Method to improve beam quality by compensating spherical aberrations in master oscillator power amplifier laser systems Zhibin Ye,1 Yi Wang,1 Zhigang Zhao,2 Chong Liu,1,* and Zhen Xiang1 1
2
State Key Laboratory of Modern Optical Instrumentation, Department of Optical Engineering, Zhejiang University, Hangzhou 310027, China
Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan *Corresponding author: [email protected] Received 19 August 2014; revised 25 October 2014; accepted 26 October 2014; posted 28 October 2014 (Doc. ID 221182); published 18 November 2014
A method is presented for beam quality improvement, in master oscillator power amplifier laser systems. Intensive study was first carried out with the beam wavefront evolution in a laser resonator. When the laser beam propagates inside the resonator, the spherical aberration coefficient of the beam wavefront can change sign, i.e., the negative spherical aberration coefficient can turn to positive, and vice versa. This process also occurs when the beam propagates outside the resonator in a free space. The laser beam, from an oscillator with negative spherical aberration, was found to be well-compensated by the positive spherical aberration of a strongly pumped laser rod in a laser amplifier. The laser beam quality M 2 factor has been significantly improved, from 2.2 to 1.4, while the output power has been scaled from 31 W up to 60 W. © 2014 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (140.3280) Laser amplifiers; (140.3410) Laser resonators; (140.3580) Lasers, solid-state. http://dx.doi.org/10.1364/AO.53.007963
1. Introduction
Bulk-crystal based high-power lasers are still widely used, due to their high efficiency, compact package, and high damage threshold [1–3]. A key factor that limits their applications is the beam quality degradation, due to the thermal effects in the gain medium. The gain medium can be described (ideally) as a perfect lens [4]. In practice, the temperature distribution in the cross section of the medium differs from the ideal parabolic profile, because of inhomogeneous pumping, and the temperature dependence of the thermal conductivity and thermo-optical coefficient, which leads to the spherical aberration effect of the thermal lens [5–7]. Spherical aberrations in gain mediums have been proven to be the major 1559-128X/14/337963-05$15.00/0 © 2014 Optical Society of America
reason for beam quality deterioration in laser resonators and amplifiers [5,8–11]. Some techniques have been adopted to improve the beam quality, by compensating spherical aberrations in gain mediums, in the past few years. First, the spherical aberration effect can be weakened, by properly designing the pump light distributions. In 2005, Leibush et al. [12] reported that the spherical aberration was eliminated in multi-kW, Nd:YAG, rod pump-chambers, by optimizing the pump distributions. Second, a relay-imaging telescope, with negative spherical aberration, can be used to compensate the positive spherical aberrations of gain media. Effective compensations were realized both in laser resonators, and in multi-rod amplifier systems [13]. Third, by using aspherical phase-plates, which introduce spherical aberrations with the same absolute value (but opposite sign) as the laser beams, the 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
7963
beam quality can be significantly improved, inside or outside the cavity [14–16]. However, there are still some drawbacks in the three methods mentioned previously. The coupling system, as well as the doping concentration, of the laser crystal should be specially designed in the first method, which is not easy to manage in experiments. Additional optical elements are required in the second and third method. Moreover, aspherical phase-plates are found to be expensive and complicated to fabricate, for large spherical aberrations. In this paper, we show a master oscillator power amplifier (MOPA) system, designed to compensate spherical aberrations without additional optical elements. The beam from the master oscillator has a wavefront with negative spherical aberration. The wavefront is compensated by the positive spherical aberration of the Nd:YVO4 rod in the laser amplifier, which results in a significant improvement in the beam quality after the amplifier. In Section 2, the mathematical relationships between the beam quality factor M 2, and spherical aberration coefficient, are briefly reviewed. In Section 3, experimental measurements are performed for the beam wavefront evolution in the master oscillator. The results show that the beam wavefront aberration changes when the beam passes through the gain medium, which results in the beam quality difference of the intracavity beams. A laser amplifier, which compensates the spherical aberration of the incident beam, is proposed in Section 4, and conclusions are given in Section 5. 2. Brief Review of the Relationship between the Spherical Aberration Coefficient, and the Beam Quality Factor M 2
M2
