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Simulations of conjugate Dammann grating based 2D coherent solid-state laser array combination
Optics Communications 290 (2013) 126–131
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Simulations of conjugate Dammann grating based 2D coherent solid-state laser array combination Bing Li, Enwen Dai, Aimin Yan, Xiaoyu Lv, Yanan Zhi, Jianfeng Sun n, Liren Liu Key Laboratory of Space Laser Communication and Testing Technology, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, No. 390, Qinghe Road, Jiading District, Shanghai 201800, People’s Republic of China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 June 2012 Received in revised form 25 September 2012 Accepted 11 October 2012 Available online 30 October 2012
Conjugate Dammann grating based coherent beam combing technique is an efficient way of combining phase-locked laser array into one high-power high beam-quality laser output. In this paper, this technique is applied to the combination of two-dimensional phase-locked array of solid state laser beams. Square apertures that could increase the areal fill factor of 21.5% are adopted. Numerical and experimental simulations of the combinations of 4 4 and 5 5 2D coherent laser arrays are carried out. The influences of manufacturing and positioning errors of the conjugate Dammann gratings to the beam combining efficiency are also discussed in detail. & 2012 Elsevier B.V. All rights reserved.
Keywords: Coherent beam combination Laser array Solid-state laser Dammann grating Fourier transform Fill factor
1. Introduction Lasers systems with high-average-power and high-beamquality have a wide application in fields such as laser cutting, laser welding, laser radar, and laser weapons [1]. Advantages such as compact all-solid setup, large gain medium size, long lifetime, and often very good beam quality make solid-state lasers a perfect candidate in these areas. However, the output power from a single solid-state laser is largely limited by thermo-optic distortions of the solid-state gain media that also lower the beam quality [2,3]. The effects of these distortions can be mitigated by the choice of pumping and extraction architectures. Thin disks, rods, and zigzag slabs have all demonstrated near-diffraction-limited powers at or above 1 kW, but further scaling of the single-aperture devices continues to be limited by thermal effects [2,3]. An alternative method to scale solid-state lasers to higher power levels is to coherently combine multiple lower-power beams to form a single, monolithic output beam without loss of beam-quality [2,3]. Jingwen et al. demonstrated the phase-locking of a twoelement Nd:YAG laser array using evanescent coupling [4]. Menard et al. used a self-imaging confocal resonator to obtain the phase-locking of two emitted beams generated by a Nd:YAG laser [5]. However, a significant amount of energy is distributed in
n
Corresponding author. E-mail address: [email protected] (J. Sun).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.10.019
the sidelobes of the far field of the phase-locked laser array causing declination of the beam quality and waste of energy. One way of overcoming this is the aperture filling technique. Of many of the existing aperture filling techniques such as a phase corrector [6], multi-order aperture filling [7] and Talbot selfimaging [8], diffraction-optics-based coherent beam combining technique, especially Dammann grating based technique, is an effective one that combines and aperture-fills the phase-locked laser array simultaneously [9–13]. In Refs. [10,12], beam combination of a 2D fiber array using conjugate Dammann grating based technique is carried out with an array period of 300 mm and a sub-beam aperture of 60 mm. In this paper, by adjusting the period of the conjugate DG and the focal length of the Fourier lens, conjugate Dammann grating based coherent beam combining technique is applied to the combination of large aperture phase-locked solid state laser arrays. The effective areas of the emitters with square, circle apertures are discussed. Compared with the round apertures, square apertures that can increase the areal fill factor by 21.5%, are adopted. Numerical and experimental simulations of the CDG based combinations of 4 4 and 5 5 2D phase-locked solid laser arrays are carried out. Far-field distributions of the combined beam from the 4 4 and 5 5 solid-state laser array are calculated and compared with the experimental results. The influences of manufacturing and positioning error of the conjugate Dammann grating to the beam combining efficiency are also experimentally studied.
