Transformation of a High Order Mode Intensity Distribution to a Nearly Gaussian beam

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1 Transformation of a High Order Mode Intensity Distribution to a Nearly Gaussian beam G. Machavariania, N. Davidsona, A. A. Ishaaya', A.A. Friesema, E. Hasman" adepament of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel boptical Engineering Laboratory, Faculty of Mechanical Engineering, Technion- Israel Institute of Technology, Haifa 32000, Israel ABSTRACT A simple method for obtaining a nearly Gaussian laser beam from a high order Hermite-Gaussian mode is presented. The method is based on separating the equal lobes of the high order mode and combining them together coherently. The method was experimentally verified with an arrangement of three mirrors, a 50% beam splitter and a phase tuning plate. The beam quality factor calculated in x-direction for the resulting output beam is 1.045, being very close to that of ideal Gaussian beam. The calculated power leakage is only 1.5 %. The experimental near-field and farfield intensity distributions of the output beam have nearly Gaussian cross sections in both the x and the y directions, with M1.34 and Ml.32. With some modifications, it is possible to obtain an output beam with M1.15 and no power leakage. Keywords: beam shaping, laser beams, beam quality, mode selection, phase elements, interference. 1. INTRODUCTION The quality of a laser beam is normally given by the product of the beam waist and beam divergence, and is usually denoted as M2 1,2 The optimal beam quality of M21 can be obtained from a laser operating with the fundamental mode of a Gaussian shape. Unfortunately, in such lasers only a small volume of a gain medium is used, and this leads to a significant reduction of the output power with respect to a laser operating with a higher mode. In order to increase the output power and also retain good beam quality, one should operate with a single, high order mode and then transform the emerging beam to a Gaussian beam. This is possible because of the well-defined amplitude and phase distributions of a high order mode,'. In principle, such a transformation can be performed by means of a two specially designed external phase elements. However, the calculation and design procedure for forming such elements is not simple, and up to now an efficient transformation of a high order mode distribution to a Gaussian beam has not been reported. High order Laguerre-Gaussian (LG) and Hermite-Gaussian (HG) modes often consist of several bright spots (lobes), separated by dark interfaces with very low intensity. The adjacent lobes of the field distribution in such a mode normally have opposite phases (it phase shift), and the intensity distribution of each lobe is rather close to that of the Gaussian beam. Thus, our approach is based on separating and then sunmiing coherently the individual lobes of a high order mode. Specifically, we present and demonstrate a simple method for obtaining a nearly Gaussian beam from a beam originating from a laser operating with a single HG10 mode, by means of dividing the mode into two symmetric parts and sunming them coherently. We calculated conditions for best performance of such a transformation and calculate the beam quality factor, and then verify the proposed method experimentally with cwnd- YAG laser. Proceedings of SPIE Vol ALT'02 International Conference on Advanced Laser Technologies, edited by Heinz P. Weber, Vitali I. Konov, Thomas Graf, (2003, Bellingham, WA, 2003) X/03/$

2 2. THE TRANSFORMATION ARRANGEMENT One possible arrangement for separating the two lobes of the HG10 mode and then combining them together coherently is shown in Fig. 1. As shown, a sharp edge of a mirror is aligned so that the two lobes of the incident pure HG10 mode are separated. One lobe is reflected by the sharp mirror towards another mirror and then reflected towards a 50% beam splitter. The other lobe is directed to a different mirror and also reflected towards the same beam splitter. At the beam splitter, the distributions of the two lobes are combined coherently. With the appropriate phase relation between the two lobe distributions, as adjusted by the phase tuning plate, complete constructive interference occurs, resulting in a single intense output beam of nearly Gaussian distribution. Phase tuning plate Beam splitter 50% Output beam HG10 mode Sharp mirror Figure 1. One possible arrangement for transforming an HG10 mode distribution to a nearly Gaussian beam. With the arrangement shown in Fig.1, the field distributions of the two lobes do not coincide. Effectively, one lobe of the HG10 mode is shifted toward the other, as shown in Fig. 2, where the field of each lobe is drawn by dashed line, while theirs sum is shown by solid line. The horizontal coordinate is in the units of the waist parameter w, where one lobe of the mode is shifted by the shift parameter xjw. As the two lobes are summed coherently, it is desired that a maximal power is obtained in one output direction, and the power leakage (the power outgoing in the other direction) will be minimal. The relative power leakage AP IF can be written as J{U1(x) - U2(x- x0)}2 = P, (1) 2 JU12(x)+U22(x_x0)dx where Uj(x) and U2(x-xo) are the electric fields of the two halves of the HG10 mode, which have the same phase. We have calculated the relative power leakage (4) as a function of the shift parameter x0iw,and found that the minimal power leakage of 1.5% is at x01.6 w. We also determined the beam propagation factor M2. The factor M2 describes the beam quality, as a product of the beam width and the beam divergence, normalized to the corresponding value of Gaussian beam 1,2 In the x direction, it is written as M2 =4,roo-, (2) 272 Proc. of SPIE Vol. 5147

