Abstract

This study aims to demonstrate the effectiveness of sets of microlens arrays to homogenize and finely shape focal spots, for optimizing optical pumps of interest for Yb-doped slabs in the field of high-power lasers, or for use in materials processing and industrial applications. Homogenization basically involves optical integration in the Fourier plane of a large focusing lens, the input light source being consistently sampled. The generic capabilities of three complementary optical designs are demonstrated thanks to combining transfer-matrix computations with highly resolved ray-tracing modeling. This helps to benchmark a variety of experimental results. Fully generic trends are evidenced in terms of the shaping capabilities, for highly efficient focusing, in the form either of a single spot or of multiple sub-spots. Peak power densities as elevated as some hundreds of kW/cm2 or a few MW/cm2, respectively, may be produced by a couple of multi-kilowatt diode stacks.

© 2019 Optical Society of America

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References

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2018 (3)

W. Yuan, L.-H. Li, W.-B. Lee, and C.-Y. Chan, “Fabrication of microlens array and its application: a review,” Chin. J. Mech. Eng. 31, 16 (2018).
[Crossref]

X. Chen, Y. Song, W. Zhang, M. Sulaman, S. Zhao, B. Guo, Q. Hao, and L. Li, “Imaging method based on the combination of microlens arrays and aperture arrays,” Appl. Opt. 57, 5392–5398 (2018).
[Crossref]

A. Laskin, J. Volpp, V. Laskin, and A. Ostrun, “Beam shaping of focused radiation of multimode lasers,” Proc. SPIE 10525, 1052507 (2018).
[Crossref]

2017 (4)

2016 (1)

W. Choi, R. Shin, J. Lim, and S. Kang, “Design methodology for a confocal imaging system using an objective microlens array with an increased working distance,” Sci. Rep. 6, 33278 (2016).
[Crossref]

2015 (2)

Z. Wang, G. Zhu, Y. Huang, X. Zhu, and C. Zhu, “Analytical model of microlens array system homogenizer,” Opt. Laser Technol. 75, 214–220 (2015).
[Crossref]

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

2014 (1)

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

2012 (1)

2010 (1)

R. Stevens and T. Miyashita, “Review of standards for microlenses and microlens arrays,” J. Imaging Sci. 58, 202–212 (2010).
[Crossref]

2008 (2)

R. Voelkel and K. J. Weible, “Laser beam homogenizing: limitations and constraints,” Proc. SPIE 7102, 71020J (2008).
[Crossref]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[Crossref]

2007 (3)

2006 (1)

2004 (1)

K. F. Kleine and K. G. Watkins, “Pulse shaping for micro cutting applications of metals with fiber lasers,” Proc. SPIE 5339, 510–517 (2004).
[Crossref]

2002 (1)

N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A 4, S1–S9 (2002).
[Crossref]

2001 (1)

N. Lindlein and H. P. Herzig, “Design and modeling of a miniature system containing micro-optics,” Proc. SPIE 4437, 1–13 (2001).
[Crossref]

2000 (1)

L. Li, “The advances and characteristics of high-power diode laser materials processing,” Opt. Lasers Eng. 34, 231–253 (2000).
[Crossref]

Arai, J.

Aschke, L.

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[Crossref]

Bayer, A.

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[Crossref]

Beyrau, F.

S. Pfadler, M. Löffler, F. Beyrau, and A. Leipertz, “Improvement of planar laser diagnostics by the application of a beam homogenizer,” J. Phys. Conf. Ser. 85, 012010 (2007).
[Crossref]

Bich, A.

M. Zimmermann, M. Schmidt, A. Bich, and R. Voelkel, “Refractive micro-optics for multi-spot and multi-line generation,” in 9th International Symposium on Laser Precision Microfabrication (LPM) (2008).

Brauer, A.

Caley, A. J.

Cao, J.

Chan, C.-Y.

W. Yuan, L.-H. Li, W.-B. Lee, and C.-Y. Chan, “Fabrication of microlens array and its application: a review,” Chin. J. Mech. Eng. 31, 16 (2018).
[Crossref]

Chen, X.

Cheng, Y.

Choi, W.

W. Choi, R. Shin, J. Lim, and S. Kang, “Design methodology for a confocal imaging system using an objective microlens array with an increased working distance,” Sci. Rep. 6, 33278 (2016).
[Crossref]

Dannberg, P.