The beam quality factor is often used to characterize that how well an arbitrary laser beam can be focused [17,18]. For a coherent laser source, the mathematical expression of the M 2 factor is related to the beam intensity distributions, and wavefront aberrations, independently [19] M2
q M 2diff 2 M 2ab 2 ;
(1)
where M 2diff represents the amplitude term, and M 2ab represents the phase term. Taking an aberrated Gaussian beam as an example, the complex amplitude can be expressed as
r2 2π ~ ur exp − 2 j W ; λ ω0
(2)
with an aberrated wavefront of W C0 C2 r2 C4 r4 ;
(3)
where ω0 is the beam radius, λ is the wavelength, and C2 and C4 are coefficients (which represent the focusing term and spherical aberrations term, 7964
APPLIED OPTICS / Vol. 53, No. 33 / 20 November 2014
respectively). The component M 2ab in Eq. (1) is related to the coefficient C4 , according to [19] M 2ab
2π jC jω4 λ 4 0
r 3 : 2
(4)
The beam intensity distribution is Gaussian, and we have the component M 2diff in Eq. (1) of M 2diff 1:
(5)
In our experiments, and also in some literature, the wavefront is expressed by a set of Zernike functions [20] Wρ; θ
N X n1
cn Zn ρ; θ;
(6)
where Zn ρ; θ is the nth Zernike polynomial (with cn being its coefficient), ρ r∕r0 ; r and θ are polar axes, and r0 is the normalized radius (which equals the beam radius p ω0 ). The 13th Zernike polynomial Z13 ρ; θ 56ρ4 − 6ρ2 1, which is related to the spherical aberration, is the main item of the thermally induced aberrations. We only considered the spherical aberration term in the M 2 factor calculap tions, i.e., 6 5c13 ρ4 . From Eq. (3) and the 13th Zernike polynomial expression, we have the relationship between C4 and c13 as p 6 5c13 C4 : ω40
(7)
The contribution of the spherical aberration coefficient c13 to the M 2 factor can be calculated, using Eqs. (1), (4), (5), and (7) M2
q 1 270k2 c213 ;
(8)
where k 2π∕λ is the wavenumber. Eq. (8) describes the relationship between the spherical aberration and M 2 factor, when the beam intensity is a Gaussian distribution. Useful conclusions can be deduced from Eq. (8). The stronger the spherical aberration, the higher the M 2 factor (i.e., the worse beam quality), provided that the beam intensity keeps a Gaussian distribution (M 2diff ≈ 1). This gives a proper explanation of the beam quality degradation in multi-stage MOPA laser systems. Expanding telescopes are always used between the amplifiers, to achieve mode volume matching in the MOPA systems. These expanding telescopes are always relay optics, which image the near field of the laser beam from one stage to the next. This solution has the advantage of keeping the beam intensity as a Gaussian distribution, which is effective for eliminating beam quality degradation, by minimizing the component M 2diff . However, the spherical aberrations in each amplifier
are gradually superposed to the beam wavefront [21], which degrades the beam quality by increasing the component M 2ab . 3. Spherical Aberration Measurements inside the Resonator
An asymmetrical plano–plano dynamically stable resonator (DSR) [22,23] is built up as the master oscillator (shown in the lower part of Fig. 1). Two high-power fiber-coupled diode lasers (from DILAS) are used for pumping. Each diode laser, with 808 nm central wavelength, delivers a maximum output power of 50 W in CW mode. The pump beam is focused into the crystal from two end surfaces, with a beam diameter of 0.8 mm at the incident surface. The two dichroic mirrors (M1 and M2) are high transmittance (HT), coated at 808 nm, and high reflection (HR), coated at 1064 nm, for light at an incidence angle of 45°. The mirror M3 is also high-reflection coated at 1064 nm. The output couple (OC) has a reflectivity of 50%. The laser is operated in fundamental mode. An average power of 31 W is achieved, with a pump power of 75 W. The beam quality is measured for the output beam from the OC, using International Organization for Standardization (ISO) standardized methods [24]. It has an M 2 factor of 2.2. The M 2 factor of the beam that leaks from the HR mirror is also measured, which is 1.3. We exchanged the OC and HR mirror in the experiments. In this condition, the beam from the OC has an M 2 factor of 1.3, while the beam from the HR mirror has an M 2 factor of 2.0. It is an important result that the beam from the long arm shows a better beam quality than that from the short arm [the laser cavity is divided into two parts by the laser medium, and we define the one with longer distance L2 (320 mm) as the long arm, and the shorter one L1 (117 mm) as the short arm]. This phenomenon has been observed in [23]. The beam quality difference of intra-cavity beams can be explained by the evolution of beam wavefront aberrations. The beam intensity and wavefront distribution, in any plane inside the resonator, can be obtained using the method introduced by [23]. The laser fields especially attract attention in the four planes of A1, A2, B1, and B2, shown in Fig. 1. Labels A and B represent the planes in the left and right end of the crystal, respectively. Labels 1 and 2 represent the forward and backward beams, respectively (we
Fig. 1. Configuration of the MOPA system; LD, laser diode; M1, M2, M4, and M5, dichroic mirrors; M3, 45° high-reflection mirror.
