New Physics: Sae Mulli, Vol. 65, No. 12, December 2015, pp. 1234 1240 DOI: 10.3938/NPSM.65.1234 Optimizations of the Thickness and the Operating Temperature of LiB 3 O 5, BaB 2 O 4, and KTiOPO 4 Crystals for Second Harmonic Generation Doo Jae Park Department of Physics, Hallym University, Chuncheon 24252, Korea Hong Chu Laseroptek Co. LTD, Sungnam 13212, Korea Won Bae Cho BioMed Research Section, Electronics and Telecommunications Research Institute (ETRI), Daejeon 34129, Korea Soo Bong Choi Department of Physics, Incheon National University, Incheon 22012, Korea (Received 5 August 2015 : revised 27 August 2015 : accepted 27 August 2015) We demonstrate a theoretical study for determining the optimal crystal thickness and operating temperature when generating a second harmonic of a pulse laser with pulsewidths ranging from a few tens of femtosecond to a few nanosecond by using commonly-used nonlinear crystals of lithium barium borate, beta barium borate, and potassium titanyl phosphate. The optimal thicknesses of those crystals to avoid any pump pulse depletion for a fundamental-mode laser pulse having a high intensity and a long pulsewidth was calculated as a function of the intensity and the pulsewidth of the fundamental mode. Also, for short-pulse operation, thickness limits are calculated for conditions under which no pulse dispersion due to group velocity mismatch is observed. Finally, the fluctuation of second-harmonic yield due to temperature variations which introduce a refractive-index change is calculated, and the effective temperature ranges are demonstrated for room-temperature operation. PACS numbers: 42.65.Ky, 42.65.Re, 42.79.Nv Keywords: Harmonic generation, Ultrafast process, Optical frequency converter I. INTRODUCTION A virtue of pulse laser is its high peak intensity which is sufficient to introduce various nonlinear optical effect such as second harmonic generation (SHG), sumfrequency generation, difference frequency generation, electro-optic effect and etc. Specifically, SHG enables us to double the frequency of fundamental pulse laser simply introducing a second-harmonic generating nonlinear crystals as Lithium Barium Borate (LBO), beta- Barium Borate (BBO), and Potassium Titanyl Phosphate (KTP), without requiring any complicated opti- E-mail: sbchoi@incheon.ac.kr cal alignments. Considering that wavelengths of frequently used pulse lasers such as Ti:Sapphire laser and q-switched Nd:YAG laser are centered at near-infrared region, SHG provides a handy method to push those to visible frequency region, for application in biomedical imaging, curing, and therapy. In those biomedical applications, supply of optimal condition for highly efficient and stable generation of SHG in terms of choice of second-harmonic generating crystals and its thickness without disturbing pulseshape of generated second harmonic pulse is important. Additionally, because properties of those nonlinear crystals are highly affected by ambient conditions (especially temperature), it is also critical to suggest an optimal This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Optimizations of the Thickness and the Operating Temperature of Doo Jae Park et al. 1235 Fig. 1. (Color online) Calculated normalized SHG efficiency as a function of thickness for LBO (black curve), KTP (red curve), and BBO (blue curve). operating temperature. Even though there is a lot of experimental, theoretical studies are performed through more than half of a century [1 7], surprisingly a systematic study for giving optimal thickness of SHG crystal for high intensity operation and temperature condition was not given so far. In this report, we demonstrate a theoretical research to give a proper choice of SHG crystals for high intensity operation for medical therapy and surgery, including the thickness. Also, a specification for temperature controller which assures a stable emission of second harmonic when operating SHG apparatus in near room temperature is suggested. II. OPTIMAL THICKNESS DUE TO PUMP LASER DEPLETION In ideal case, while fundamental pulse is injected into SHG crystal, energy carried by pulse is either consumed during second harmonic generation or dissipated by heating or scattering, finally diminished inside crystal with a certain travel length, which is referred to as a nonlinear interaction thickness L NL. Consequently, of a nonlinear crystal L is thicker than L NL, some part of crystal never contributes in SHG. Hence, finding out the optimal thickness based on an exact determination of L NL is critical. It is well known that L NL is dependent on nonlinear conversion efficiency d eff, refractive index for fundamental (n ω ) and index for second harmonic (n 2ω ) Fig. 2. (Color online) Optimal thickness as function of incident pump energy with fixed wavelength of 1064 nm and fixed pulsewidth of 750 ps for LBO (black curve), KTP (red curve), and BBO (blue curve). of crystal, incident fundamental wavelength λ, and incident intensity of fundamental I ω (0). Here, nonlinear interaction thickness can be given as [1] 1 2ε 0 n 2 ω n 2ω cλ L NL = 2, (1) I ω (0) 4πd eff with ε 0 being vacuum permittivity. Because of fundamental depletion, SHG efficiency becomes reduced while fundamental pulse