B. Li et al. / Optics Communications 290 (2013) 126–131
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2. Principles and experimental setup The schematic of the proposed coherent beam combination scheme of 4 4 solid-state-laser array is shown in Fig. 1 [10,12]. The phase values of the beams from the coherent solid-state-laser array (CSSLA), which is coherently phase-locked, are initialized by the phase plate (PP), the phase values of which is designed according to that of the diffraction orders of the Dammann grating. Then, the beams are combined into one beam by the Fourier lens (FL). The conjugate Dammann grating (CDG), which is the complex-conjugate of the Dammann grating, is placed in the image focal plane to eliminate the phase value variations of the combined beam. A coherently combined beam with single farfield lobe is obtained eventually. The same scheme is used for the combination of the 5 5 laser array. For detailed description of the combining principles, please refer to Refs. [10,12]. Since the wave function in the image focal plane is the Fourier Transform of the amplitude transmission function of the DG, the period of the laser array t and the period of the DG T follow the equation: t¼
lf T
,
ð1Þ
where l is the wavelength of the incident light and f is the focal length of the Fourier lens. The aperture L2 in the image focal plane of the Fourier lens for a plane incident wave with an aperture of L1 will be L2 ¼ 0:61
lf L1
,
ð2Þ
According to Eq. (1), by adjusting the DG period T and the Fourier lens focal length f, this CDG based beam combining technique can be applied to the combination of laser arrays with different array periods and duty ratios. Also, in order to eliminate the phase value variations of the combined beam, enough periods of the CDG must be illuminated by the combined beam. And according to Eq. (2), the spot size of the combined beam in the image focal plane is defined by the focal length of the Fourier lens and the aperture size of the incident light. The DG period T and the Fourier lens focal length f are decided by taking all the parameters into consideration. By increasing the Fourier lens focal length f, the same grating in Ref. [10] is used here for the combination of the 5 5 laser array with a much larger beam aperture (T¼ 1 mm). 2.1. Areal fill factors for square and round apertures on a square grid For a square array consisting of emitters with a period of t, the areal fill factor FA will be the area of one emitter divided by the unit cell area t t [14]. The array of emitting areas could be constructed by simply packing the emitter shapes AeL edge-toedge, then shrinking them by a factor of FL to get the actual configuration Ae as shown in Fig. 2. And FL is the ratio of emitter diameter to the array period t. (1 FL)*t is the structure between emitters for pumping and cooling. Thus the areal fill factor will be F A At2e . And for round aperture, F A, Round p4 F 2L 0:785F 2L , and for square aperture F A, Square F 2L .
Fig. 2. Round and square emitting apertures Ae on a t t grid [14].
Suppose that both round and square aperture emitters have the same factor FL, then the area fill factor of square aperture array is 21.5% larger than that of round aperture array. For square apertures, the intensity of the far-field central lobe can be increased by 21.5% at most, as the effective emitting area is increased. From Ref. [10], the field after the inverse conjugate Dammann grating at the image focal plane can be written as E2 ðx,yÞ ¼ 1 IfE1 ðf x ,f y Þg T nDG ðx,yÞ. Then, in the far-field, the field will be jlf Efar ðu,vÞ ¼
M X N 1 X X 1 1 I E2 ðx,yÞ ¼ es ðu,vÞ** 2 jlz l zf m n m1 ¼ 1 1 X C mn C nm1n1 d uðm1mÞ=T x , vðn1nÞ=T y Þ n1 ¼ 1
from which we can see that every single diffraction in the far filed is convoluted with the same single field es( u, v), which is exactly the same field function of the emitter of the input array. Thus, the beam combining efficiency of our system, which is defined as the intensity ratio of far-field zero order and all the order, has nothing to do with the beam shape, aperture or intensity distribution of the individual emitter es( u, v). That is also why we can use uniform beams instead of real beams with complex intensity distribution in the following numerical and experimental simulations. 2.2. Numerical combining simulations of arrays with squareaperture emitters Based on the theoretical analysis mentioned above, numerical simulations of the beam combination of a 4 4 solid-state laser array and a 5 5 solid-state laser array are carried out. The simulated 4 4 coherent solid-state laser array consists of 16 square beams with a square aperture of 1 mm and a period of 7.6 mm and the simulated 5 5 coherent solid-state laser array consists of 25 square beams with a square aperture of 1 mm and a period of 3.47 mm. And the focal length of the Fourier lens used here is 1.8 m. The wave length is 0.6328 mm. Far-field intensity distribution patterns of the combined beams are presented in Fig. 3 and Fig. 4. To have an overall impression, only the sectional (y¼0) patterns of the far-field intensity distribution are given here. Fig. 3 shows the simulated and normalized far-field intensity distribution patterns of the combined beam of the 4 4 coherent solid-state laser array and Fig. 4 shows the 5 5 coherent solidstate laser array. The far-field main lobes contain most of the input energy and the side lobes are greatly restrained. In addition, the main and the side lobes of the far-field intensity patterns, as seen from Figs. 3 and 4, are rectangle. 2.3. Experimental setup of the simulations
Fig. 1. Scheme of the CDG based CSSLA beam combination.
