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In this paper, we propose a quantitative approach to analyze the influence of pupil truncation on the phase-only modulation laser beam shaping system, based on the near-field phase and the far-field metric functions. First, the relationship between near-field phase and pupil radius is studied by Lagrange multiplier method. Result indicates that both the peak-to-valley and the root-mean-square of the near-field phase increase approximately linearly with the pupil radius. Second, the influence of pupil radius on a beam shaping system is investigated. To quantify the performance of the beam shaping system, the correlation coefficient (C) and the mean square difference (MSD) are introduced as the metric functions. Then, by comparing the metric functions at different pupil radius, it is shown that the pupil radius influences the performance of focal beam shaping distinctly at the lower pupil radius, whereas the influence trails off, and both the C and the MSD get close to the theoretical limit as the pupil radius continuously increases. Third, the mathematical models of the C and the MSD are proposed to reveal the relationship among the metric functions, pupil radius and target intensity's size, as it is difficult to obtain the explicit expressions on the basis of metric functions' definition. And the three coefficients in each model are ascertained by surface fitting method based on the sampling data. In addition, SSE (sum of square due to error), RMSE (root mean square error) and R-square (coefficient of determination) are adopted to determine the fitting precision. For both the metric functions, the precision of SSE and RMSE can reach 10-2 and the R-square is shown to be more than 97%. The SSE, RMSE and R-square verify the proposed mathematical models. Finally, according to the models, we analyze when the influence of pupil truncation becomes negligible for the rectangle or circle target intensity. In practice, the size of target intensity is determined first. Sequentially, by combining the mathematical models and their first-order partial differentials, the changing regularity of metric functions with respect to pupil radius is studied. Meanwhile, the regularity helps us to find the beginning points for rectangle target and circle target intensities respectively. For the rectangle target intensity, when the pupil radius is 2.5 times that of the Gaussian waist radius, the metric functions become stable. The C with a value of 0.997 and the MSD with a value of 410-4 are both close to the theoretical limit. In the meantime, the influence of pupil truncation tends to be minimal as expected. For circle target intensity, when the pupil radius is 3 times that of the Gaussian waist radius, the first-order partial differentials of the C and the MSD decrease to about 10-3. This means that the metric functions begin to converge and that the influence of pupil truncation tends to be minimal at this point. Consequently, it is effective and meaningful to determine the best pupil radius using the proposed models in the article when designing a beam shaping system. Moreover, the models can also be used to evaluate the performance of a laser beam shaping system.
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Keywords:
- laser beam shaping /
- metric functions /
- fitting formula /
- pupil radius
[1] Lan X J 2005 Laser Technology (2nd Ed.) (Beijing: Science Press) p1 (in Chinese) [蓝信钜 2005 激光技术(第二版) (北京:科学出版社) 第1页]
[2] Xie Y J, Liu J R, Zhao X Q 2001 Laser Technol. 25 454 (in Chinese) [谢永杰, 刘晶儒, 赵雪庆 2001 激光技术 25 454]
[3] Walker E P, Milster T D 2001 Proc. SPIE 4443 73
[4] Dickey F M, Holswade S C, Shealy D L 2006 Laser Beam Shaping Applications (Boca Raton: CRC Press) pp182-208
[5] Rapp L, Constantinescu C, Larmande Y, Diallo A K, Videlot-Ackermann C, Delaporte P, Alloncle A P 2015 Sensor. Actuat. A: Phys. 224 111
[6] Mauclair C, Pietroy D, Maio Y D, Baubeau E, Colombier J P, Stoian R, Pigeon F 2015 Opt. Laser Eng. 67 212
[7] Frieden B R 1965 Appl. Opt. 4 1400
