A simple method for astigmatic compensation of folded resonator without Brewster window

A folded resonator requires an oblique angle of incidence on the folded curved mirror, which introduces astigmatic distortions that limit the performance of the lasers. We present a simple method to compensate the astigmatism of folded resonator without Brewster windows for the first time to the best of our knowledge. Based on the theory of the propagation and transformation of Gaussian beams, the method is both effective and reliable. Theoretical results show that the folded resonator can be compensated astigmatism completely when the following two conditions are fulfilled. Firstly, when the Gaussian beam with a determined size beam waist is obliquely incident on an off-axis concave mirror, two new Gaussian beam respectively in the tangential and sagittal planes are formed. Another off-axis concave mirror is located at another intersection point of the two new Gaussian beams. Secondly, adjusting the incident angle of the second concave mirror or its focal length can make the above two Gaussian beam coincide in the image plane of the second concave mirror, which compensates the astigmatic aberration completely. A side-pumped continues-wave (CW) passively mode locked Nd:YAG laser was taken as an example of the astigmatically compensated folded resonators. The experimental results show good agreement with the theoretical predictions. This method can be used effectively to design astigmatically compensated cavities resonator of high-performance lasers. ©2014 Optical Society of America OCIS codes: (140.3410) Laser resonators; (140.4780) Optical resonators; (010.3310) Laser beam transmission; (070.2590) ABCD transforms; (140.3560) Lasers, ring References and links 1. H. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972). 2. D. Kane, “Astigmatism compensation in off-axis laser resonators with two or more coupled foci,” Opt. Commun. 71(3–4), 113–118 (1989). 3. N. Jamasbi, J.-C. Diels, and L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35(12), 1891–1906 (1988). 4. T. Skettrup, T. Meelby, K. Faerch, S. L. Frederiksen, and C. Pedersen, “Triangular laser resonators with astigmatic compensation,” Appl. Opt. 39(24), 4306–4312 (2000). 5. T. Skettrup, “Rectangular laser resonators with astigmatic compensation,” J. Opt. A Pure Appl. Opt. 7(11), 645– 654 (2005). 6. S. Yefet, V. Jouravsky, and A. Pe’er, “Kerr lens mode locking without nonlinear astigmatism,” J. Opt. Soc. Am. B 30(3), 549–551 (2013). #199869 $15.00 USD Received 23 Oct 2013; revised 19 Dec 2013; accepted 21 Dec 2013; published 28 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002309 | OPTICS EXPRESS 2309 7. U. Keller, D. A. Miller, G. D. Boyd, T. H. Chiu, J. F. Ferguson, and M. T. Asom, “Solid-state low-loss intracavity saturable absorber for Nd:YLF lasers: an antiresonant semiconductor Fabry-Perot saturable absorber,” Opt. Lett. 17(7), 505–507 (1992). 8. D. Burns, M. Hetterich, A. Ferguson, E. Bente, M. Dawson, J. Davies, and S. Bland, “High-average-power (> 20-W) Nd:YVO lasers mode locked by strain-compensated saturable Bragg reflectors,” J. Opt. Soc. Am. B 17(6), 919–926 (2000). 9. R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, and U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000). 10. S. W. Tsai, Y. P. Lan, S. C. Wang, K. F. Huang, and Y. F. Chen, “High-power diode-end-pumped passively mode-locked Nd: YVO4 laser with a relaxed saturable Bragg reflector,” Proc. SPIE 4630, 17–23 (2002). 11. K. K. Li, A. Dienes, and J. R. Whinnery, “Stability and astigmatic compensation analysis of five-mirror cavity for mode-locked dye lasers,” Appl. Opt. 20(3), 407–411 (1981). 12. K. K. Li, “Stability and astigmatic analysis of a six-mirror ring cavity for mode-locked dye lasers,” Appl. Opt. 21(5), 967–970 (1982). 13. E. Cojocaru, T. Julea, and N. Herisanu, “Stability and astigmatic compensation analysis of fiveand sixor seven-mirror cavities for mode-locked dye lasers,” Appl. Opt. 28(13), 2577–2580 (1989). 14. H. Kegelnik and T. Li, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 40 (1965). 15. D. G. Lancaster and J. M. Dawes, “Thermal-lens measurement of a quasi steady-state repetitively flashlamppumped Cr, Tm, Ho:YAG laser,” Opt. Laser Technol. 30(2), 103–108 (1998). 16. F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, and N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002). 17. A. Geng, C. Zhao, B. Yong, Y. Lu, and Z. Xu, “A method for measuring thermal focal length of LD-side pumped laser crystal (in Chinese),” Acta Phys. Sin. 57, 6987–6991 (2008).


