## Abstract

Thermal lensing in diode-side-pumped Nd:YAG laser has been measured quantitatively using digital holographic interferometry. A series of holograms, carrying the information of the laser rod under different pump currents, are recorded with a CCD and reconstructed numerically. The optical path difference induced by the thermal lensing and the corresponding evolution process under different currents are obtained accordingly. Further, the thermal lensing diopters, induced aberrations, and its Zernike coefficients are calculated. The proposed method can be applied in the thermal lensing measurement and the optimization design of a laser resonator.

© 2016 Optical Society of America

## 1. Introduction

A diode-pumped solid-state laser (DPSSL) with high efficiency, high output power, good spatial beam quality and stability is highly desired in scientific and industrial applications. Compared with end-pumped one, the diode-side-pumped Nd:YAG laser is more simple, robust, reliable, and usually used to achieve high power output. In diode-side-pumped Nd:YAG laser, the pump power is focused directly into the center Nd:YAG active medium and deposited as heat to generate a temperature gradient along the radial direction. Then a strong thermal loading occurs in the Nd:YAG rod, and this thermal gradient results in a refractive index variation in the material. Meanwhile, the thermal gradient also results in a nonuniform thermal expansion leading to significant stress distribution and refractive index variations due to the photoelastic effect. Consequently, this leads to thermal effects, such as thermal lens, thermally induced birefringence, and spherical aberration in the laser rod. For an induced purely parabolic phase profile, thermal effects result in an aberration-free thermal lens that changes the resonator stability domains and modifies the laser beam characteristics. For a not strictly parabolic phase profile, the lens aberrations lead to losses and degradation in beam quality. In addition, the stress-induced birefringence creates depolarization losses, defocusing, and polarization-dependent astigmatism on the laser beam [1, 2].

Given the importance of thermal lensing, it is extremely significant to get the accurate parameters of the thermal lens used to aid resonator design and optimization. Numerical finite element methods can be applied in the calculation of the thermal effects [3, 4]. Many analytical models for thermal effects inside laser rod have also been proposed [5–7]. However, these studies are all based on ideal theoretical model and cannot accurately reflect the real situation. Due to the complex mechanisms of thermal lensing, experimental determination is often the only accurate method for its measurement. One simple way is to make a probe beam travel through the laser rod and measure the axial shift of the focal point position [8,9]. The main drawback of this method is low accuracy with a relative precision only 20-30%. It’s also very difficult for weak thermal lensing measurement. By monitoring the laser output and recording the thermal lensing effect that causes the instability of the laser cavity, the focal length of thermal lens can also be determined [10]. This method does not enable the aberration measurement, which can be solved by use of Shack–Hartmann wavefront sensor, lateral shearing interferometry or iterative algorithm. The drawback of Shack–Hartmann wavefront sensor is the low cut-off spatial frequency, primarily caused by the two-dimensional sampling of the wavefront performed by the microlens array [11, 12]. Lateral shearing interferometer is simpler to implement and insensitive to vibration. However, its cost is high and the algorithm is complex [13, 14]. Two or more images are requisite in an iterative reconstruction process to iterate the wavefront, which will affect the instantaneity [15].

In this paper, we adopt digital holographic interferometry (DHI) for quantitative measurement of the thermal lensing effect in diode-side-pumped Nd:YAG laser. DHI is a full field, non-destructive, high-resolution and real time technique that captures the quantitative amplitude and phase information of specimens. It can achieve quantitative phase imaging, numerical refocusing, real-time detection, and numerical correction of aberrations. In the thermal lensing measurement, by continuously recording a series of holograms of the crystal rod under different pump currents and numerically reconstructing them, we can obtain the optical path difference (OPD) distributions, which represents the wavefront changes of the object beam induced by thermal lensing effects, and the evolution of the thermal lens with the current changes. Further, we can also obtain the thermal lensing diopters, the induced aberrations, and its Zernike coefficients. The experiment results show the feasibility of DHI in the quantitative measurement of thermal lensing.

