma in physics from the Heinrich-Heine- Universität, Düsseldorf, in 21. He is currently pursuing the Ph.D. degree in electrical engineering at Stanford University, Stanford, CA. Currently, he is a Research Assistant in the Ginzton Laboratory at Stanford University. His research interests include generation and detection of ultrashort optical pulses, nonlinear optics, and terahertz spectroscopy.