Introduction
The resonator is an important part of lasers.The design quality for a resonator largely determines the performance of lasers, especially for some delicate lasers such as mode-locked lasers and the lasers with the frequency nonlinear transformation function.In the highperformance lasers, there is needful for a resonator design that provides intracavity focuses where the small spots size are highly concentrated.These small spots are required by pumping, mode locking and frequency nonlinear transforming.For the purpose of achieving several small intracavity focal spots, high performance lasers is mostly based on multi-mirror folded resonators, in an effort to minimize the dispersion and the volume of the lasers.However, a folded resonator requires an oblique angle of incidence at the curved mirror and introduces astigmatic distortions that limit the performance of the system.
In order to improve the performance of the lasers, many researchers [1][2][3][4][5][6] focused on the study of the astigmatism compensation.H. Kogelnik et.al [1,2].first presented a method to compensate the astigmatism of a tight focus continues-wave (CW) dye-laser by using the Brewster angled plate.For many practical applications, there are many cavities without using Brewster angle.Consequently, their astigmatic compensation method is invalid at this time.Moreover, utilizing their astigmatically compensating method, it can get good results only under the certainly approximate condition, and their method is not suitable for all cavities.The study of N. Jamasbi et al. [3] employed ABCD matrix (or ABCD laws) to compensate the astigmatism in only one arm of the resonator without a Brewster angled plate.Similarly, ABCD matrix (or ABCD laws) was employed to achieve astigmatism compensation in the specific region of the special shaped lasers, such as in one arm of the symmetric triangular laser [4], two arms simultaneous in the symmetric rectangular resonator [5], or kerr lens mode locking non-planar resonator with symmetrical incidence angle on two curved mirrors [6].Using ABCD matrix, it needs tedious calculations to implement the calculations of the roundtrip propagation matrix both in sagittal and tangential planes.Furthermore, the methods above are only suitable for a specific shaped cavity, and it is not intuitively clear when they are used to design the astigmatically compensated cavity.In a word, although there are many progresses in the astigmatic compensation, there is still lacking a simple and powerful method for designing resonators, in which the astigmatism in two or more arms is compensated.In this paper, we present a simple method to compensate the astigmatism of folded resonator without Brewster windows.The method is based on the theory of the propagation and transformation of Gaussian beams.Using this method, it is easy to design the astigmatically compensated folded cavity without Brewster windows.A side pumped CW passively mode locked Nd:YAG laser is taken as an example of the astigmatically compensated folded resonators.