## 2. Theory

In DPSSL, the laser diode pump module is the engine of the laser, affecting critical system parameters, such as beam quality, pump efficiency, and harmonic generation efficiency, etc. The thermal lensing effect in DPSSL is determined by the combined effects of the temperature- and stress-dependent variation of the refractive index and the distortion of the end-face curvature of the rod. The expression of the total focal length *f* of thermal lens is given by [1]

*K*is the thermal conductivity,

*A*is the cross sectional rod area,

*P*is the total heat dissipated in the rod,

_{h}*C*is the photo-elastic coefficient,

_{r,φ}*α*is the thermal coefficient of expansion, and

*r*

_{0}and

*l*are the radius and length of the rod, respectively. For the Nd:YAG crystal, temperature-dependent variation of the refractive index constitutes the major contribution of the thermal lensing. The stress-dependent variation of the refractive index modifies the focal length about 20%. The effect of end-face deformation of the rod is less than 6%, which is usually ignored. As a result, the thermal lensing effect influences the distribution of the detection light field.

In this paper, DHI is implemented to achieve quantitative measurement of the thermal lensing effect in the laser rod. Considering the object wave *O*(*x*,*y*) and reference wave *R*(*x*,*y*) interfere on the CCD target plane, the intensity of the interferogram can be given by

*x*,

*y*are the rectangular coordinates on the CCD target plane. This hologram can then be numerically reconstructed by use of convolution method, and the corresponding reconstructed object wave field

*U*(

*ξ*,

*η*) is expressed as

*g*(

*ξ*,

*η*;

*x*,

*y*) is the impulse response function,

*FFT*and

*IFFT*represent the Fourier and inverse Fourier transform operations, respectively, and

*C*(

*x*,

*y*) is the numerical reference wave.

Assuming that we separately record two digital holograms *H*_{1} and *H*_{2} for the laser rod under different currents, and the reconstructed object waves in two different states are *U*_{1}(*ξ*,*η*) and *U*_{2}(*ξ*,*η*). Based on double-exposure holographic interferometry [16–20], the phase change Δ*φ* between *U*_{1}(*ξ*,*η*) and *U*_{2}(*ξ*,*η*) can be calculated as

As we know, ignoring the end-face deformation of the rod, the refractive index change inside the laser rod induced by thermal lensing effects leads to the change of OPD Δ*l*(*x*, *y*) and thereby to an interference phase change between two light waves passing through the rod before and after the change. Then the phase change Δ*φ*(*x*,*y*) due to the index change is given by

*λ*is the wavelength,

*L*is the rod length,

*n*

_{0}is the initial refractive index of the rod under unperturbed state and

*n*(

*x*,

*y*,

*z*) is the final refractive index distribution [22]. The light beam passes through the Nd:YAG rod in

*z*direction and the integration is taken along the propagation direction. Because of the symmetry of the rod and its wraparound sleeve, the refractive index is supposed to be constant along

*z*direction over the entire length of the rod, and the phase map representing the phase change directly reflects the variation of refractive index of the rod.

## 3. Experimental set-up

Figure 1 shows the experimental setup for measuring the thermal lensing effects in diode-side-pumped Nd:YAG laser by use of DHI.

In our DPSSL system, the laser diode pump module consists of a Nd:YAG rod, a cooling sleeve, a diffusive optical reflector chamber, and three diode array modules as illuminated in the bottom left inset of Fig. 1. The crystal rod has a fine ground barrel to minimize the light loss that occur within the round crystal surface. The rod with a diameter of 1.8mm and length of 65mm is cut in Brewster angle (151.2° for 1064nm) and fixed by an antireflection optical glue in the center of the chamber. The pumping diodes are set to make the pumping light incident to barrel surface of the laser rod horizontally with a total power of 105 W at 35 A input current. This ensures good light coupling of the three diodes directly into the laser crystal rod. The design is to reduce the birefringence effect of the Nd:YAG crystal, make additional loss to the vertical polarized light and increase the horizontal polarization ratio of the laser output. The use of elongated laser crystal is to act as an aperture for the laser cavity to suppress the higher mode operation.