The application of the above mentioned CDG based coherent beam combining technique in fiber laser array is already confirmed in Ref. [10]. Here, we use the same experimental setup to
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test the feasibility of the application of this technique in the coherent combination of solid-state laser arrays, as is shown in Fig. 5. For an actual coherent laser array, the complicated phase control systems are required to establish and maintain the complete coherence of every element in the arrays. In this experiment, the phase-locked and coherent solid-state laser arrays are simulated by illuminating aperture masks (AM) with expanded and collimated He–Ne laser beams [10,12]. To simplify the design process, the 2D DGs are obtained by crossing two 1D DGs in the x and y directions [15–17], respectively, in a way of variable separation f(x,y)¼ f(x)f(y) where f(x) and f(y) could be the same 1D grating. For the 5 5 beam combination, the 1 5 DG is obtained from Ref. [17]. The phase values of the phase plate in the object focal plane of the Fourier lens is shown in Table 1 where ‘Calculated’ means theoretical and ‘Experimental’ means the values used in the experiment. The calculated values are simplified into the experimental values to ease the fabrication. The differences between the calculated and the experimental phase values will certainly influence the beam combining efficiency, which is discussed in detailed in the following parts. For the 4 4 beam combination, the 1 4 DG is
Fig. 3. The sectional (y¼ 0) far-field intensity distribution of the combined beam with 4 4 beamlets.
specially designed so that it can produce 1 4 equal-intensity sub beams at diffraction orders { 3, 1, þ1, þ3} with the phase values of exactly { p/2, p/2, p/2, p/2}. As there is only one phase step, this grating is easy to fabricate. The period of the CDG used in the experiment is 0.3 mm with the smallest sub period feature size as 16.2 mm. The phase state of the phase plate (PP) in the object focal plane of the Fourier lens is {0, 0, p, p} without simplification of the grating phase values.
3. Experimental results and discussion Simulated experiments based on the above mentioned setup are carried out. The actual combining efficiencies are measured and compared with the theoretical values. The influences of the manufacturing error and positioning error of the CDG to the beam combining efficiency are analyzed in detail. Figs. 6 and 7 show the experimental photographs of the nearfield and far-field patterns of the combined beams for the simulated 4 4 and 5 5 CSSLAs, respectively. The near-field patterns are obtained in the position immediately after the CDG while the far-field patterns are obtained in a position 20 m away from the CDG. As expected, strong peaks in the center with low side-lobes can be found in the combined beams. And the far-field patterns of the combined beams are rectangle, which tally with the calculated results shown in Figs. 3 and 4. The combining efficiency Z is defined here as the ratio of the power of the main spot and that of the whole combined beam Z ¼Pmain/Ptotal in the far field. For the 4 4 combination, the measured combining efficiency is Z ¼25.1%, while the theoretical value is 49.99%. And for the 5 5 combination, the measured combining efficiency is Z ¼37%, while the theoretical value is 59.88%. The round aperture beam combining efficiency (8%) in Ref. [10] is the ratio of the main lobe energy in the far field and the input energy of the 25 round apertures while the beam combining efficiency (49%) in Ref. [12] has the same definition as this paper. The causes of the difference between the measured and theoretical beam combining efficiency are discussed in the following parts. 3.1. Manufacturing error Figs. 8 and 9 are, respectively, the sectional surface profiles of the CDG used in our experiments for 4 4 and 5 5 beam combination, which are measured with a Taylor–Hobson Talysurf (S3F). The refraction index of the DG material at 633 nm is 1.52, so the depth for the phase value p should be 608.5 nm. Fig. 7 shows that the depth of the CDG for the 4 4 combination is Table 1 Calculated and experimental phase values of the 1 5 phase plate [8]. Phase values
Fig. 4. The sectional (y¼ 0) far-field intensity distribution of the combined beam with 5 5 beamlets.
Calculated Experimental
Diffraction orders 2
1
0
1
2
0.91p
0.05p 0
0 0
0.05p 0
0.91p
p
Fig. 5. Schematic diagram of experimental setup for the beam combination for the 4 4 simulated coherent solid-state laser array [8].
p
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Fig. 6. Experimental photographs of (a) the near-field pattern and (b) far-field mainlobe of the combined beam of the simulated 4 4 CSSLA.
Fig. 7. Experimental photographs of (a) the near-field pattern and (b) far-field mainlobe of the combined beam of the simulated 5 5 CSSLA.
Fig. 8. Measured surface profile of the CDG for the simulated 4 4 CSSLA combination.
approximately 574.3 nm. The phase error is about 5%, which, according to our calculation, reduces the beam combining efficiency by 3%. Fig. 9 shows that the depth of the CDG for the 5 5 combination which is approximately 608 nm much better than the 4 4 CDG. Fig. 10 is the sectional surface profiles of the PP used in our experiments for 5 5 beam combination. The depth of the step is 704 nm which means the PP depth error is 15%, which reduces the beam combining efficiency by 5%. These manufacturing errors can be reduced by employing better machine and operators.