Method for compensating astigmatism
A high-performance laser, such as a semiconductor saturable absorber mirror (SESAM) passively mode-locked laser, is mostly formed with a multi-mirror folded resonator.The cavity shown in Fig. 1 is widely adopted [7][8][9][10], with the combination of a short focal length spherical mirror (M 1 ) and a longer one (M 2 ).It can be easily to have a very small focal spot on the SESAM and an appropriate spot size in the crystal simultaneously.The disadvantage of the oblique angles of incidence occurring at M 1 and M 2 , results in astigmatic aberration that limits the performance of the system.The conventional method to investigate a cavity astigmatism mainly uses ABCD matrix method [4,5,[11][12][13].The product of 12 ABCD matrices is needed to represent a complete cavity round-trip in a plane for the cavity shown in Fig. 1.The calculations of the round-trip propagation matrices both in sagittal and tangential planes are quite complicated and not intuitive.The mode parameters are obtained by repeatedly changing the cavity geometry parameter, and it is inefficient and tedious.In the following, we will use a simple method to compensate simultaneously the astigmatism in the two terminal arms in Fig. 1.We apply the numerical calculation of the Gaussian beam to compensate the astigmatism of the cavity, which is different from traditional methods.In our approach, the cavity mode parameters of the designed resonator are firstly given, then the geometric parameters of the astigmatically compensated cavity is determined by the cavity mode parameters.This approach is similar to the general procedure of designing resonator, and it is particularly suitable for designing the astigmatically compensated cavities of highperformance lasers.In order to study the astigmatism, a special transformation relationship of astigmatismfree Gaussian laser beam propagation through two off-axis concave mirrors, will be described firstly in the following section.As shown in Fig. 2, the transformations of sagittal and tangential plane are drawn in a plane for a purpose of showing the difference between the beam transformations of the two orthogonal planes intuitively.The beam size in the sagittal plane is indicated by a solid red line and in tangential plane by a dashed blue line.The lenses F 1 and F 2 in Fig. 2  effective focal lengths, which are related to the focal length f of the mirrors, in both the sagittal plane and tangential plane are given by [1]: (2) Where, θ is the angle of incidence, and the sign with subscript s and t represent for the sagittal and tangential planes, respectively.When a round astigmatism-free Gaussian beam, which has a beam waist radius w 1 and a distance l 1 away from the lens F 1 , passes through the lens F 1 , a new Gaussian beam is formed.The transformation meets the following formulas given by H. Kogelnik [14]: where the new Gaussian beam is characterized by the parameters of the new beam waist radius 1 w′ and the spacing 1 l′ between the new beam waist position and the lens F 1 .The subscript 1 is that the Gaussian beam propagates through the 1st lens F 1 .The sign with superscript ' represents in image space, while the sign without superscript represents in object space.w′ , 1s l′ and 1t w′ , 1t l′ , respectively.There are two discrepant Gaussian beams after the mirror M 1 , and it is apparent that the oblique angle of the incidence at M 1 results in an elliptical transformed beam.
The beam radius w(z) at a distance z from beam waist can be obtained by the equation for free-space propagation of a Gaussian beam [14]: Using above formula, we obtain that the two discrepant Gaussian beams intersect at two locations, which means this elliptical beam has two round spots inevitably.One is at the lens F 1 when lens F 1 is assumed to be a thin lens, and the other one is at certain point after F 1. For the numerical example given above, one locates in F 1 , another one locates away from the beam waist 2s 50., we obtain that another round spot locates in 78 cm away from F 1 .For the goal of attaining an astigmatism-free transformed beam by the lens F 2 , the lens F 2 must be located in another round spot on account of the continuity of the transformation of the Gaussian beam.This is an essential condition but not full condition for compensating astigmatism.This means that meeting this condition is no sufficient to vanish the astigmatism in the image space of the lens F 2 .In order to further study astigmatism, we mainly investigate the differences of the beam waist parameters (including the beam waist radius and its position) between in the sagittal and tangential plane.The astigmatic values, which were defined as the absolute values of  The above results denote that the method for simultaneously compensating astigmatism in two terminal arms of a folded resonator without Brewster windows, is very effective and simple.A folded resonator can be compensated astigmatism completely when the following two conditions are fulfilled.Firstly, when the Gaussian beam with a determined size beam waist is obliquely incident on an off-axis concave mirror, two new Gaussian beam respectively in the sagittal and tangential planes are formed.Another off-axis concave mirror shall be located at another intersection point of the two new Gaussian beams.Secondly, adjusting the incident angle of the second surface mirror or its focal length can make the above two Gaussian beam coincide in the image plane of the second concave mirror.

Experimental study
According to the sketch in Fig. 2, a Z-type folded resonator as illustrated in Fig. 1 was well designed in order to compensate astigmatism.A side-pumped module (GTPC-75S, GT LASER CO.,LTD, China), which consists of a Ф3 × 65 (mm) Nd:YAG crystal and sidepumped by laser diode bars, was used in our experiment.M 1 and M 2 are concave mirrors, with radii of curvature of 100 mm and 500 mm respectively.M 3 and M 4 are both plane mirrors.Based on the results of our theoretical research above, when the concave mirror M 1 is located away from M 3 5.3 cm, and the oblique angle of incidence at the curved mirror M 1 is 7°, another off-axis concave mirror M 2 should be located away from M 1 78 cm, which satisfies the first condition of astigmatism compensation.According to Fig. 3(a), it can be seen that when the oblique angle of incidence at the curved mirror M 2 is approximately 12°, the second condition is satisfied and the astigmatism is compensated completely.Based on above calculations based on Fig. 2, we get 2 2 32.6 . Taking into account the thermal lens effect of the laser crystal, we use two flat mirrors cavity stability condition method [15][16][17] to measure the focal length of the laser media.The focal length was 89.3 cm when the laser diode driving current was 7.9 A. Due to the thermal lens, the distance between M 2 and M 4 change into 34.3cm.At this time, the astigmatic aberration is eliminated completely in the both terminal arms.
Figure 4 plots the intensity profile of the two terminal arms as measured by a laser beam analyzer (SP503U, Ophir Optronics Solutions Ltd, Israel).Figures 4(a  In the following section, we study the mode locking operation of the astigmatically compensated cavity.A SESAM (provided by Institute of Semiconductors, Chinese Academy of Science) adhered to a copper heat sink by silicon grease, was used to substitute the plane mirrors at M 3 location.A plane output coupler with a partial transmittance of 5% instead of the other plane mirrors at M 4 location.We set up a laser diode side-pumped Nd:YAG modelocked laser based on SESAM as shown in Fig. 1.The gain medium is in a terminal arm between the M 2 and M 4 , SESAM is placed in the other one between the M 1 and M 3 .Although some astigmatism still remains in the middle arm, considering there is no optical element in this arm, we can neglect the influence of the astigmatism.This method can avoid the introduction of additional element to compensate the astigmatism.The output characteristics were monitored and analyzed by a 1 GHz analog bandwidth high speed digital oscilloscope (DPO4104B, Tektronix, Inc., USA) and a 1 ns rise time fast photodetector (PIN2-11-12, Hi-Tech Optoelectronics Co., Ltd., China).When the driving current of the module increases to 7.9 A, stable CW mode locked pulse train are obtained.Figure 5 shows a typical oscilloscope trace of the pulse envelope including both the long term pulse trains (10us/div) and short term pulse trains (10ns/div) in the inset.The experimental results show that the astigmatically compensated cavity designed by us, is very suitable for SESAM mode-locked laser.