travels inside crystal, so that the total conversion efficiency η 2ω can be given as [1] η 2ω = tan h 2 L. (2) L NL By using this formula and utilizing those optical constants of LBO, BBO, and KTP from literature [3, 8], SHG efficiency can be plotted as a function of crystal thickness as depicted in Fig. 1. In this plot, fundamental (hereafter, pump) laser pulse was assumed to have 1064 nm center wavelength, 750 ps pulsewidth, and 300 mj pulse energy with 5 mm beam diameter, which is a typical value emitted from Nd:YAG q-switched laser. In calculation, all crystals are assumed to have a cut for perfect phase matching condition. As depicted in this plot, SHG efficiencies are saturated at thickness of about 3 7 mm for all crystals, which denotes that thicker crystals, i.e., >10 mm, is not needed to achieve maximal SHG. If we regards an optimal thickness L opt as a thickness having 90% conversion compared to maximum conversion when thickness is infinite, we can find that L opt for LBO, BBO, and KTP is read as 2 mm, 4.5 mm, and 5.5 mm,
1236 New Physics: Sae Mulli, Vol. 65, No. 12, December 2015 Fig. 3. (Color online) Optimal thickness as function of pump puslewidth with fixed wavelength of 1064 nm and fixed pulse energy of 200 mj for LBO (black curve), KTP (red curve), and BBO (blue curve). respectively. It is found that L opt for LBO is remarkably smaller than other crystals, which originates from high nonlinear conversion efficiency of LBO, which suggests that LBO is the best candidates for high intensity SHG. Optimal thicknesses as a function of incident pump energy is plotted in Fig. 2 with fixed wavelength of 1064 nm and pulsewidth of 750 ps. As expected, optimal thickness was larger for smaller pump energy, and becomes smaller while an increase of pump energy. However, in our simulation energy region, optimal thickness was in a few mm range. Pulsewidth dependence was also monitored as plotted in Fig. 3. With increase of pulsewidth, optimal thicknesses are slowly grow, because of the peak intensity decrease. Here, pump wavelength and pulse energy was fixed as 1064 nm and 200 mj, respectively. III. BANDWIDTH LIMIT WITHOUT INTRODUCING SECOND HARMONIC PULSE DISPERSION In wide bandwidth incidence (in other words, short pulsewidth incidence), it is well known that a phase matching should be kept both to achieve maximal conversion efficiency and minimal broadening of SHG pulse [1,3]. For almost every materials, refractive indices varies with wavelength, which introduces inequality of refractive index for pump and SHG. This introduces a phase mismatch between pump and SHG inside the crystal, finally generates destructive interferences between those Fig. 4. (Color online) Bandwidth limit with no pulse broadening as a function of crystal thickness for various pump wavelength of 755 nm (black curve), 800 nm (red curve), and 1064 nm (blue curve) for (a) LBO crystal and (b) BBO crystal. Upper dotted line denotes bandwidth corresponding to 100 fs and bottom dotted line denotes bandwidth corresponding to 10 fs pulsewidth. second harmonic pulses generated at different site of crystal. This results in a decrease of total SHG intensity. Additionally, such mismatch of refractive indices also introduce group velocity mismatch (GVM) between pump pulse and second harmonic pulse, which introduces a delayed generation of second harmonic and finally cause a second harmonic pulse broadening. Hence, it is necessary to match refractive indices at the pump wavelength and second harmonic wavelength, by using a birefringence nature of crystal and corresponding cut of crystal. Even with this well-known phase matching techniques, pulse dispersion still occurs in wide bandwidth operation, because refractive index difference in spectral band of pump pulse and that in second harmonic pulse prevents a perfect phase matching, resulting in introducing a second harmonic pulse dispersion. Hence, the only so-
Optimizations of the Thickness and the Operating Temperature of Doo Jae Park et al. 1237 lution to avoid such dispersion is to use thinner crystals with sacrificing SHG intensity. An amount of group velocity mismatch inside crystal in phase-matched condition is given as follows: GV M = c λ (n(λ) 1 n(λ/2)). (3) 2 Applying this relation to broadband laser pulse, bandwidth limit for a crystal having thickness of L can be given as follows: 0.44λ/L δλ F W HM = n(λ) 1 (4) 2n(λ/2), with an assumptions of Fourier-limited pulse and Gaussian distribution of pulse spectrum [1]. Based on this, bandwidth limit in various pump wavelengths for LBO and BBO crystals are plotted in Fig. 4. refractive indices, In obtaining a Sellmeier equation was utilized for ordinary axis. A calculation for KTP was omitted because phase matching condition cannot be found for this crystal due to inappropriate index contrasts between ordinary axis and extraordinary axis in near-ir region. In figures, upper dotted line near 100 nm denotes bandwidth of Fourier-limited pulse corresponding to 10 fs pulsewidth, and lower dotted line near 10 nm denotes 100 fs pulsewidth. As can be seen in figure, bandwidth limits for different wavelengths rapidly decrease with increasing thickness of crystal, which denotes that pulse broadening is highly sensitive