3.2. Positioning errors In order to eliminate the phase variations of the combined beam in the image focal plane, the CDG must be carefully positioned so that it matches the sub-period phase structures of the combined beam precisely. We measured the variation of the beam combining efficiency when the CDGs move along the y direction as shown in Figs. 11 and 12 where the discrete ones are the experimental results and the solid lines represent the theoretical results. The combining efficiency varies periodically with the
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Fig. 9. Measured surface profile of the CDG for the simulated 5 5 CSSLA combination.
Fig. 10. Measured surface profile of the PP for the simulated 5 5 CSSLA combination.
Fig. 11. Normalized beam combining efficiency variation as a function of displacement: theoretical (line) and experimental (dot) results for the simulated 4 4 CSSLA combination.
Fig. 12. Normalized beam combining efficiency variation as a function of displacement: theoretical (line) and experimental (asterisk) results for the simulated 5 5 CSSLA combination.
translation distance. And the experimental results match quite well with the theoretical results. To have an experimental efficiency that is above 90% of the peak value, the displacement error has to be less than, for the 4 4 combination 0.03 T and for
the 5 5 combination 0.1 T, respectively. Notice that the minima of the experimental results are much higher than that of the theoretical results. This indicates the existence of background noises in the experiments.
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4. Conclusions
Fig. 13. Experimental beam combining efficiency versus rotation angle for the simulated 4 4 CSSLA combination.
Theoretical analysis and experimental results of the simulations show that conjugate Dammann grating based coherent beam combining technique is completely suitable for coherent solid-state laser array combination. By adjusting the period of the grating and the focal length of the Fourier lens, this technique can be used for beam combining systems of different array periods and beam apertures. Square aperture that could increase the central lobe intensity by 21.5% at most is used here. Error analysis shows that the manufacturing and positioning of the conjugate grating have a great influence on the beam combining efficiency, however, this influence can be eliminated effectively by proper mechanical design and careful operation. The work described in this paper will prove valuable for the development of all-solid high power laser systems.
Acknowledgments The authors acknowledge the support of the National Nature Science Foundation of China (No. 60907006 and No. 61108069) and the Key Laboratory of Space Laser Communication and Testing Technology of Chinese Academy of Sciences.
References
Fig. 14. Experimental beam combining efficiency versus rotation angle for the simulated 5 5 CSSLA combination.
Also, when the CDG has an angle to the image focal plane of the FL, its projection on the focal plane is smaller than the actual size of the properly designed CDG. The phase variations will not be eliminated completely which leads to a reduction in the beam combining efficiency. Figs. 13 and 14 are the experimental results of the beam combining efficiency for 4 4 and 5 5 simulations, respectively, when the CDGs have a slight angle to the image focal plane. The beam combining efficiency drops concussively when the angle of the CDGs to the image focal plane increases. To have an experimental efficiency that is above 90% of the peak value, the angle error have to be less than, for the 4 4 combination 0.001 rad and for the 5 5 combination 0.0005 rad, respectively. Finally, when the CDGs move back and forth along the optical axis away from the focal plane for about 2 cm, and since the system has a large scene depth, there is no significant change of the beam combining efficiency. These positioning errors can be reduced by proper mechanical design of the supporting structures.