Conclusion
In conclusion, we proposed a new method for achieving astigmatism compensation in folded cavity.Based on the propagation and transformation of Gaussian beam the method is very convenient and intuitive.A folded resonator can be compensated astigmatism completely when the following two conditions are fulfilled.Firstly, place another off-axis concave mirror at the intersection point of the two Gaussian beams in the sagittal and tangential planes.Second, adjust the incident angle of the second surface mirror or its focal length, which can make the astigmatic aberration be eliminated completely.A Z-type cavity for SESAM passively mode-locked laser was designed.The experimental results show that the astigmatic aberration is eliminated completely in the both terminal arms.Our designed cavity can achieve a small focal circular spot on SESAM and long cavity length.The mode-locking operation can work steadily in an astigmatically compensated cavity.

Fig. 2 .
Fig. 2. A special transformation of Gaussian beam pass through two off-axis lenses sequence.Due to the different focal length between in the sagittal and tangential planes of F 1 , formulas (3) and (4) must be calculated respectively in the two orthogonal planes.According to the actual application, a beam waist 40 um is chosen to place a SESAM, and the others parameters are l 1 = 5.3 cm, f 1 = 50 mm, 1 7 θ = °.Referring to the formulas (1)-(4), one gets

Figure 3
shows that the astigmatic values vary with the focal length of the M 2 and the angle 2 θ of incidence at M 2 , respectively.Figures3(a) and 3(b) indicate that the astigmatism can be completely compensated through choosing an appropriate angle of incidence, when the focal length f 2 has been determined.Likewise, the astigmatism can be completely compensated by selecting an appropriate focal length as shown in Figs.3(c) and 3(d), when the oblique angle of the incidence at M 2 has been determined.It also can be seen from Figs.3(a) and 3(c) that a round beam waist f 2 is equal to 200 mm, the best value of incidence angle 2 θ is 10.7° as shown in Figs.3(b) and3(d).It can also be seen that the best value of 2 θ changes with f 2 .The existence of a round beam waist of the transformed beam indicates the astigmatism introduced at the folded mirror M 1 has been compensated by M 2 .
) and 4(b) are the output spot intensity profiles from the arm between M 2 and M 4 on location close to M 4 and far from M 4 , respectively.It can be seen that from Figs. 4(a) and 4(b) that the output spot profiles on different observation locations are both circular spots.It indicates that the astigmatism on the arm between M 2 and M 4 is completely compensated.Similarly, Figs.4(c) and 4(d)show the output spot intensity profiles from the arm between M 1 and M 3 at the place close to M 3 and far from M 3 , respectively.The output spot profiles on different observation locations are both circular spots as shown in Figs.4(c) and 4(d), which indicates that the astigmatism on the arm between M 1 and M 3 is also completely compensated.The experimental results show that the astigmatic aberration is eliminated completely in the both terminal arms.

Fig. 4 .
Fig. 4. output spot Intensity profiles from the terminal arms on different observation locations: (a) output spot intensity profiles from the arm between M 2 and M 4 at location close to M 4 , (b) output spot intensity profiles from the arm between M 2 and M 4 at location far from M 4 , (c) output spot intensity profiles from the arm between M 1 and M 3 at location close to M 3 , (d) output spot intensity profiles from the arm between M 1 and M 3 at location far from M 3 .