In the experiment, the DPSSL is under non-lasing condition driven with different pump currents. A narrow beam from a He-Ne laser with *λ* = 632.8nm is collimated to a plane wave by the collimator. Then the plane wave is divided into two parts by the first beam splitter. One beam is reflected by the mirror and the second beam splitter to form the reference beam. The other beam is reflected by the mirror and travels into the laser rod with an incident angle *i*_{1} = 61.34° as shown in the upper right inset of Fig. 1. This beam travels along the rod and is modulated by the generated thermal lensing effects acting as the object beam. The object beam and the reference beam interfere with each other on the CCD target plane. Since the pump diodes emit 808nm bright light affecting the digital hologram recording, a narrow-band pass filter for 632.8nm is placed in front of the CCD target. To avoid the twin images problem and to eliminate the zero order diffraction, off-axis holograms are recorded with a monochorme CCD camera (The Imaging Source DMK41AU02 with 1280 × 1960 pixels and pixel size of 4.65μm × 4.65μm). The distance between the camera and the end of the laser rod is 185mm. Since the illumination beam is refracted on the Brewster-angled surface of the rod, the round beam is skewed to an elliptical beam.

## 4. Experiment results and discussions

Initially, when the input current of laser diode is 0 A, we record an initial hologram of the measured rod. Then we gradually increase the current to 40 A and record a series of holograms in this process. Figure 2 shows the recorded digital hologram under 0 A current and the corresponding reconstructed phase map (unit: rad), respectively. Because the cross sectional area of the rod is much less than the reference beam, the effective measurement region concentrates on the central part of Fig. 2(b). The elliptical phase region is nearly flat due to the absence of thermal gradient inside the rod.

Based on double-exposure holographic interferometry, each individual hologram is numerically reconstructed separately to obtain a series of sequential phase maps of the object beam for the pump currents from 0 A to 40 A. Figure 3 shows the reconstructed 2D wrapped phase maps (unit: rad) under different currents, in which the effective elliptical measurement region has been extracted. It clearly shows that the wrapped phase fringes become denser with the current increase. The change of the phase fringes is asymmetrical in horizontal and vertical directions, representing an uneven refractive index distribution in the rod. In laser diode pump module, most of the pump light energy is concentrated in the middle horizontal region as the horizontal setting diode, resulting in the heat absorption mainly in this section, and thus a large thermal gradient is generated in the vertical direction leading to asymmetric thermal lensing.

After further phase unwrapping operation, the unwrapped phase map can be obtained and the thermal lensing induced OPD can be calculated according to Eq. (5). This OPD also represents the wavefront variation after the beam passing through the rod. By seriating these maps, the shape and evolution of wavefront (unit: μm) under different pump currents can be obtained. Figure 4 shows this evolution (Visualization 1). It is found that the change of OPD is not obvious when the pump current is less than 8 A, which is mainly due to the minimum required excitation current threshold of the diodes. When the pump current exceeds this threshold value, the output power of laser diodes linearly increases with the increasing current and up to 35 W when the current is equal to 35 A. Under this circumstance, the DPSSL works at a stable state with a good beam quality, pump-to-laser efficiency, and harmonic generation efficiency. With the currents from 8 A to 35 A, the change of OPD gradually increases, but again not obvious for a more than 35 A current due to the heat inside the rod reaching a dynamically balanced state. Although the pump energy gradually increases with the increasing current, the heat absorbed by the rod remains unchanged, as some heat is conducted away by the cooling sleeve. Therefore, the thermal gradient in the rod is almost unaltered and the OPD is also unchanged. As the recording and reconstruction of the digital holograms are almost real time in DHI, the quantitative measurement of thermal lensing is real time too.

The measured 2D phase distribution of wavefront (unit: rad) under 35 A pump current is shown in Fig. 5(a). Compared with the flat phase distribution in Fig. 2(b), significant phase changes occurred in Fig. 5(a), which indicates that the phase is delayed by the refractive index difference introduced by the thermal lensing under 35 A pump current. The wavefront variation in vertical direction is more obvious due to the large thermal gradient. As the crystal rod is cylindrically symmetric, this can be attributed to the highly elongated nature of the pumped region. Since the incident beam is a plane wave, and the rod is equivalent to a single lens, this measured result characterizes the thermal lens’s modulation capabilities to the light. Assuming that the initial refractive index of the Nd:YAG rod is 1.8295 and the refractive index vibrations along axial direction of the rod are homogeneous in the heating process, the maximum refractive index difference introduced by the temperature gradient can be calculated as Δ*n* = 4.79 × 10^{−5} according to Eq. (5). Further according to the thermal lensing theory, we can get the heat transfer parameters in the pumping process.