define the direction from OC to HR as a forward direction, while the reverse one is the backward direction). The spherical aberration coefficients c13 are measured, for the four fields, by a Hartman–Shack wavefront sensor (WFS 150-5C, Thorlabs), which are −0.017 μm (A1), 0.005 μm (B1), 0.019 μm (A2), and −0.007 μm (B2), respectively. The beam intensity is nearly a Gaussian distribution for all the four fields. Eq. (8) can thus be used to calculate the M 2 factors of the four fields, which are 1.93 (A1), 1.11 (B1), 2.10 (A2), and 1.21 (B2), respectively. It should be noticed that the M 2 factors should be the same for the fields in the planes A1 and A2, because beam quality does not change when a beam propagates in a free space, or is reflected by a planar mirror. The inconsistency of the calculated M 2 factor value, in the planes A1 and A2, is mainly due to the measurement errors of the spherical aberration coefficient. The same result can be concluded for the fields in the planes B1 and B2. In spite of this, these calculated values of M 2 factors agree well with the measured values, which are 2.2 for the fields in planes A1 and A2, and 1.3 for the fields in planes B1 and B2. The measured values are a little bit larger, which is mainly due to the fact that the beam intensity is not an ideal Gaussian distribution in the four planes, and the component M 2diff is larger than 1. The evolution of the beam wavefront aberration can be described as follows, according to the previously mentioned measurements. The spherical aberration coefficient c13 is nearly zero for the field in the plane B2. The value of c13 increases to 0.019 μm when the beam passes through the gain medium to the plane A2, because the medium introduces a positive spherical aberration, with a coefficient of ∼0.02 μm. The value of c13 keeps almost the same absolute value, but changes the sign, when the beam propagates from the plane A2 to A1. Then the beam wavefront, with negative c13 in the plane A1, is compensated by the medium with positive spherical aberration. The value of c13 becomes nearly zero again for the field in the plane B1. There are two conclusions that should be emphasized, especially for the resonator. First, the value of c13 can be positive or negative for the intra-cavity beams, even if the gain medium always introduces positive spherical aberration. It can even flip the sign when the laser beam propagates a distance. The beam in the short arm acts as an example. The spherical aberration coefficient flips its sign, while it keeps almost the same absolute value when the beam propagates a distance of 2L1, from the plane A2 to the plane A1. Second, the beam wavefront with a negative c13 can be compensated by the laser medium. The value of c13 changes from −0.017 to 0.005 μm when the forward beam passes through the medium, from the plane A1 to the plane B1, which leads to the beam quality improvement. These two conclusions support the design of the laser amplifier, for beam quality improvement in the next section. 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
7965
4. Spherical Aberration Compensation in Laser Amplifiers
The schematic of the MOPA system is shown in Fig. 1. The amplifier (in the upper part of Fig. 1) has the same pumping parameters, and laser medium, as the master oscillator, which ensure the identical spherical aberration effect of the two crystals. No isolator and no coupling systems are used, between the amplifier and the oscillator. The laser beam from the OC has a large divergence angle of about 10 mrad [23]. It propagates a distance of L1 to the incident surface of the gain medium of the amplifier (the plane A1’ in Fig. 1), where L1 is the short arm length of the master oscillator. In this way, the mode volume matching, between the pump volume and laser mode in the amplifier, are realized. Another advantage of this configuration is that the amplitude and phase distribution of the beam, in the plane A1’, are the same as that in the plane A1. The higher-order Zernike coefficients of the beam wavefront, and the beam intensity in the plane A1’, are shown in Fig. 2(a). The value of c13 is −0.017 μm, which is the same as that in the plane A1. The medium in the amplifier introduces the same positive spherical aberration as that in the oscillator, with a coefficient of ∼0.02 μm. Therefore, the incident beam (with negative spherical aberration) can be compensated by the gain medium in laser amplifier. Figs. 2(a) and 2(b) compare the Zernike coefficients of the wavefront and intensity distributions, in the incident surface and the exit surface of the amplifier. It is shown that the defocus (the fifth Zernike coefficient) and the spherical aberration are both compensated in the exit surface, while the intensity distributions remain almost the same as that in the incident surface. The two fields should have almost the same value of M 2diff , but a different value of M 2ab, hence different M 2 factor. The beam after amplification has an M 2 factor
Fig. 2. Higher-order Zernike coefficients of the beam wavefront, and the beam intensity in the amplifier; (a) before the gain medium in the plane A1’; (b) after the gain medium. 7966
APPLIED OPTICS / Vol. 53, No. 33 / 20 November 2014
of 1.4, which shows better beam quality than that from the master oscillator. This process of spherical aberration compensation (in the laser amplifier) is similar to the beam quality improvement (from the plane A1 to B1 in the oscillator). The laser power is scaled up to 60 W by the power amplifier, with the decreasing M 2 factor. A negative coefficient c13 of the incident beam is a requirement in the MOPA system for beam quality improvement, because the medium in the amplifier introduces positive spherical aberrations. Fortunately, such a beam can always be obtained from a DSR. The value of c13 can be positive, negative, or nearly zero for the intra-cavity beams in a DSR. It is important to image the intra-cavity field with negative c13 , to the incident surface of the medium in the amplifier. Otherwise, the wavefront-compensation solution does not work. It should be noted that if a laser system contains only an oscillator without an amplifier, the beam should be coupled out from the resonator with a smaller spherical aberration coefficient (c13 ≈ 0) [22,23], which hence shows better beam quality. The MOPA system, with two or three stage amplifiers, will be carried out experimentally in the next step, the schematic of which is shown in Fig. 3. The pumping parameters and laser medium units are the same as those in Fig. 1. The basic principle of beam quality management in these systems is also to control the evolution of the wavefront spherical aberrations in amplifiers. We will show this solution, taking the MOPA system with two stage amplifiers in Fig. 3(a), as an example. The laser beam from the oscillator has a beam quality factor M 2 of about 1.2. When it propagates a distance L2 (the long arm length of the DSR), and then passes through the first amplifier, the positive spherical aberration is superposed to the beam wavefront. The beam quality deteriorates in the exit surface of the medium (the plane A2’). This positive coefficient would flip its sign when the beam propagates to the left end of the second amplifier (the plane A1’), provided that the distance between the two amplifiers is 2L1 (2× the short
Fig. 3. Schematic of MOPA laser systems with multi-stage amplifiers. All the pumping systems are omitted; (a) with two stage amplifiers; (b) with three stage amplifiers.