[1] G.D. Goodno, C.P. Asman, J. Anderegg, S. Brosnan, E.C. Cheung, D. Hammons, H. Injeyan, H. Komine, W.H. Long, M. McClellan, S.J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S.B. Weiss, M. Wickham, IEEE Journal of Selected Topics in Quantum Electronics 13 (2007) 460. [2] G.D. Goodno, H. Komine, S.J. McNaught, S.B. Weiss, S. Redmond, W. Long, R. Simpson, E.C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, H. Injeyan, Optics Letters 31 (2006) 1247. [3] T.Y. Fan, IEEE Journal of Selected Topics in Quantum Electronics 11 (2005) 567. [4] J. Xu, S. Li, K.K. Lee, Y.C. Chen, Optics letters 18 (1993) 513. [5] S. Me´nard, M. Vampouille, B. Colombeau, C. Froehly, Optics Letters 21 (1996) 1996. [6] J.R. Leger, G.J. Swanson, M. Holz, Applied Physics Letters 50 (1987) 1044. [7] M. Khajavikhan, A. Hoyer-Leitzel, J.R. Leger, Optics Letters 33 (2008) 2377. [8] L. Liren, Z. Yiyang, Chinese Journal of Lasers 16 (1989) 37. [9] E.C. Cheung, J.G. Ho, G.D. Goodno, R.R. Rice, J. Rothenberg, P. Thielen, M. Weber, M. Wickham, Optics Letters 33 (2008) 354. [10] A. Yan, L. Liu, E. Dai, J. Sun, Y. Zhou, Optics Letters 35 (2010) 1251. [11] D. Pab uf, F. Emaury, S. de Rossi, R. Mercier, G. Lucas-Leclin, P. Georges, Optics Letters 35 (2010) 1515. [12] B. Li, A. Yan, L. Liu, E. Dai, J. Sun, B. Shen, X. Lv, Y. Wu, Journal of Optics 13 (2011) 055706. [13] P.A. Thielen, J.G. Ho, D.A. Burchman, G.D. Goodno, J.E. Rothenberg, M.G. Wickham, A. Flores, C.A. Lu, B. Pulford, C. Robin, A.D. Sanchez, D. Hult, K.B. Rowland, Optics Letters 37 (2012) 3741. [14] I.W. Smith, SPIE 8052 (2011) 80520P. [15] H. Dammann, E. Klotz, Optica Acta: International Journal of Optics 24 (1977) 505. [16] Q.Y. Bi, C.H. Zhou, J.J. Zheng, W. Jia, X.L. Xie, Z.Q. Lin, P.S. Chung, Optics Communications 282 (2009) 742. [17] C. Zhou, L. Liu, Applied Optics 34 (1995) 5961.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Simulations of conjugate Dammann grating based 2D coherent solid-state laser array combination Bing Li, Enwen Dai, Aimin Yan, Xiaoyu Lv, Yanan Zhi, Jianfeng Sun n, Liren Liu Key Laboratory of Space Laser Communication and Testing Technology, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, No. 390, Qinghe Road, Jiading District, Shanghai 201800, People’s Republic of China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 June 2012 Received in revised form 25 September 2012 Accepted 11 October 2012 Available online 30 October 2012
Conjugate Dammann grating based coherent beam combing technique is an efficient way of combining phase-locked laser array into one high-power high beam-quality laser output. In this paper, this technique is applied to the combination of two-dimensional phase-locked array of solid state laser beams. Square apertures that could increase the areal fill factor of 21.5% are adopted. Numerical and experimental simulations of the combinations of 4 4 and 5 5 2D coherent laser arrays are carried out. The influences of manufacturing and positioning errors of the conjugate Dammann gratings to the beam combining efficiency are also discussed in detail. & 2012 Elsevier B.V. All rights reserved.
Keywords: Coherent beam combination Laser array Solid-state laser Dammann grating Fourier transform Fill factor
1. Introduction Lasers systems with high-average-power and high-beamquality have a wide application in fields such as laser cutting, laser welding, laser radar, and laser weapons [1]. Advantages such as compact all-solid setup, large gain medium size, long lifetime, and often very good beam quality make solid-state lasers a perfect candidate in these areas. However, the output power from a single solid-state laser is largely limited by thermo-optic distortions of the solid-state gain media that also lower the beam quality [2,3]. The effects of these distortions can be mitigated by the choice of pumping and extraction architectures. Thin disks, rods, and zigzag slabs have all demonstrated near-diffraction-limited powers at or above 1 kW, but further scaling of the single-aperture devices continues to be limited by thermal effects [2,3]. An alternative method to scale solid-state lasers to higher power levels is to coherently combine multiple lower-power beams to form a single, monolithic output beam without loss of beam-quality [2,3]. Jingwen et al. demonstrated the phase-locking of a twoelement Nd:YAG laser array using evanescent coupling [4]. Menard et al. used a self-imaging confocal resonator to obtain the phase-locking of two emitted beams generated by a Nd:YAG laser [5]. However, a significant amount of energy is distributed in
n
Corresponding author. E-mail address: [email protected] (J. Sun).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.10.019
the sidelobes of the far field of the phase-locked laser array causing declination of the beam quality and waste of energy. One way of overcoming this is the aperture filling technique. Of many of the existing aperture filling techniques such as a phase corrector [6], multi-order aperture filling [7] and Talbot selfimaging [8], diffraction-optics-based coherent beam combining technique, especially Dammann grating based technique, is an effective one that combines and aperture-fills the phase-locked laser array simultaneously [9–13]. In Refs. [10,12], beam combination of a 2D fiber array using conjugate Dammann grating based technique is carried out with an array period of 300 mm and a sub-beam aperture of 60 mm. In this paper, by adjusting the period of the conjugate DG and the focal length of the Fourier lens, conjugate Dammann grating based coherent beam combining technique is applied to the combination of large aperture phase-locked solid state laser arrays. The effective areas of the emitters with square, circle apertures are discussed. Compared with the round apertures, square apertures that can increase the areal fill factor by 21.5%, are adopted. Numerical and experimental simulations of the CDG based combinations of 4 4 and 5 5 2D phase-locked solid laser arrays are carried out. Far-field distributions of the combined beam from the 4 4 and 5 5 solid-state laser array are calculated and compared with the experimental results. The influences of manufacturing and positioning error of the conjugate Dammann grating to the beam combining efficiency are also experimentally studied.