Doblerd, M.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

Eiblc, F.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

Esser, D.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Ferrarioa, F.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

Fritsche, H.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

González, A. B.

A. B. González and J. Pozo, “Optical beam shaping: unmet needs in laser materials processing,” Opt. Photon. 12, 15–17 (2017).
[Crossref]

Graf, T.

Griesa, W.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

Grohea, A.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

Guo, B.

Han, S.

Hao, Q.

Herzig, H. P.

N. Lindlein and H. P. Herzig, “Design and modeling of a miniature system containing micro-optics,” Proc. SPIE 4437, 1–13 (2001).
[Crossref]

Höfer, M.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Hoffmann, D.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Homburg, O.

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[Crossref]

Huang, Y.

Z. Wang, G. Zhu, Y. Huang, X. Zhu, and C. Zhu, “Analytical model of microlens array system homogenizer,” Opt. Laser Technol. 75, 214–220 (2015).
[Crossref]

Kang, S.

W. Choi, R. Shin, J. Lim, and S. Kang, “Design methodology for a confocal imaging system using an objective microlens array with an increased working distance,” Sci. Rep. 6, 33278 (2016).
[Crossref]

Kawai, H.

Kleine, K. F.

K. F. Kleine and K. G. Watkins, “Pulse shaping for micro cutting applications of metals with fiber lasers,” Proc. SPIE 5339, 510–517 (2004).
[Crossref]

Kocha, R.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

Kohld, S.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

Kruschea, B.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

Laskin, A.

A. Laskin, J. Volpp, V. Laskin, and A. Ostrun, “Beam shaping of focused radiation of multimode lasers,” Proc. SPIE 10525, 1052507 (2018).
[Crossref]

Laskin, V.

A. Laskin, J. Volpp, V. Laskin, and A. Ostrun, “Beam shaping of focused radiation of multimode lasers,” Proc. SPIE 10525, 1052507 (2018).
[Crossref]

Lee, W.-B.

W. Yuan, L.-H. Li, W.-B. Lee, and C.-Y. Chan, “Fabrication of microlens array and its application: a review,” Chin. J. Mech. Eng. 31, 16 (2018).
[Crossref]

Leipertz, A.

S. Pfadler, M. Löffler, F. Beyrau, and A. Leipertz, “Improvement of planar laser diagnostics by the application of a beam homogenizer,” J. Phys. Conf. Ser. 85, 012010 (2007).
[Crossref]

Li, L.

Li, L.-H.

W. Yuan, L.-H. Li, W.-B. Lee, and C.-Y. Chan, “Fabrication of microlens array and its application: a review,” Chin. J. Mech. Eng. 31, 16 (2018).
[Crossref]

Lim, J.

W. Choi, R. Shin, J. Lim, and S. Kang, “Design methodology for a confocal imaging system using an objective microlens array with an increased working distance,” Sci. Rep. 6, 33278 (2016).
[Crossref]

Lindlein, N.

N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A 4, S1–S9 (2002).
[Crossref]

N. Lindlein and H. P. Herzig, “Design and modeling of a miniature system containing micro-optics,” Proc. SPIE 4437, 1–13 (2001).
[Crossref]

Liu, J.

Löffler, M.

S. Pfadler, M. Löffler, F. Beyrau, and A. Leipertz, “Improvement of planar laser diagnostics by the application of a beam homogenizer,” J. Phys. Conf. Ser. 85, 012010 (2007).
[Crossref]

Löhring, J.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Loosen, P.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Luttmann, J.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Meinschien, J.

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[Crossref]

Meissner, A.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Michalowski, A.

Mitra, T.

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[Crossref]

Miyashita, T.

R. Stevens and T. Miyashita, “Review of standards for microlenses and microlens arrays,” J. Imaging Sci. 58, 202–212 (2010).
[Crossref]

Ni, X.

Okano, F.

Ostrun, A.

A. Laskin, J. Volpp, V. Laskin, and A. Ostrun, “Beam shaping of focused radiation of multimode lasers,” Proc. SPIE 10525, 1052507 (2018).
[Crossref]

Pahla, U.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

Pauwels, J.

Petrov, N. I.

Petrova, G. N.

Pfadler, S.

S. Pfadler, M. Löffler, F. Beyrau, and A. Leipertz, “Improvement of planar laser diagnostics by the application of a beam homogenizer,” J. Phys. Conf. Ser. 85, 012010 (2007).
[Crossref]

Pfluegerb, S.