arm length of the DSR). Finally, the negative spherical aberration would be compensated by the gain medium in the second amplifier, and the beam quality would be enhanced. The wavefront evolution in amplifiers is quite similar to that in the oscillator. MOPA systems with multi-stage amplifiers can also be set up by this principle, to manage the beam quality. 5. Conclusion
Intensive study was carried out with the beam wavefront evolution in an asymmetrical DSR. The spherical aberration coefficients of the beam wavefront have different values in the end surfaces of the gain medium, which can be positive, negative, or nearly zero. It is demonstrated that wavefront spherical aberrations are the major contributors to the beam quality difference of intra-cavity beams. A MOPA system is set up experimentally, in which the beam from the oscillator has a wavefront with negative spherical aberration. The wavefront is well-compensated by the medium in the laser amplifier. The beam quality is enhanced, with the M 2 factor decreasing from 2.2 to 1.4. Solutions are also given for beam quality management in MOPA systems with multi-stage amplifiers. These findings are useful for designing high-power and high-beamquality MOPA systems. This paper was supported in part by the National Nature Science Foundation of China (No. U1230101). This paper was also supported by the foundation of the Key Laboratory of Science and Technology on High Energy Laser, CAEP (No. 2014HEL04). References 1. F. Levine, “TEM00 enhancement in CW Nd-YAG by thermal lensing compensation,” IEEE J. Quantum Electron. 7, 170–172 (1971). 2. J. E. Bernard and A. J. Alcock, “High-efficiency diode-pumped Nd:YVO4 slab laser,” Opt. Lett. 18, 968–970 (1993). 3. C. Scurtescu, Z. Y. Zhang, J. Alcock, R. Fedosejevs, M. Blumin, I. Saveliev, S. Yang, H. Ruda, and Y. Y. Tsui, “Quantum dot saturable absorber for passive mode locking of Nd:YVO4 lasers at 1064 nm,” Appl. Phys. B 87, 671–675 (2007). 4. J. A. Alcock, D. J. Gendron, and S. K. Nikumb, “Further development of a diode-pumped Nd:YVO4/Nd:YAG hybrid oscillator,” Proc. SPIE 3491, 1115–1118 (1998). 5. N. Hodgson and H. Weber, “Influence of spherical aberration of the active medium on the performance of Nd:YAG lasers,” IEEE J. Quantum Electron. 29, 2497–2507 (1993). 6. A. Montmerle Bonnefois, M. Gilbert, P. Y. Thro, and J. M. Weulersse, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223–235 (2006). 7. D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” IEEE J. Quantum Electron. 34, 2383–2392 (1998).
8. J. Bourderionnet, A. Brignon, J. P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode of high power rod solid-state lasers,” Opt. Commun. 204, 299–310 (2002). 9. C. Liu, T. Riesbeck, X. Wang, J. Ge, Zh. Xiang, J. Chen, and H. J. Eichler, “Influence of spherical aberrations on the performance of dynamically stable resonators,” Opt. Commun. 281, 5222–5228 (2008). 10. R. Martinez-Herrero, P. M. Mejias, N. Hodgson, and H. Weber, “Beam-quality changes generated by thermally-induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995). 11. I. Buske and U. Wittrock, “Diffraction analysis of aberrated laser resonators,” Appl. Phys. B 83, 229–233 (2006). 12. E. Leibush, S. M. Jackel, S. Goldring, I. Moshe, Y. Tzuk, and A. Meir, “Elimination of spherical aberration in multi-kW, Nd:YAG, rod pump-chambers by pump-distribution control,” in Advanced Solid-State Photonics (Optical Society of America, 2005), paper MB45. 13. I. Moshe, S. Jackel, and A. Meir, “Correction of spherical and azimuthal aberrations in radially polarized beams from strongly pumped laser rods,” Appl. Opt. 44, 7823–7827 (2005). 14. Y. Lumer, I. Moshe, S. Jackel, and A. Meir, “Use of phase corrector plates to increase the power of radially polarized oscillators,” J. Opt. Soc. Am. B 27, 1337–1342 (2010). 15. A. M. Bonnefois, M. Gilbert, P. Y. Thro, and J. M. Weulersse, “Novel method to improve the performance of Nd:YAG highpower low-divergence lasers using a passive compensation of the spherical aberration inside the resonator,” in Lasers and Applications in Science and Engineering (International Society for Optics and Photonics, 2005), pp. 362–369. 16. J. J. Kasinski and R. L. Burnham, “Near-diffraction-limited, high-energy, high-power, diode-pumped laser using thermal aberration correction with aspheric diamond-turned optics,” Appl. Opt. 35, 5949–5954 (1996). 17. A. E. Siegman, “New developments in laser resonators,” in New Developments in Laser Resonators, Los Angeles, CA, January 14–19 (International Society for Optics and Photonics, 1990), pp. 2–14. 18. A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32, 5893– 5901 (1993). 19. B. J. Neubert and B. Eppich, “Influences on the beam propagation ratio M2,” Opt. Commun. 250, 241–251 (2005). 20. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980). 21. M. Ostermeyer, P. Kappe, R. Menzel, and V. Wulfmeyer, “Diode-pumped Nd:YAG master oscillator power amplifier with high pulse energy, excellent beam quality, and frequencystabilized master oscillator as a basis for a next-generation lidar system,” Appl. Opt. 44, 582–590 (2005). 22. Z. Zhao, S. Pan, Z. Xiang, Y. Dong, J. Ge, C. Liu, and J. Chen, “Influences of spherical aberration on resonator’s stable zones and fundamental mode output power scaling of solid state laser oscillators,” Opt. Express 20, 10605–10616 (2012). 23. Z. Ye, Z. Zhao, S. Pan, X. Zhang, C. Wang, Y. Qi, C. Liu, Z. Xiang, and J. Ge, “Beam profile evolution and beam quality changes inside a diode-end-pumped laser oscillator,” IEEE J. Quantum Electron. 50, 62–67 (2014). 24. International Organization for Standardization, “Lasers and laser-related equipment–test methods for laser beam parameters–beam widths, divergence angle and beam propagation factor,” Standard 11146, 1999.