B. Li et al. / Optics Communications 290 (2013) 126–131
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2. Principles and experimental setup The schematic of the proposed coherent beam combination scheme of 4 4 solid-state-laser array is shown in Fig. 1 [10,12]. The phase values of the beams from the coherent solid-state-laser array (CSSLA), which is coherently phase-locked, are initialized by the phase plate (PP), the phase values of which is designed according to that of the diffraction orders of the Dammann grating. Then, the beams are combined into one beam by the Fourier lens (FL). The conjugate Dammann grating (CDG), which is the complex-conjugate of the Dammann grating, is placed in the image focal plane to eliminate the phase value variations of the combined beam. A coherently combined beam with single farfield lobe is obtained eventually. The same scheme is used for the combination of the 5 5 laser array. For detailed description of the combining principles, please refer to Refs. [10,12]. Since the wave function in the image focal plane is the Fourier Transform of the amplitude transmission function of the DG, the period of the laser array t and the period of the DG T follow the equation: t¼
lf T
,
ð1Þ
where l is the wavelength of the incident light and f is the focal length of the Fourier lens. The aperture L2 in the image focal plane of the Fourier lens for a plane incident wave with an aperture of L1 will be L2 ¼ 0:61
lf L1
,
ð2Þ
According to Eq. (1), by adjusting the DG period T and the Fourier lens focal length f, this CDG based beam combining technique can be applied to the combination of laser arrays with different array periods and duty ratios. Also, in order to eliminate the phase value variations of the combined beam, enough periods of the CDG must be illuminated by the combined beam. And according to Eq. (2), the spot size of the combined beam in the image focal plane is defined by the focal length of the Fourier lens and the aperture size of the incident light. The DG period T and the Fourier lens focal length f are decided by taking all the parameters into consideration. By increasing the Fourier lens focal length f, the same grating in Ref. [10] is used here for the combination of the 5 5 laser array with a much larger beam aperture (T¼ 1 mm). 2.1. Areal fill factors for square and round apertures on a square grid For a square array consisting of emitters with a period of t, the areal fill factor FA will be the area of one emitter divided by the unit cell area t t [14]. The array of emitting areas could be constructed by simply packing the emitter shapes AeL edge-toedge, then shrinking them by a factor of FL to get the actual configuration Ae as shown in Fig. 2. And FL is the ratio of emitter diameter to the array period t. (1 FL)*t is the structure between emitters for pumping and cooling. Thus the areal fill factor will be F A At2e . And for round aperture, F A, Round p4 F 2L 0:785F 2L , and for square aperture F A, Square F 2L .
Fig. 2. Round and square emitting apertures Ae on a t t grid [14].
Suppose that both round and square aperture emitters have the same factor FL, then the area fill factor of square aperture array is 21.5% larger than that of round aperture array. For square apertures, the intensity of the far-field central lobe can be increased by 21.5% at most, as the effective emitting area is increased. From Ref. [10], the field after the inverse conjugate Dammann grating at the image focal plane can be written as E2 ðx,yÞ ¼ 1 IfE1 ðf x ,f y Þg T nDG ðx,yÞ. Then, in the far-field, the field will be jlf Efar ðu,vÞ ¼
M X N 1 X X 1 1 I E2 ðx,yÞ ¼ es ðu,vÞ** 2 jlz l zf m n m1 ¼ 1 1 X C mn C nm1n1 d uðm1mÞ=T x , vðn1nÞ=T y Þ n1 ¼ 1
from which we can see that every single diffraction in the far filed is convoluted with the same single field es( u, v), which is exactly the same field function of the emitter of the input array. Thus, the beam combining efficiency of our system, which is defined as the intensity ratio of far-field zero order and all the order, has nothing to do with the beam shape, aperture or intensity distribution of the individual emitter es( u, v). That is also why we can use uniform beams instead of real beams with complex intensity distribution in the following numerical and experimental simulations. 2.2. Numerical combining simulations of arrays with squareaperture emitters Based on the theoretical analysis mentioned above, numerical simulations of the beam combination of a 4 4 solid-state laser array and a 5 5 solid-state laser array are carried out. The simulated 4 4 coherent solid-state laser array consists of 16 square beams with a square aperture of 1 mm and a period of 7.6 mm and the simulated 5 5 coherent solid-state laser array consists of 25 square beams with a square aperture of 1 mm and a period of 3.47 mm. And the focal length of the Fourier lens used here is 1.8 m. The wave length is 0.6328 mm. Far-field intensity distribution patterns of the combined beams are presented in Fig. 3 and Fig. 4. To have an overall impression, only the sectional (y¼0) patterns of the far-field intensity distribution are given here. Fig. 3 shows the simulated and normalized far-field intensity distribution patterns of the combined beam of the 4 4 coherent solid-state laser array and Fig. 4 shows the 5 5 coherent solidstate laser array. The far-field main lobes contain most of the input energy and the side lobes are greatly restrained. In addition, the main and the side lobes of the far-field intensity patterns, as seen from Figs. 3 and 4, are rectangle. 2.3. Experimental setup of the simulations
Fig. 1. Scheme of the CDG based CSSLA beam combination.