[8] Dickey F M, Holswade S C 2000 Laser Beam Shaping: Theory and Techniques (New York: Marcel Dekker Inc) pp1-4
[9] Mohammed W, Gu X J 2009 Appl. Opt. 48 2249
[10] Chen H X, Sui Z, Chen Z P, An B, Li M Z 2001 Acta Opt. Sin. 21 1107 (in Chinese) [陈怀新, 隋展, 陈祯陪, 安波, 李明中 2001 光学学报 21 1107]
[11] Hoffnagle J A, Jefferson C M 2000 Appl. Opt. 39 5488
[12] Hoffnagle J A, Jefferson C M 2003 Opt. Eng. 42 3090
[13] Jahan S R, Karim M A 1989 Opt. Laser Technol. 21 27
[14] Geng Y C, Liu L Q, Wang W Y, Zhang Y, Huang W Q, Su J Q, Li P 2013 Acta Phys. Sin. 62 145201 (in Chinese) [耿远超, 刘兰琴, 王文义, 张颖, 黄晚晴, 粟敬钦, 李平 2013 物理学报 62 145201]
[15] Jiang X J, Zhou S L, Lin Z Q 2007 J. Appl. Phys. 101 023109
[16] Yin Z Y, Wang Y F, Yin S Y, Qiang J P, Lei C Q, Sun X H, Yang K 2013 Chin. J. Lasers 40 0602016 (in Chinese) [殷智勇, 汪越峰, 尹韶云, 强继平, 雷呈强, 孙秀辉, 杨凯 2013 中国激光 40 0602016]
[17] Eismann M T, Tai A M, Cederquist J N 1989 Appl. Opt. 28 2641
[18] Liu J S, Taghizadeh M R 2002 Opt. Lett. 27 1463
[19] Ji Y, Zhang J J, Yao D C, Chen Y S 1996 Acta Phys. Sin. 45 2027 (in Chinese) [姬扬, 张静娟, 姚德成, 陈岩松 1996 物理学报 45 2027]
[20] Nemoto K, Fujii T, Goto N, Nayuki T 1996 Opt. Lett. 21 168
[21] Romero L A, Dickey F M 1996 J. Opt. Soc. Am. A 13 751
[22] Dickey F M, Holswade S C 1996 Opt. Eng. 35 3285
[23] Han J, Bayanheshig, Li W H 2012 Acta Phys. Sin. 61 084202 (in Chinese) [韩建, 巴音贺希格, 李文昊 2012 物理学报 61 084202]
[24] Zhao T Y, Liu Q X, Yu F H 2012 Chin. Phys. B 21 064203
[25] Born M, Wolf E (translated by Yang J S) 2006 Principles of Optics (Beijing: Electronic Industry Press) pp740-746 (in Chinese) [玻恩 M, 沃耳夫E 著 (杨葭荪 译) 2006 光学原理 (北京: 电子工业出版社) 第740746页]
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[1] Lan X J 2005 Laser Technology (2nd Ed.) (Beijing: Science Press) p1 (in Chinese) [蓝信钜 2005 激光技术(第二版) (北京:科学出版社) 第1页]
[2] Xie Y J, Liu J R, Zhao X Q 2001 Laser Technol. 25 454 (in Chinese) [谢永杰, 刘晶儒, 赵雪庆 2001 激光技术 25 454]
[3] Walker E P, Milster T D 2001 Proc. SPIE 4443 73
[4] Dickey F M, Holswade S C, Shealy D L 2006 Laser Beam Shaping Applications (Boca Raton: CRC Press) pp182-208
[5] Rapp L, Constantinescu C, Larmande Y, Diallo A K, Videlot-Ackermann C, Delaporte P, Alloncle A P 2015 Sensor. Actuat. A: Phys. 224 111
[6] Mauclair C, Pietroy D, Maio Y D, Baubeau E, Colombier J P, Stoian R, Pigeon F 2015 Opt. Laser Eng. 67 212
[7] Frieden B R 1965 Appl. Opt. 4 1400
[8] Dickey F M, Holswade S C 2000 Laser Beam Shaping: Theory and Techniques (New York: Marcel Dekker Inc) pp1-4
[9] Mohammed W, Gu X J 2009 Appl. Opt. 48 2249
[10] Chen H X, Sui Z, Chen Z P, An B, Li M Z 2001 Acta Opt. Sin. 21 1107 (in Chinese) [陈怀新, 隋展, 陈祯陪, 安波, 李明中 2001 光学学报 21 1107]
[11] Hoffnagle J A, Jefferson C M 2000 Appl. Opt. 39 5488
[12] Hoffnagle J A, Jefferson C M 2003 Opt. Eng. 42 3090
[13] Jahan S R, Karim M A 1989 Opt. Laser Technol. 21 27
[14] Geng Y C, Liu L Q, Wang W Y, Zhang Y, Huang W Q, Su J Q, Li P 2013 Acta Phys. Sin. 62 145201 (in Chinese) [耿远超, 刘兰琴, 王文义, 张颖, 黄晚晴, 粟敬钦, 李平 2013 物理学报 62 145201]
[15] Jiang X J, Zhou S L, Lin Z Q 2007 J. Appl. Phys. 101 023109
[16] Yin Z Y, Wang Y F, Yin S Y, Qiang J P, Lei C Q, Sun X H, Yang K 2013 Chin. J. Lasers 40 0602016 (in Chinese) [殷智勇, 汪越峰, 尹韶云, 强继平, 雷呈强, 孙秀辉, 杨凯 2013 中国激光 40 0602016]
[17] Eismann M T, Tai A M, Cederquist J N 1989 Appl. Opt. 28 2641
[18] Liu J S, Taghizadeh M R 2002 Opt. Lett. 27 1463
[19] Ji Y, Zhang J J, Yao D C, Chen Y S 1996 Acta Phys. Sin. 45 2027 (in Chinese) [姬扬, 张静娟, 姚德成, 陈岩松 1996 物理学报 45 2027]
[20] Nemoto K, Fujii T, Goto N, Nayuki T 1996 Opt. Lett. 21 168
[21] Romero L A, Dickey F M 1996 J. Opt. Soc. Am. A 13 751
[22] Dickey F M, Holswade S C 1996 Opt. Eng. 35 3285
[23] Han J, Bayanheshig, Li W H 2012 Acta Phys. Sin. 61 084202 (in Chinese) [韩建, 巴音贺希格, 李文昊 2012 物理学报 61 084202]
[24] Zhao T Y, Liu Q X, Yu F H 2012 Chin. Phys. B 21 064203
[25] Born M, Wolf E (translated by Yang J S) 2006 Principles of Optics (Beijing: Electronic Industry Press) pp740-746 (in Chinese) [玻恩 M, 沃耳夫E 著 (杨葭荪 译) 2006 光学原理 (北京: 电子工业出版社) 第740746页]
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