H. Fritsche, F. Ferrarioa, R. Kocha, B. Kruschea, U. Pahla, S. Pfluegerb, A. Grohea, W. Griesa, F. Eiblc, S. Kohld, and M. Doblerd, “Direct diode lasers and their advantages for material processing and other applications,” Proc. SPIE 9356, 93560I (2015).
[Crossref]

Poprawe, R.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Pozo, J.

A. B. González and J. Pozo, “Optical beam shaping: unmet needs in laser materials processing,” Opt. Photon. 12, 15–17 (2017).
[Crossref]

Qin, Y.

Russbueldt, P.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Sartorius, T.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Schmidt, M.

M. Zimmermann, M. Schmidt, A. Bich, and R. Voelkel, “Refractive micro-optics for multi-spot and multi-line generation,” in 9th International Symposium on Laser Precision Microfabrication (LPM) (2008).

Shin, R.

W. Choi, R. Shin, J. Lim, and S. Kang, “Design methodology for a confocal imaging system using an objective microlens array with an increased working distance,” Sci. Rep. 6, 33278 (2016).
[Crossref]

Sinzinger, S.

Song, Y.

Stevens, R.

R. Stevens and T. Miyashita, “Review of standards for microlenses and microlens arrays,” J. Imaging Sci. 58, 202–212 (2010).
[Crossref]

Sulaman, M.

Taghizadeh, M. R.

Thomson, M. J.

Traub, M.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Ullah, N.

Verschaffelt, G.

Voelkel, R.

R. Voelkel and K. J. Weible, “Laser beam homogenizing: limitations and constraints,” Proc. SPIE 7102, 71020J (2008).
[Crossref]

M. Zimmermann, M. Schmidt, A. Bich, and R. Voelkel, “Refractive micro-optics for multi-spot and multi-line generation,” in 9th International Symposium on Laser Precision Microfabrication (LPM) (2008).

Volpp, J.

A. Laskin, J. Volpp, V. Laskin, and A. Ostrun, “Beam shaping of focused radiation of multimode lasers,” Proc. SPIE 10525, 1052507 (2018).
[Crossref]

Waddie, A. J.

Wang, L.

Wang, Z.

Z. Wang, G. Zhu, Y. Huang, X. Zhu, and C. Zhu, “Analytical model of microlens array system homogenizer,” Opt. Laser Technol. 75, 214–220 (2015).
[Crossref]

Watkins, K. G.

K. F. Kleine and K. G. Watkins, “Pulse shaping for micro cutting applications of metals with fiber lasers,” Proc. SPIE 5339, 510–517 (2004).
[Crossref]

Weber, R.

Weible, K. J.

R. Voelkel and K. J. Weible, “Laser beam homogenizing: limitations and constraints,” Proc. SPIE 7102, 71020J (2008).
[Crossref]

Weitenberg, J.

P. Russbueldt, D. Hoffmann, M. Höfer, J. Löhring, J. Luttmann, A. Meissner, J. Weitenberg, M. Traub, T. Sartorius, D. Esser, R. Wester, P. Loosen, and R. Poprawe, “Innoslab amplifiers,” IEEE J. Sel. Top. Quantum Electron. 21, 3100117 (2014).
[Crossref]

Wester, R.

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Z. Wang, G. Zhu, Y. Huang, X. Zhu, and C. Zhu, “Analytical model of microlens array system homogenizer,” Opt. Laser Technol. 75, 214–220 (2015).
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M. Zimmermann, M. Schmidt, A. Bich, and R. Voelkel, “Refractive micro-optics for multi-spot and multi-line generation,” in 9th International Symposium on Laser Precision Microfabrication (LPM) (2008).

Appl. Opt. (3)

Chin. J. Mech. Eng. (1)

W. Yuan, L.-H. Li, W.-B. Lee, and C.-Y. Chan, “Fabrication of microlens array and its application: a review,” Chin. J. Mech. Eng. 31, 16 (2018).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

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Z. Wang, G. Zhu, Y. Huang, X. Zhu, and C. Zhu, “Analytical model of microlens array system homogenizer,” Opt. Laser Technol. 75, 214–220 (2015).
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M. Zimmermann, M. Schmidt, A. Bich, and R. Voelkel, “Refractive micro-optics for multi-spot and multi-line generation,” in 9th International Symposium on Laser Precision Microfabrication (LPM) (2008).