The application of the above mentioned CDG based coherent beam combining technique in fiber laser array is already confirmed in Ref. [10]. Here, we use the same experimental setup to
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test the feasibility of the application of this technique in the coherent combination of solid-state laser arrays, as is shown in Fig. 5. For an actual coherent laser array, the complicated phase control systems are required to establish and maintain the complete coherence of every element in the arrays. In this experiment, the phase-locked and coherent solid-state laser arrays are simulated by illuminating aperture masks (AM) with expanded and collimated He–Ne laser beams [10,12]. To simplify the design process, the 2D DGs are obtained by crossing two 1D DGs in the x and y directions [15–17], respectively, in a way of variable separation f(x,y)¼ f(x)f(y) where f(x) and f(y) could be the same 1D grating. For the 5 5 beam combination, the 1 5 DG is obtained from Ref. [17]. The phase values of the phase plate in the object focal plane of the Fourier lens is shown in Table 1 where ‘Calculated’ means theoretical and ‘Experimental’ means the values used in the experiment. The calculated values are simplified into the experimental values to ease the fabrication. The differences between the calculated and the experimental phase values will certainly influence the beam combining efficiency, which is discussed in detailed in the following parts. For the 4 4 beam combination, the 1 4 DG is
Fig. 3. The sectional (y¼ 0) far-field intensity distribution of the combined beam with 4 4 beamlets.
specially designed so that it can produce 1 4 equal-intensity sub beams at diffraction orders { 3, 1, þ1, þ3} with the phase values of exactly { p/2, p/2, p/2, p/2}. As there is only one phase step, this grating is easy to fabricate. The period of the CDG used in the experiment is 0.3 mm with the smallest sub period feature size as 16.2 mm. The phase state of the phase plate (PP) in the object focal plane of the Fourier lens is {0, 0, p, p} without simplification of the grating phase values.
3. Experimental results and discussion Simulated experiments based on the above mentioned setup are carried out. The actual combining efficiencies are measured and compared with the theoretical values. The influences of the manufacturing error and positioning error of the CDG to the beam combining efficiency are analyzed in detail. Figs. 6 and 7 show the experimental photographs of the nearfield and far-field patterns of the combined beams for the simulated 4 4 and 5 5 CSSLAs, respectively. The near-field patterns are obtained in the position immediately after the CDG while the far-field patterns are obtained in a position 20 m away from the CDG. As expected, strong peaks in the center with low side-lobes can be found in the combined beams. And the far-field patterns of the combined beams are rectangle, which tally with the calculated results shown in Figs. 3 and 4. The combining efficiency Z is defined here as the ratio of the power of the main spot and that of the whole combined beam Z ¼Pmain/Ptotal in the far field. For the 4 4 combination, the measured combining efficiency is Z ¼25.1%, while the theoretical value is 49.99%. And for the 5 5 combination, the measured combining efficiency is Z ¼37%, while the theoretical value is 59.88%. The round aperture beam combining efficiency (8%) in Ref. [10] is the ratio of the main lobe energy in the far field and the input energy of the 25 round apertures while the beam combining efficiency (49%) in Ref. [12] has the same definition as this paper. The causes of the difference between the measured and theoretical beam combining efficiency are discussed in the following parts. 3.1. Manufacturing error Figs. 8 and 9 are, respectively, the sectional surface profiles of the CDG used in our experiments for 4 4 and 5 5 beam combination, which are measured with a Taylor–Hobson Talysurf (S3F). The refraction index of the DG material at 633 nm is 1.52, so the depth for the phase value p should be 608.5 nm. Fig. 7 shows that the depth of the CDG for the 4 4 combination is Table 1 Calculated and experimental phase values of the 1 5 phase plate [8]. Phase values
Fig. 4. The sectional (y¼ 0) far-field intensity distribution of the combined beam with 5 5 beamlets.