3 0 cd 0 I- 0 Ui x/w Figure 2. The absolute values ofthe two halves ofthe HG10 mode. The left half(dashed line) is shifted to the write half(dashed line) by the shift parameter xoi'w (The x coordinate is in the units ofthe waist parameter w). The sum ofthe two fields is drown by solid line. where cy and are the standard deviations of the beam intensity profile in x direction, calculated in the near- and farfield, with 5 as the spatial frequency that is related to the propagation angle 0 by s=sino/x O/X. The amplitudes in the far field are found by numerically solving the Bessel- Fourier transformation7 of that in the near field. The calculated M factor of the beam, obtained by coherent summation of the two lobes of the HG10 mode, as a function of x/w, revealed that at the optimal value of x0=1. 6 w, the M factor is equal to So, we have a significant reduction of M2 value, from 3 for a HG10 mode' to Since the M2 factor (in y direction) remains 1, the effective 'cylindrical' M2 value can be calculated as following: M2 = (M+M)/2 8,1 yielding Consequently, we obtain a nearly Gaussian beam with high energy. It is interesting to note that each lobe of the HG10 mode distribution has an value of 1.15, which is larger than the optimal value obtained for the sum of the two lobes. It seems that the combined summation of the two lobes has a smoother and more symmetric shape than each separate lobe (see Fig. 3). Another possible arrangement for the transformation of a HG10 mode distribution to a nearly Gaussian beam is presented in Fig. 3. Because of an inversion change to one of the lobes, there will be exact matching between the two lobe distributions (as evident from the inversion symmetry of the HG10 mode). This arrangement should completely eliminate power leakage, but does introduce slight degradation in the beam quality. Specifically, the combined output beam now will have Ml. 15, identical to that of each lobe distribution separately, and somewhat larger than that obtained with the arrangement of Fig.l. 3. EXPERIMENTAL PROCEDURE AND RESULTS In our experiments we used an Nd YAG cw laser that contained intra-cavity discontinuous phase element (DPE), that provided a sufficiently pure HG10 mode distribution. This HG10 mode distribution was then introduced as the input to the arrangement shown in Fig.1. By a fine adjustment of the mirrors and the phase tuning plate we reached optimal overlap of the lobes and absence of interference fringes, in both the near field and the far field. This procedure ensured that the directions of the two beams were completely matched and that there was no phase difference between them, so that the two lobe distributions would sum coherently. It should be noted that the coherent summation should be made sufficiently close to the original beam waist, in order to minimize wavefront curvature of either beam. In our experiment, the entire optical distance from the laser output coupler was 35 cm, while the Rayleigh distance was 3m. Proc. of SPIE Vol

4 Output beam Beam splitter HG10 mode Phase tuning plate Figure 3. Another possible arrangement for transforming an HG10 mode distribution to a nearly Gaussian beam. The experimental results detected with a CCD camera are shown in Fig. 4. Figure 4(a) shows the input HG10 intensity distribution. Figure 4(b) shows the intensity distribution of the output beam, obtained in the near-field, while Fig.4(c) shows the intensity distribution of the output beam, obtained in the far-field. The intensity cross sections along the x and y axis are shown at the bottom and left-hand sides of the figures 4(b) and 4(c). The near field intensity distribution of the output beam has the expected shape of one bright spot, with nearly Gaussian cross sections in the two directions. The corresponding far field intensity distribution was obtained by focusing the output beam with a spherical lens (f101 cm). Here we see again one bright spot with the cross sections being close to that of the Gaussian beam. Figure 4. Experimental near- and far-field intensity distributions. (a): Input HG1O mode distribution; (b) near-field intensity distribution of the combined output beam; (c) far-field intensity distribution of the combined output beam. The experimental crosssections in the x and y directions are shown in the low and the left parts ofthe pictures. From these experimental data, we calculated the experimental beam quality factor M2. We obtained that M= 1.34 for the output beam, somewhat higher than the expected value of In the y direction, we obtained M= 1.32, somewhat higher than the expected diffraction limit (M2 1). We attribute these discrepancies to a possible impurity of the original mode and to possible phase aberrations introduced by the optical elements. Still, there is reasonable agreement between the predicted and experimental results, proving the validity of our approach. The experimental M2 factor, calculated for one half of the HG01 mode, gives Also, we calculated the experimental M and M values for the input HG01 mode distribution as M=3.21 and M= Proc. of SPIE Vol. 5147

5 4. CONCLUDING REMARKS A new simple method for improving the beam quality of a high order mode was presented. By dividing of a high order mode into symmetric parts and summing them together coherently, it is possible to obtain a nearly Gaussian beam with very high quality. The proposed method is verified experimentally with HG10 mode. Our method may be readily extended to other high order modes, for example, the i mode, which is composed of four identical lobes and has M=M=3 8 In this case, the arrangement of Fig. 1 can be applied twice to fold the mode in the two perpendicular directions and to obtain a nearly Gaussian beam with M, =M= and total power leakage of ' '3%. Also, it is possible to use twice the modified arrangement (Fig. 2) and obtain output beam with M = M= and no power leakage. Alternatively, our transformation arrangements can be used to transform a Gaussian beam to one of a pure higher order mode with extremely high efficiency by simply reversing the direction ofthe beams. ACKNOWLEDGEMENTS The authors would like to thank Impala Advanced Laser Solutions Ltd. and Pamot Venture Capital Fund for their support. REFERENCES 1. A.E. Siegman, "New developments in laser resonators", Optical Resonators, Proc. SPIE 1224, 2-14, S. Saghafi and C.J.R. Sheppard, "The beam propagation factor for higher order Gaussian beams", Optics Comm. 153, , T. Grafand J.E. Balmer, "Laser beam quality, entropy, and the limits ofbeam shaping", Opt. Comm. 169, , R. Oron, Y. Danziger, N. Davidson and A.A. Friesem, "Discontinuous phase elements for transverse mode selection in laser resonators", Appl. Phys. Left. 74, , H. Kogelnik and T. Li, "Laser beams and resonators", Proc. IEEE 54, , A.E. Siegman, Lasers, University Science Books, Mill Valley, California, J.W. Goodman, Introduction to Fourier Optics, Chapter 3, McGraw-Hill, New York, J.B. Murphy, "Phase space conservation and selection rules for astigmatic mode converters", Opt. Comm, 165, 11-18, Proc. of SPIE Vol