20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
7967
2
State Key Laboratory of Modern Optical Instrumentation, Department of Optical Engineering, Zhejiang University, Hangzhou 310027, China
Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan *Corresponding author: [email protected] Received 19 August 2014; revised 25 October 2014; accepted 26 October 2014; posted 28 October 2014 (Doc. ID 221182); published 18 November 2014
A method is presented for beam quality improvement, in master oscillator power amplifier laser systems. Intensive study was first carried out with the beam wavefront evolution in a laser resonator. When the laser beam propagates inside the resonator, the spherical aberration coefficient of the beam wavefront can change sign, i.e., the negative spherical aberration coefficient can turn to positive, and vice versa. This process also occurs when the beam propagates outside the resonator in a free space. The laser beam, from an oscillator with negative spherical aberration, was found to be well-compensated by the positive spherical aberration of a strongly pumped laser rod in a laser amplifier. The laser beam quality M 2 factor has been significantly improved, from 2.2 to 1.4, while the output power has been scaled from 31 W up to 60 W. © 2014 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (140.3280) Laser amplifiers; (140.3410) Laser resonators; (140.3580) Lasers, solid-state. http://dx.doi.org/10.1364/AO.53.007963
1. Introduction
Bulk-crystal based high-power lasers are still widely used, due to their high efficiency, compact package, and high damage threshold [1–3]. A key factor that limits their applications is the beam quality degradation, due to the thermal effects in the gain medium. The gain medium can be described (ideally) as a perfect lens [4]. In practice, the temperature distribution in the cross section of the medium differs from the ideal parabolic profile, because of inhomogeneous pumping, and the temperature dependence of the thermal conductivity and thermo-optical coefficient, which leads to the spherical aberration effect of the thermal lens [5–7]. Spherical aberrations in gain mediums have been proven to be the major 1559-128X/14/337963-05$15.00/0 © 2014 Optical Society of America
reason for beam quality deterioration in laser resonators and amplifiers [5,8–11]. Some techniques have been adopted to improve the beam quality, by compensating spherical aberrations in gain mediums, in the past few years. First, the spherical aberration effect can be weakened, by properly designing the pump light distributions. In 2005, Leibush et al. [12] reported that the spherical aberration was eliminated in multi-kW, Nd:YAG, rod pump-chambers, by optimizing the pump distributions. Second, a relay-imaging telescope, with negative spherical aberration, can be used to compensate the positive spherical aberrations of gain media. Effective compensations were realized both in laser resonators, and in multi-rod amplifier systems [13]. Third, by using aspherical phase-plates, which introduce spherical aberrations with the same absolute value (but opposite sign) as the laser beams, the 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
7963
beam quality can be significantly improved, inside or outside the cavity [14–16]. However, there are still some drawbacks in the three methods mentioned previously. The coupling system, as well as the doping concentration, of the laser crystal should be specially designed in the first method, which is not easy to manage in experiments. Additional optical elements are required in the second and third method. Moreover, aspherical phase-plates are found to be expensive and complicated to fabricate, for large spherical aberrations. In this paper, we show a master oscillator power amplifier (MOPA) system, designed to compensate spherical aberrations without additional optical elements. The beam from the master oscillator has a wavefront with negative spherical aberration. The wavefront is compensated by the positive spherical aberration of the Nd:YVO4 rod in the laser amplifier, which results in a significant improvement in the beam quality after the amplifier. In Section 2, the mathematical relationships between the beam quality factor M 2, and spherical aberration coefficient, are briefly reviewed. In Section 3, experimental measurements are performed for the beam wavefront evolution in the master oscillator. The results show that the beam wavefront aberration changes when the beam passes through the gain medium, which results in the beam quality difference of the intracavity beams. A laser amplifier, which compensates the spherical aberration of the incident beam, is proposed in Section 4, and conclusions are given in Section 5. 2. Brief Review of the Relationship between the Spherical Aberration Coefficient, and the Beam Quality Factor M 2
M2
The beam quality factor is often used to characterize that how well an arbitrary laser beam can be focused [17,18]. For a coherent laser source, the mathematical expression of the M 2 factor is related to the beam intensity distributions, and wavefront aberrations, independently [19] M2
q M 2diff 2 M 2ab 2 ;
(1)
where M 2diff represents the amplitude term, and M 2ab represents the phase term. Taking an aberrated Gaussian beam as an example, the complex amplitude can be expressed as
r2 2π ~ ur exp − 2 j W ; λ ω0
(2)
with an aberrated wavefront of W C0 C2 r2 C4 r4 ;
(3)
where ω0 is the beam radius, λ is the wavelength, and C2 and C4 are coefficients (which represent the focusing term and spherical aberrations term, 7964
APPLIED OPTICS / Vol. 53, No. 33 / 20 November 2014
respectively). The component M 2ab in Eq. (1) is related to the coefficient C4 , according to [19] M 2ab
2π jC jω4 λ 4 0
r 3 : 2
(4)
The beam intensity distribution is Gaussian, and we have the component M 2diff in Eq. (1) of M 2diff 1:
(5)