Calculated Experimental
Diffraction orders 2
1
0
1
2
0.91p
0.05p 0
0 0
0.05p 0
0.91p
p
Fig. 5. Schematic diagram of experimental setup for the beam combination for the 4 4 simulated coherent solid-state laser array [8].
p
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Fig. 6. Experimental photographs of (a) the near-field pattern and (b) far-field mainlobe of the combined beam of the simulated 4 4 CSSLA.
Fig. 7. Experimental photographs of (a) the near-field pattern and (b) far-field mainlobe of the combined beam of the simulated 5 5 CSSLA.
Fig. 8. Measured surface profile of the CDG for the simulated 4 4 CSSLA combination.
approximately 574.3 nm. The phase error is about 5%, which, according to our calculation, reduces the beam combining efficiency by 3%. Fig. 9 shows that the depth of the CDG for the 5 5 combination which is approximately 608 nm much better than the 4 4 CDG. Fig. 10 is the sectional surface profiles of the PP used in our experiments for 5 5 beam combination. The depth of the step is 704 nm which means the PP depth error is 15%, which reduces the beam combining efficiency by 5%. These manufacturing errors can be reduced by employing better machine and operators.
3.2. Positioning errors In order to eliminate the phase variations of the combined beam in the image focal plane, the CDG must be carefully positioned so that it matches the sub-period phase structures of the combined beam precisely. We measured the variation of the beam combining efficiency when the CDGs move along the y direction as shown in Figs. 11 and 12 where the discrete ones are the experimental results and the solid lines represent the theoretical results. The combining efficiency varies periodically with the
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Fig. 9. Measured surface profile of the CDG for the simulated 5 5 CSSLA combination.
Fig. 10. Measured surface profile of the PP for the simulated 5 5 CSSLA combination.
Fig. 11. Normalized beam combining efficiency variation as a function of displacement: theoretical (line) and experimental (dot) results for the simulated 4 4 CSSLA combination.
Fig. 12. Normalized beam combining efficiency variation as a function of displacement: theoretical (line) and experimental (asterisk) results for the simulated 5 5 CSSLA combination.
translation distance. And the experimental results match quite well with the theoretical results. To have an experimental efficiency that is above 90% of the peak value, the displacement error has to be less than, for the 4 4 combination 0.03 T and for
the 5 5 combination 0.1 T, respectively. Notice that the minima of the experimental results are much higher than that of the theoretical results. This indicates the existence of background noises in the experiments.
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4. Conclusions
Fig. 13. Experimental beam combining efficiency versus rotation angle for the simulated 4 4 CSSLA combination.
Theoretical analysis and experimental results of the simulations show that conjugate Dammann grating based coherent beam combining technique is completely suitable for coherent solid-state laser array combination. By adjusting the period of the grating and the focal length of the Fourier lens, this technique can be used for beam combining systems of different array periods and beam apertures. Square aperture that could increase the central lobe intensity by 21.5% at most is used here. Error analysis shows that the manufacturing and positioning of the conjugate grating have a great influence on the beam combining efficiency, however, this influence can be eliminated effectively by proper mechanical design and careful operation. The work described in this paper will prove valuable for the development of all-solid high power laser systems.
Acknowledgments The authors acknowledge the support of the National Nature Science Foundation of China (No. 60907006 and No. 61108069) and the Key Laboratory of Space Laser Communication and Testing Technology of Chinese Academy of Sciences.
References
Fig. 14. Experimental beam combining efficiency versus rotation angle for the simulated 5 5 CSSLA combination.
Also, when the CDG has an angle to the image focal plane of the FL, its projection on the focal plane is smaller than the actual size of the properly designed CDG. The phase variations will not be eliminated completely which leads to a reduction in the beam combining efficiency. Figs. 13 and 14 are the experimental results of the beam combining efficiency for 4 4 and 5 5 simulations, respectively, when the CDGs have a slight angle to the image focal plane. The beam combining efficiency drops concussively when the angle of the CDGs to the image focal plane increases. To have an experimental efficiency that is above 90% of the peak value, the angle error have to be less than, for the 4 4 combination 0.001 rad and for the 5 5 combination 0.0005 rad, respectively. Finally, when the CDGs move back and forth along the optical axis away from the focal plane for about 2 cm, and since the system has a large scene depth, there is no significant change of the beam combining efficiency. These positioning errors can be reduced by proper mechanical design of the supporting structures.
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