The amount of thermal lensing caused by the temperature distribution in the crystal rod is an important parameter for the design of optical resonator. In a first approximation, we ignore the surface distortion effect, the thermally induced lens can be described by considering only the temperature-dependent and stress-dependent part of the refractive index, which allows to estimate the deviations of thermal lens from a perfect parabolic lens. Figure 5(b) shows the profile of OPD along the white dash line in Fig. 5(a). By fitting this retrieved wavefront with a parabolic function, we can further obtain the focal length and diopter of the thermal lens.

Thermal lensing data were successfully retrieved for the Nd:YAG rod and the measured thermal lensing diopter as a function of the pump current is shown in Fig. 6. The thermal lens diopters are clearly very different in the horizontal and vertical directions. This can only be attributed to the asymmetric side-pumped mode, as the crystal is cylindrically symmetric. Due to the larger thermal gradient in the vertical direction, the thermal lensing diopter in this direction is greater than that in the horizontal direction. We can find a linear relationship between the thermal lensing diopter and pump power as the linear relationship between the pump power and current, which is consistent with the theory analysis in Ref [1].

It should be noted that the above measurement is for Nd:YAG laser under non-lasing condition, and the two mirrors of the resonant cavity has been removed in the measurement. In fact, it’s also feasible to measure thermal lensing for laser under lasing condition. The output light wavelength of our laser system is 1064nm, and the resonant cavity mirrors are narrow-band laser line mirrors at 1064nm. Hence, we can place the resonant cavity and the crystal rod entirely to the measurement region of the experimental setup. The DHI approach still works for 632.8nm detection light.

For a perfectly parabolic distortion of the wavefront or equivalently a pure thermal lens, the thermal distortion can be easily compensated by addition in the laser cavity of the opposite divergent lens or by adjusting the distance of the different cavity elements. However, the aberrations are present when the wave front distortions are not perfectly parabolic. The thermal lens deviates strongly from being parabolic in the edge region, and is thus highly aberrated in Fig. 5(b). Thus, the compensation becomes very difficult and requires complex systems. While uncorrected, these aberrations may lead to degradation in the beam quality, and also may incur losses due to diffraction of the beam high spatial frequencies.

As we have obtained the wavefront introduced by the thermal lensing in DHI, we can further analysis its aberrations by use of Zernike polynomials. Figure 7(a) shows the 2D phase distribution of the wavefront (unit: λ) in the central circular region of Fig. 5(a). After implementing Zernike polynomial fitting, the obtained fitting wavefront and the residual error map are shown in Fig. 7(b) and 7(c), respectively. The corresponding calculated Zernike coefficients are given in Table 1. Thus, we can easily determine the third item coefficient (focus). At the same time, the wavefront also includes other aberrations (coma *x*, trefoil *y* and secondary astigmatism *x*). Some other coefficients can be negligible. Therefore, the aberrations of the thermal lensing can be analyzed systematically by use of DHI, which can be used in the optimization design of laser resonator.

## 5. Conclusions

We have presented using DHI to measure thermal lensing in diode-side-pumped Nd:YAG laser quantitatively. The OPD induced by the thermal lensing and the corresponding evolution process under different currents are obtained. Then the focal length and diopter of the thermal lens are calculated by fitting the retrieved wavefront with a parabolic function, which is consistent with theoretical analysis. The induced aberrations and its Zernike coefficients are also calculated. The experimental results show that DHI is a valuable and important approach in the measurement of thermal lensing.

## Funding

National Natural Science Foundation of China (61405164), Translational Innovation Fund grant MOE2013-TIF-2-G-011 and MOE2013-TIF-2-G-012 from the Singapore Ministry of Education (MOE).