In our experiments, and also in some literature, the wavefront is expressed by a set of Zernike functions [20] Wρ; θ
N X n1
cn Zn ρ; θ;
(6)
where Zn ρ; θ is the nth Zernike polynomial (with cn being its coefficient), ρ r∕r0 ; r and θ are polar axes, and r0 is the normalized radius (which equals the beam radius p ω0 ). The 13th Zernike polynomial Z13 ρ; θ 56ρ4 − 6ρ2 1, which is related to the spherical aberration, is the main item of the thermally induced aberrations. We only considered the spherical aberration term in the M 2 factor calculap tions, i.e., 6 5c13 ρ4 . From Eq. (3) and the 13th Zernike polynomial expression, we have the relationship between C4 and c13 as p 6 5c13 C4 : ω40
(7)
The contribution of the spherical aberration coefficient c13 to the M 2 factor can be calculated, using Eqs. (1), (4), (5), and (7) M2
q 1 270k2 c213 ;
(8)
where k 2π∕λ is the wavenumber. Eq. (8) describes the relationship between the spherical aberration and M 2 factor, when the beam intensity is a Gaussian distribution. Useful conclusions can be deduced from Eq. (8). The stronger the spherical aberration, the higher the M 2 factor (i.e., the worse beam quality), provided that the beam intensity keeps a Gaussian distribution (M 2diff ≈ 1). This gives a proper explanation of the beam quality degradation in multi-stage MOPA laser systems. Expanding telescopes are always used between the amplifiers, to achieve mode volume matching in the MOPA systems. These expanding telescopes are always relay optics, which image the near field of the laser beam from one stage to the next. This solution has the advantage of keeping the beam intensity as a Gaussian distribution, which is effective for eliminating beam quality degradation, by minimizing the component M 2diff . However, the spherical aberrations in each amplifier
are gradually superposed to the beam wavefront [21], which degrades the beam quality by increasing the component M 2ab . 3. Spherical Aberration Measurements inside the Resonator
An asymmetrical plano–plano dynamically stable resonator (DSR) [22,23] is built up as the master oscillator (shown in the lower part of Fig. 1). Two high-power fiber-coupled diode lasers (from DILAS) are used for pumping. Each diode laser, with 808 nm central wavelength, delivers a maximum output power of 50 W in CW mode. The pump beam is focused into the crystal from two end surfaces, with a beam diameter of 0.8 mm at the incident surface. The two dichroic mirrors (M1 and M2) are high transmittance (HT), coated at 808 nm, and high reflection (HR), coated at 1064 nm, for light at an incidence angle of 45°. The mirror M3 is also high-reflection coated at 1064 nm. The output couple (OC) has a reflectivity of 50%. The laser is operated in fundamental mode. An average power of 31 W is achieved, with a pump power of 75 W. The beam quality is measured for the output beam from the OC, using International Organization for Standardization (ISO) standardized methods [24]. It has an M 2 factor of 2.2. The M 2 factor of the beam that leaks from the HR mirror is also measured, which is 1.3. We exchanged the OC and HR mirror in the experiments. In this condition, the beam from the OC has an M 2 factor of 1.3, while the beam from the HR mirror has an M 2 factor of 2.0. It is an important result that the beam from the long arm shows a better beam quality than that from the short arm [the laser cavity is divided into two parts by the laser medium, and we define the one with longer distance L2 (320 mm) as the long arm, and the shorter one L1 (117 mm) as the short arm]. This phenomenon has been observed in [23]. The beam quality difference of intra-cavity beams can be explained by the evolution of beam wavefront aberrations. The beam intensity and wavefront distribution, in any plane inside the resonator, can be obtained using the method introduced by [23]. The laser fields especially attract attention in the four planes of A1, A2, B1, and B2, shown in Fig. 1. Labels A and B represent the planes in the left and right end of the crystal, respectively. Labels 1 and 2 represent the forward and backward beams, respectively (we
Fig. 1. Configuration of the MOPA system; LD, laser diode; M1, M2, M4, and M5, dichroic mirrors; M3, 45° high-reflection mirror.
define the direction from OC to HR as a forward direction, while the reverse one is the backward direction). The spherical aberration coefficients c13 are measured, for the four fields, by a Hartman–Shack wavefront sensor (WFS 150-5C, Thorlabs), which are −0.017 μm (A1), 0.005 μm (B1), 0.019 μm (A2), and −0.007 μm (B2), respectively. The beam intensity is nearly a Gaussian distribution for all the four fields. Eq. (8) can thus be used to calculate the M 2 factors of the four fields, which are 1.93 (A1), 1.11 (B1), 2.10 (A2), and 1.21 (B2), respectively. It should be noticed that the M 2 factors should be the same for the fields in the planes A1 and A2, because beam quality does not change when a beam propagates in a free space, or is reflected by a planar mirror. The inconsistency of the calculated M 2 factor value, in the planes A1 and A2, is mainly due to the measurement errors of the spherical aberration coefficient. The same result can be concluded for the fields in the planes B1 and B2. In spite of this, these calculated values of M 2 factors agree well with the measured values, which are 2.2 for the fields in planes A1 and A2, and 1.3 for the fields in planes B1 and B2. The measured values are a little bit larger, which is mainly due to the fact that the beam intensity is not an ideal Gaussian distribution in the four planes, and the component M 2diff is larger than 1. The evolution of the beam wavefront aberration can be described as follows, according to the previously mentioned measurements. The spherical aberration coefficient c13 is nearly zero for the field in the plane B2. The value of c13 increases to 0.019 μm when