to the thickness of crystal. Specifically, for short pulse operation reaching 10 fs, crystal thickness should be relatively small as 40 µm, 50 µm, and 200 µm for 755 nm, 800 nm, and 1064 nm pump wavelength, respectively for LBO, which suggests quite a lot of SHG efficiency should be sacrificed to achieve short pulse SHG, regarding the optimal thickness for high-intensity operation obtained from previous chapter. However, when pulsewidth exceeds 100 fs (<10 nm bandwidth), crystal thickness without pulse broadening is larger than 500 µm which becomes much closer to the optimal thickness without pump depletion, which denotes that such pulse dispersion phenomena is not a major concern in long pulse, high intensity SHG. In case of BBO, crystal thicknesses for 10 fs operation without pulsewidth broadening were obtained as shorter than LBO as 25 µm, 32 µm, and 150 µm for 755 nm, 800 nm, and 1064 nm operation. This comparison suggests that LBO is better choice for SHG for short pulse operation, compared to BBO. IV. TEMPERATURE DEPENDENT YIELD FLUCTUATION Since GVM and corresponding phase matching condition critically determined by refractive indices of SHG crystals, a small changes of indices due to variation of ambient condition such as vapor pressure and temperature induces a considerable fluctuation of SHG yields. For example, index variation n(x,y,z) for crystal axes x, y, and z with temperature variation T is expressed as follows for LBO [5]: n x = ( 3.76λ + 2.30) 10 6 ( T + 29.13 10 3 ( T ) 2 ) (5) n y = (6.01λ 9.70) 10 6 ( T 32.89 10 4 ( T ) 2 ) (6) n z = (1.50λ 9.70) 10 6 ( T 74.49 10 3 ( T ) 2 ) (7) When applying this relation, it is found that the refractive index for crystal axis x varies from 1.55 to 1.57 in temperature range of 0 to 300 degree Celcius at a fixed wavelength of 1000 nm. This rather small changes of refractive index introduces variation of phase matching angle as amount of 0.5 degree, as depicted in Fig. 5. Such variation of phase matching angle introduces a wavevector mismatch between fundamental and second harmonic, even a crystal having perfect phase matching condition at fixed temperature. Traditionally to minimize the amount of such wavevector mismatch due to temperature fluctuation, phase matching cut of LBO
1238 New Physics: Sae Mulli, Vol. 65, No. 12, December 2015 Fig. 5. (Color online) Bandwidth limit with no pulse broadening as a function of crystal thickness for various pump wavelength of 755 nm (black curve), 800 nm (red curve), and 1064 nm (blue curve) for (a) LBO crystal and (b) BBO crystal. Upper dotted line denotes bandwidth corresponding to 100 fs and bottom dotted line denotes bandwidth corresponding to 10 fs pulsewidth. crystal is given at a certain temperature where refractive index variation is minimized. This temperature is found as 104. 5 C, 141 C, and 148 C at wavelength of 1064 nm, 800 nm, and 755 nm, respectively, by finding local maximum of Eq. (5). Fig. 6(a) depicts wavevector mismatch of those optimally prepared crystals for different operating wavelength as a function of temperature. As can be seen in this figure, the amount of wavevector mismatch is 30 cm 1 within the temperature span of 200 C. However, those optimal temperature is relatively higher than room temperature, hence necessitates heating device for crystal which may introduce an increase of cost and complexity of SHG apparatus. To exclude such heating device, an operating temperature may be about 40 degree Celcius regarding natural heating of crystal due to pump laser irradiation. To answer such demand, same calculation was performed with assuming that the phase matching condition was met in 40 degree temperature and depicted in Fig. 6(b). Unlike to the case depicted in Fig. 6(a), the amounts of wavevector mismatch are quite different between different operating wavelength having 20 cm 1 to 40 cm 1, which implies SHG yield fluctuation should be larger at this temperature. SHG yields fluctuations due to such wavevector mismatch were calculated and depicted in Fig. 7(a) and (b), Fig. 6. (Color online) (a) Wavevector mismatch as a function of temperature for those crystals optimized for 1064 nm pump wavelength and 104 degree temperature (black curve), 800 nm pump wavelength and 141 degree temperature (red curve), 755 nm pump wavelength and 148 degree temperature (blue curve). (b) Wavevector mismatch as a function of temperature for those crystals optimized in 40 degree Celcius. with fixed crystal thickness of 2 mm. As depicted in Fig. 7(a), SHG yield almost vanishes when temperature changes more than 100 degree from optimal temperature, where wavevector mismatch is about 30 cm 1. However, a clear plateau are observed for every operation wavelength, which denotes that stable operation is available in those optimal temperature region. The span of temperature range T opt allowing 5% yield fluctuation is read as 60 degree Celcius at 1064 nm pump wavelength. However, as depicted in Fig. 7(b), T opt becomes much narrower as 32 C for 1064 nm and 10 C for 755 nm and 800 nm pump wavelength in 40 degree operation temperature. This suggests that quite precise temperature controller is require for stable operation of SHG device when trying to use in room temperature.