## References and links

**1. **W. Koechner, *Solid-State Laser Engineering* (Springer, 2013), Vol. 1.

**2. **S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. **30**(4), 89–153 (2006). [CrossRef]

**3. **R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. **11**(2-3), 245–254 (1999). [CrossRef]

**4. **M. Schmid, R. Weber, T. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. **36**(5), 620–626 (2000). [CrossRef]

**5. **W. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D Appl. Phys. **34**(16), 2381–2395 (2001). [CrossRef]

**6. **Y. S. Huang, H. L. Tsai, and F. L. Chang, “Thermo-optic effects affecting the high pump power end pumped solid state lasers: Modeling and analysis,” Opt. Commun. **273**(2), 515–525 (2007). [CrossRef]

**7. **S. Chénais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-Part I: theoretical analysis and wavefront measurements,” IEEE J. Quantum Electron. **40**(9), 1217–1234 (2004). [CrossRef]

**8. **D. C. Burnham, “Simple measurement of thermal lensing effects in laser rods,” Appl. Opt. **9**(7), 1727–1728 (1970). [CrossRef] [PubMed]

**9. **C. Hu and J. R. Whinnery, “New thermooptical measurement method and a comparison with other methods,” Appl. Opt. **12**(1), 72–79 (1973). [CrossRef] [PubMed]

**10. **D. Lancaster and J. Dawes, “Thermal-lens measurement of a quasi steady-state repetitively flashlamp-pumped Cr, Tm, Ho: YAG laser,” Opt. Laser Technol. **30**(2), 103–108 (1998). [CrossRef]

**11. **S. Ito, H. Nagaoka, T. Miura, K. Kobayashi, A. Endo, and K. Torizuka, “Measurement of thermal lensing in a power amplifier of a terawatt Ti: sapphire laser,” Appl. Phys. B **74**(4-5), 343–347 (2002). [CrossRef]

**12. **A. Nishimura, K. Akaoka, A. Ohzu, and T. Usami, “Temporal change of thermal lens effects on highly pumped ytterbium glass by wavefront measurement,” J. Nucl. Sci. Technol. **38**(12), 1043–1047 (2001). [CrossRef]

**13. **J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” J. Appl. Phys. **83**(6), 2901–2906 (1998). [CrossRef]

**14. **J. Blows, P. Dekker, P. Wang, J. Dawes, and T. Omatsu, “Thermal lensing measurements and thermal conductivity of Yb: YAB,” Appl. Phys. B **76**(3), 289–292 (2003). [CrossRef]

**15. **L. Grossard, A. Desfarges-Berthelemot, B. Colombeau, and C. Froehly, “Iterative reconstruction of thermally induced phase distortion in a Nd3+: YVO4 laser,” J. Opt. A, Pure Appl. Opt. **4**(1), 1–7 (2001). [CrossRef]

**16. **W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express **17**(22), 20342–20348 (2009). [CrossRef] [PubMed]

**17. **Q. Wang, J. Zhao, X. Jiao, J. Di, and H. Jiang, “Visual and quantitative measurement of the temperature distribution of heat conduction process in glass based on digital holographic interferometry,” J. Appl. Phys. **111**(9), 093111 (2012). [CrossRef]

**18. **J. Zhang, C. Ma, S. Dai, J. Di, Y. Li, T. Xi, and J. Zhao, “Transmission and total internal reflection integrated digital holographic microscopy,” Opt. Lett. **41**(16), 3844–3847 (2016). [CrossRef] [PubMed]

**19. **J. Zhang, J. Di, Y. Li, T. Xi, and J. Zhao, “Dynamical measurement of refractive index distribution using digital holographic interferometry based on total internal reflection,” Opt. Express **23**(21), 27328–27334 (2015). [CrossRef] [PubMed]

**20. **Y. Zhang, J. Zhao, J. Di, H. Jiang, Q. Wang, J. Wang, Y. Guo, and D. Yin, “Real-time monitoring of the solution concentration variation during the crystallization process of protein-lysozyme by using digital holographic interferometry,” Opt. Express **20**(16), 18415–18421 (2012). [CrossRef] [PubMed]

**21. **M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. **50**(33), 6214–6224 (2011). [CrossRef] [PubMed]

**22. **X. Peng, L. Xu, and A. Asundi, “Thermal lensing effects for diode-end-pumped Nd: YVO4 and Nd: YAG lasers,” Opt. Eng. **43**, 2454–2461 (2004). [CrossRef]