the beam passes through the gain medium to the plane A2, because the medium introduces a positive spherical aberration, with a coefficient of ∼0.02 μm. The value of c13 keeps almost the same absolute value, but changes the sign, when the beam propagates from the plane A2 to A1. Then the beam wavefront, with negative c13 in the plane A1, is compensated by the medium with positive spherical aberration. The value of c13 becomes nearly zero again for the field in the plane B1. There are two conclusions that should be emphasized, especially for the resonator. First, the value of c13 can be positive or negative for the intra-cavity beams, even if the gain medium always introduces positive spherical aberration. It can even flip the sign when the laser beam propagates a distance. The beam in the short arm acts as an example. The spherical aberration coefficient flips its sign, while it keeps almost the same absolute value when the beam propagates a distance of 2L1, from the plane A2 to the plane A1. Second, the beam wavefront with a negative c13 can be compensated by the laser medium. The value of c13 changes from −0.017 to 0.005 μm when the forward beam passes through the medium, from the plane A1 to the plane B1, which leads to the beam quality improvement. These two conclusions support the design of the laser amplifier, for beam quality improvement in the next section. 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
7965
4. Spherical Aberration Compensation in Laser Amplifiers
The schematic of the MOPA system is shown in Fig. 1. The amplifier (in the upper part of Fig. 1) has the same pumping parameters, and laser medium, as the master oscillator, which ensure the identical spherical aberration effect of the two crystals. No isolator and no coupling systems are used, between the amplifier and the oscillator. The laser beam from the OC has a large divergence angle of about 10 mrad [23]. It propagates a distance of L1 to the incident surface of the gain medium of the amplifier (the plane A1’ in Fig. 1), where L1 is the short arm length of the master oscillator. In this way, the mode volume matching, between the pump volume and laser mode in the amplifier, are realized. Another advantage of this configuration is that the amplitude and phase distribution of the beam, in the plane A1’, are the same as that in the plane A1. The higher-order Zernike coefficients of the beam wavefront, and the beam intensity in the plane A1’, are shown in Fig. 2(a). The value of c13 is −0.017 μm, which is the same as that in the plane A1. The medium in the amplifier introduces the same positive spherical aberration as that in the oscillator, with a coefficient of ∼0.02 μm. Therefore, the incident beam (with negative spherical aberration) can be compensated by the gain medium in laser amplifier. Figs. 2(a) and 2(b) compare the Zernike coefficients of the wavefront and intensity distributions, in the incident surface and the exit surface of the amplifier. It is shown that the defocus (the fifth Zernike coefficient) and the spherical aberration are both compensated in the exit surface, while the intensity distributions remain almost the same as that in the incident surface. The two fields should have almost the same value of M 2diff , but a different value of M 2ab, hence different M 2 factor. The beam after amplification has an M 2 factor
Fig. 2. Higher-order Zernike coefficients of the beam wavefront, and the beam intensity in the amplifier; (a) before the gain medium in the plane A1’; (b) after the gain medium. 7966
APPLIED OPTICS / Vol. 53, No. 33 / 20 November 2014
of 1.4, which shows better beam quality than that from the master oscillator. This process of spherical aberration compensation (in the laser amplifier) is similar to the beam quality improvement (from the plane A1 to B1 in the oscillator). The laser power is scaled up to 60 W by the power amplifier, with the decreasing M 2 factor. A negative coefficient c13 of the incident beam is a requirement in the MOPA system for beam quality improvement, because the medium in the amplifier introduces positive spherical aberrations. Fortunately, such a beam can always be obtained from a DSR. The value of c13 can be positive, negative, or nearly zero for the intra-cavity beams in a DSR. It is important to image the intra-cavity field with negative c13 , to the incident surface of the medium in the amplifier. Otherwise, the wavefront-compensation solution does not work. It should be noted that if a laser system contains only an oscillator without an amplifier, the beam should be coupled out from the resonator with a smaller spherical aberration coefficient (c13 ≈ 0) [22,23], which hence shows better beam quality. The MOPA system, with two or three stage amplifiers, will be carried out experimentally in the next step, the schematic of which is shown in Fig. 3. The pumping parameters and laser medium units are the same as those in Fig. 1. The basic principle of beam quality management in these systems is also to control the evolution of the wavefront spherical aberrations in amplifiers. We will show this solution, taking the MOPA system with two stage amplifiers in Fig. 3(a), as an example. The laser beam from the oscillator has a beam quality factor M 2 of about 1.2. When it propagates a distance L2 (the long arm length of the DSR), and then passes through the first amplifier, the positive spherical aberration is superposed to the beam wavefront. The beam quality deteriorates in the exit surface of the medium (the plane A2’). This positive coefficient would flip its sign when the beam propagates to the left end of the second amplifier (the plane A1’), provided that the distance between the two amplifiers is 2L1 (2× the short
Fig. 3. Schematic of MOPA laser systems with multi-stage amplifiers. All the pumping systems are omitted; (a) with two stage amplifiers; (b) with three stage amplifiers.