Optimizations of the Thickness and the Operating Temperature of Doo Jae Park et al. 1239 Fig. 7. (Color online) (a) SHG yield fluctuation as a function of temperature for those LBO crystals optimized for 1064 nm pump wavelength and 104 degree temperature (black curve), 800 nm pump wavelength and 141 degree temperature (red curve), 755 nm pump wavelength and 148 degree temperature (blue curve). (b) SHG yield fluctuation as a function of temperature for those LBO crystals optimized in 40 degree Celcius. In case of BBO application, scenario becomes quite different because of different response of refractive index to temperature variation [2]. In BBO, refractive index change as a function of temperature variation is given as a linear function in the temperature range from -10 to 300 degree, hence no optimal temperature region is available. Consequently, wavevector mismatch is also linearly dependent to the temperature for every observed operating wavelength, as can be seen Fig. 8(a). The only difference is its slope, which is smallest for 1064 nm wavelength. This implies that the stable operation may possible for 1064 nm pump wavelength. Fig. 8(b) depicts SHG yield as a function of temperature when phase matching condition is fixed as 40 degree operation. Here, it is clearly seen that T opt is largest having 50 C Fig. 8. (Color online) (a) Wavevector mismatch as a function of temperature for BBO crystals optimized in 40 degree Celcius. (b) SHG yield fluctuation as a function of temperature for BBO crystals optimized in 40 degree Celcius. for 1064 nm pump incident as expected, while halved for other wavelength i.e., T opt 25 C. However, compare to the case of LBO application at 40 C operation, T opt is larger for all operating wavelength, which suggests that BBO is better choice for stable operation than LBO in room temperature operation. V. CONCLUSION In conclusion, we have demonstrated a series of calculation to supply optimal choice of crystal and optimal condition in second harmonic generation. We found that optimal thickness which is given from pump depletion is order of few millimeter, and most efficient crystal was found as LBO. Also, we found that the maximum thickness of SHG crystals without introducing dispersion was
1240 New Physics: Sae Mulli, Vol. 65, No. 12, December 2015 order of few tens of microns in 10 fs operation condition, however, this thickness becomes acceptably larger having few millimeter when pulsewidth becomes larger than 100 fs. Finally, for stable frequency doubled-pulse generation free from temperature fluctuation BBO which has relatively big T opt can be the best choice in room temperature operation. We believe our study helps in optimizing SHG devices for various applications. ACKNOWLEDGEMENTS This research is supported by the Industrial Strategic technology development program (No. 10048690) funded By the Ministry of Trade, industry & Energy (MI, Korea). REFERENCES [1] R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, Inc., 2003). [2] D. Eimerl, L. Davis, S. Velsko, E. K. Graham and A. Zalkin, J. Appl. Phys. 62, 1968 (1987). [3] R. W. Boyd, Nonlinear Optics (Academic Press, 1992). [4] S. P. Velsko, M. Webb, L. Davis and C. Huang, IEEE J. Quantum Electron. 27, 2182 (1991). [5] K. Kato, IEEE J. Quantum Electron. 30, 2950 (1994). [6] K. Kato and E. Takaoka, Appl. Opt. 41, 5040 (2002). [7] D. N. Nikogosyan, Appl. Phys. A 58, 181 (1994). [8] R. Eckardt, H. Masuda, Y. X. Fan and R. L. Byer, IEEE J. Quantum Electron. 26, 922 (1990).