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Figures (13)

Fig. 1.
Fig. 1. Variations of the optical setup and notations along the pump axis, to define MLA-based, homogenized pumps with focusing capabilities along the directions of fast and slow axes of diode stacks (FA, and/or SA).
Fig. 2.
Fig. 2. Cylindrical MLAs of interest, provided (a) f=7.2mm from the top side and (b) f=19.4mm from perspective with magnification X3.
Fig. 3.
Fig. 3. SID-based RTM results to determine (a) the variations of the spot size and divergence, for (b) optimized or (c) partially unmatched seed configurations in the presence of residual misalignment (σ2ndMLA).
Fig. 4.
Fig. 4. RID-based RTM results to determine the variations of the (a) spot size and (b) divergence versus (c) the additional degrees of freedom provided {di,f3}, and the specification of MLAs in the front end.
Fig. 5.
Fig. 5. Non-homogenized focal spot of reference with (a) FoFA=15cm and FoFA=25cm; NID-based focal spots using a MLA with f=7.2mm, (b) along the SA and (c) along the FA; and (d) associated profiles.
Fig. 6.
Fig. 6. (a) Non-homogenized focal spot of reference with FoFA=15cm and FoFA=25cm; (b) SID-based homogenization along the FA with fFA=7.2mm, or (c) of SA with fSA=19.4mm; (d) orthogonally combined SID-based homogenization along the FA and the SA with {fFA=7.2mm,fSA=19.4mm}.
Fig. 7.
Fig. 7. RID along the SA with f1SA,f2SA=7.2mm and f3SA=19.4mm, varying the distance di between the second and third MLAs in the range of (a) 1 cm, (b) 2 cm, and (c) 5 cm, and (d) associated profiles.
Fig. 8.
Fig. 8. RID along the FA with f1SA,f2SA=7.2mm and f3SA=19.4mm, varying di between the second and the third MLAs from 10 to 30 mm to produce series of (a) three or of (b) six sub-spots, and (c) associated profiles.
Fig. 9.
Fig. 9. (a) Meshing conditions and (b) RTM-based maps with SID, while ranging Fo from 16.8 to 9.6 cm to ensure a single focal spot, or (c) with RID using {Fo=16.8cm,d3=24cm} to evidence conditions to produce multiple sub-spots in the Fourier plane.
Fig. 10.
Fig. 10. Variations of NID-based focal spots, accordingly to RTM, when varying the seed divergence from (a) zero to (b) 8 mrad (b).
Fig. 11.
Fig. 11. Variations of SID-based focal spots, accordingly to RTM, when varying the seed divergence from (a) zero to (b) 8 mrad.
Fig. 12.
Fig. 12. Zoom on RID-based, former RTM results with Fo=16.8cm, evidencing the characteristics of sub-spots along the pump axis, provided (a) d3=1cm, (b) d3=1.5cm, (c) d3=3cm, (d) d3=5cm, (e) d3=10cm, and (f) d3=20cm. For scaling, refer to the size and spacing of cascaded screens, respectively 7.2 mm (vertical) and 2 mm (horizontal).
Fig. 13.
Fig. 13. Variations of the transverse profile across the pump spot versus d3, besides the focal plane, as computed from ray-tracing results for the RID option (a) with and (b) without stack divergence.

Equations (11)

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ΔxCentral_microlensNID=pFof.
ΔxCentral_microlensID=pFo(f1+f2dID)f1f2.
M1=[10Fo1],M2=[11Fo01],MFourier=[10Fo1],VS=[0np].
Vin=[αinxin]=[p2fp2],Vout=[αoutxout],
VoutSID=MOut·[VS+MOpticalSystemID·VinSID],MOpticalSystemSID=M4·M5·M4,
VoutRID=MOut·[VS+MOpticalSystemRID·VinRID],MOpticalSystemRID=M4·M5·M6·M7·M6.
αoutSID=1Fo[np+f(nσdoαstack)]+p2f(DFo1),
xoutSID=pFo2f.
αoutSID=p[1Fo(n12)+12f(DFo1)].
αoutRID=pf(1DFO)[12(1dif3)(1dif3f+1f3)fff3]pFof[dinf+(1dif)(f+12)],
xoutRID=pFo2f[1dif32(1dif3f+1f3)ff3].