arm length of the DSR). Finally, the negative spherical aberration would be compensated by the gain medium in the second amplifier, and the beam quality would be enhanced. The wavefront evolution in amplifiers is quite similar to that in the oscillator. MOPA systems with multi-stage amplifiers can also be set up by this principle, to manage the beam quality. 5. Conclusion
Intensive study was carried out with the beam wavefront evolution in an asymmetrical DSR. The spherical aberration coefficients of the beam wavefront have different values in the end surfaces of the gain medium, which can be positive, negative, or nearly zero. It is demonstrated that wavefront spherical aberrations are the major contributors to the beam quality difference of intra-cavity beams. A MOPA system is set up experimentally, in which the beam from the oscillator has a wavefront with negative spherical aberration. The wavefront is well-compensated by the medium in the laser amplifier. The beam quality is enhanced, with the M 2 factor decreasing from 2.2 to 1.4. Solutions are also given for beam quality management in MOPA systems with multi-stage amplifiers. These findings are useful for designing high-power and high-beamquality MOPA systems. This paper was supported in part by the National Nature Science Foundation of China (No. U1230101). This paper was also supported by the foundation of the Key Laboratory of Science and Technology on High Energy Laser, CAEP (No. 2014HEL04). References 1. F. Levine, “TEM00 enhancement in CW Nd-YAG by thermal lensing compensation,” IEEE J. Quantum Electron. 7, 170–172 (1971). 2. J. E. Bernard and A. J. Alcock, “High-efficiency diode-pumped Nd:YVO4 slab laser,” Opt. Lett. 18, 968–970 (1993). 3. C. Scurtescu, Z. Y. Zhang, J. Alcock, R. Fedosejevs, M. Blumin, I. Saveliev, S. Yang, H. Ruda, and Y. Y. Tsui, “Quantum dot saturable absorber for passive mode locking of Nd:YVO4 lasers at 1064 nm,” Appl. Phys. B 87, 671–675 (2007). 4. J. A. Alcock, D. J. Gendron, and S. K. Nikumb, “Further development of a diode-pumped Nd:YVO4/Nd:YAG hybrid oscillator,” Proc. SPIE 3491, 1115–1118 (1998). 5. N. Hodgson and H. Weber, “Influence of spherical aberration of the active medium on the performance of Nd:YAG lasers,” IEEE J. Quantum Electron. 29, 2497–2507 (1993). 6. A. Montmerle Bonnefois, M. Gilbert, P. Y. Thro, and J. M. Weulersse, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223–235 (2006). 7. D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” IEEE J. Quantum Electron. 34, 2383–2392 (1998).
8. J. Bourderionnet, A. Brignon, J. P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode of high power rod solid-state lasers,” Opt. Commun. 204, 299–310 (2002). 9. C. Liu, T. Riesbeck, X. Wang, J. Ge, Zh. Xiang, J. Chen, and H. J. Eichler, “Influence of spherical aberrations on the performance of dynamically stable resonators,” Opt. Commun. 281, 5222–5228 (2008). 10. R. Martinez-Herrero, P. M. Mejias, N. Hodgson, and H. Weber, “Beam-quality changes generated by thermally-induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995). 11. I. Buske and U. Wittrock, “Diffraction analysis of aberrated laser resonators,” Appl. Phys. B 83, 229–233 (2006). 12. E. Leibush, S. M. Jackel, S. Goldring, I. Moshe, Y. Tzuk, and A. Meir, “Elimination of spherical aberration in multi-kW, Nd:YAG, rod pump-chambers by pump-distribution control,” in Advanced Solid-State Photonics (Optical Society of America, 2005), paper MB45. 13. I. Moshe, S. Jackel, and A. Meir, “Correction of spherical and azimuthal aberrations in radially polarized beams from strongly pumped laser rods,” Appl. Opt. 44, 7823–7827 (2005). 14. Y. Lumer, I. Moshe, S. Jackel, and A. Meir, “Use of phase corrector plates to increase the power of radially polarized oscillators,” J. Opt. Soc. Am. B 27, 1337–1342 (2010). 15. A. M. Bonnefois, M. Gilbert, P. Y. Thro, and J. M. Weulersse, “Novel method to improve the performance of Nd:YAG highpower low-divergence lasers using a passive compensation of the spherical aberration inside the resonator,” in Lasers and Applications in Science and Engineering (International Society for Optics and Photonics, 2005), pp. 362–369. 16. J. J. Kasinski and R. L. Burnham, “Near-diffraction-limited, high-energy, high-power, diode-pumped laser using thermal aberration correction with aspheric diamond-turned optics,” Appl. Opt. 35, 5949–5954 (1996). 17. A. E. Siegman, “New developments in laser resonators,” in New Developments in Laser Resonators, Los Angeles, CA, January 14–19 (International Society for Optics and Photonics, 1990), pp. 2–14. 18. A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32, 5893– 5901 (1993). 19. B. J. Neubert and B. Eppich, “Influences on the beam propagation ratio M2,” Opt. Commun. 250, 241–251 (2005). 20. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980). 21. M. Ostermeyer, P. Kappe, R. Menzel, and V. Wulfmeyer, “Diode-pumped Nd:YAG master oscillator power amplifier with high pulse energy, excellent beam quality, and frequencystabilized master oscillator as a basis for a next-generation lidar system,” Appl. Opt. 44, 582–590 (2005). 22. Z. Zhao, S. Pan, Z. Xiang, Y. Dong, J. Ge, C. Liu, and J. Chen, “Influences of spherical aberration on resonator’s stable zones and fundamental mode output power scaling of solid state laser oscillators,” Opt. Express 20, 10605–10616 (2012). 23. Z. Ye, Z. Zhao, S. Pan, X. Zhang, C. Wang, Y. Qi, C. Liu, Z. Xiang, and J. Ge, “Beam profile evolution and beam quality changes inside a diode-end-pumped laser oscillator,” IEEE J. Quantum Electron. 50, 62–67 (2014). 24. International Organization for Standardization, “Lasers and laser-related equipment–test methods for laser beam parameters–beam widths, divergence angle and beam propagation factor,” Standard 11146, 1999.
20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
7967
