Abstract

We report on a beam characterization method that is based on the simultaneous measurement of the focus field and the farfield, thus avoiding problems with beam fluctuations during the measurement. By using reflections from both sides of a planoconvex lens, the method implements two branches of an optical system working simultaneously. Also, by letting the planoconvex lens be antireflection treated, and by allowing for both of the reflected fields to fill large and approximately equal areas on a camera detector array, the method significantly lowers the intensity onto the detector array, thus minimizing the need for additional disturbing attenuation filters to avoid camera saturation. In the numerical retrieval of the phase distribution, based on the measured intensity distributions of the focus and farfield, iterative propagation between the two branches is performed. The phase retrieval uses the two-step algorithm for the numerical field propagation conveniently providing an arbitrary choice of sampling distance in each plane.

© 2011 Optical Society of America

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    [CrossRef]
  4. J. Chilla, Q.-Z. Shu, H. Zhou, E. Weiss, M. Reed, and L. Spinelli, “Recent advances in optically pumped semiconductor lasers,” Proc. SPIE 6451, 645109 (2007).
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    [PubMed]
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    [CrossRef]
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2010

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

2008

2007

J. Chilla, Q.-Z. Shu, H. Zhou, E. Weiss, M. Reed, and L. Spinelli, “Recent advances in optically pumped semiconductor lasers,” Proc. SPIE 6451, 645109 (2007).
[CrossRef]

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack–Hartmann wavefront sensor and M2-sensor: Comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[CrossRef]

2006

2003

L. Bruel, “Numerical phase retrieval from beam intensity measurements in three planes,” Proc. SPIE 4932, 590–598(2003).
[CrossRef]

2000

1999

S. Matsuoka and K. Yamakawa, “Wavefront reconstruction from intensity measurements using Fresnel phase retrieval method,” Jpn. J. Appl. Phys. 38, L1183–L1185 (1999).
[CrossRef]

1998

M. G. Lofdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II telescope,” Proc. SPIE 3356, 1190–1201 (1998).
[CrossRef]

T. Johnston, Jr., “Beam propagation (M2) measurement made as easy as it gets: The four-cuts method,” Appl. Opt. 37, 4840–4850 (1998).
[CrossRef]

1997

1996

J. Sandusky and S. Brueck, “A CW external-cavity surface-emitting laser,” IEEE Photonics Technol. Lett. 8, 313–315(1996).
[CrossRef]

1993

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” Proc. SPIE 1868, 2 (1993).
[CrossRef]

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef] [PubMed]

1992

A. Dooghin, N. Kundikova, and B. Zel’dovich, “Phase retrieval from laser beam intensity profiles,” Opt. Commun. 91, 193–196 (1992).
[CrossRef]

1990

1982

1972

R. Gerchberg and W. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Acton, D. S.

M. G. Lofdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II telescope,” Proc. SPIE 3356, 1190–1201 (1998).
[CrossRef]

Bellancourt, A.-R.

Bengtsson, J.

Bihari, B.

Brueck, S.

J. Sandusky and S. Brueck, “A CW external-cavity surface-emitting laser,” IEEE Photonics Technol. Lett. 8, 313–315(1996).
[CrossRef]

Bruel, L.

L. Bruel, “Numerical phase retrieval from beam intensity measurements in three planes,” Proc. SPIE 4932, 590–598(2003).
[CrossRef]

Burns, D.

P. Roth, A. Maclean, A. Kemp, S. Calvez, M. Dawson, and D. Burns, “Efficiency and beam quality analysis of a semiconductor disk laser,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies 2008 (CLEO 2008), Technical Digest (Optical Society of America, 2008), paper CWD3.
[PubMed]

Caley, A.

J. Liu, A. Caley, and M. Taghizadeh, “Symmetrical iterative Fourier-transform algorithm using both phase and amplitude freedoms,” Opt. Commun. 267, 347–355 (2006).
[CrossRef]

Calvez, S.

P. Roth, A. Maclean, A. Kemp, S. Calvez, M. Dawson, and D. Burns, “Efficiency and beam quality analysis of a semiconductor disk laser,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies 2008 (CLEO 2008), Technical Digest (Optical Society of America, 2008), paper CWD3.
[PubMed]

Chatterjee, S.

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

Chen, R. T.

Cherezova, T. Y.

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack–Hartmann wavefront sensor and M2-sensor: Comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[CrossRef]

Chernikov, A.

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

Chilla, J.

J. Chilla, Q.-Z. Shu, H. Zhou, E. Weiss, M. Reed, and L. Spinelli, “Recent advances in optically pumped semiconductor lasers,” Proc. SPIE 6451, 645109 (2007).
[CrossRef]

Dawson, M.

P. Roth, A. Maclean, A. Kemp, S. Calvez, M. Dawson, and D. Burns, “Efficiency and beam quality analysis of a semiconductor disk laser,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies 2008 (CLEO 2008), Technical Digest (Optical Society of America, 2008), paper CWD3.
[PubMed]

Deng, X.

Dooghin, A.

A. Dooghin, N. Kundikova, and B. Zel’dovich, “Phase retrieval from laser beam intensity profiles,” Opt. Commun. 91, 193–196 (1992).
[CrossRef]

Duncan, A. L.

M. G. Lofdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II telescope,” Proc. SPIE 3356, 1190–1201 (1998).
[CrossRef]

Fienup, J. R.

Gan, J.

Gerchberg, R.

R. Gerchberg and W. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Gini, E.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Hader, J.

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

Harwit, A.

M. G. Lofdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II telescope,” Proc. SPIE 3356, 1190–1201 (1998).
[CrossRef]

Hoffmann, M.

Johnston, T.

Jonas, S.

S. Jonas, A. Kuczewski, and C. Thorn, “An improved phase retrieval algorithm for coherent beams,” in Conference on Lasers and Electro-Optics (CLEO 2001), Technical Digest (Optical Society of America, 2001), paper CThL35.

Kaneda, Y.

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

Keller, U.

Kemp, A.

P. Roth, A. Maclean, A. Kemp, S. Calvez, M. Dawson, and D. Burns, “Efficiency and beam quality analysis of a semiconductor disk laser,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies 2008 (CLEO 2008), Technical Digest (Optical Society of America, 2008), paper CWD3.
[PubMed]

Kendrick, R. L.

M. G. Lofdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II telescope,” Proc. SPIE 3356, 1190–1201 (1998).
[CrossRef]

Koch, S. W.

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

Kuczewski, A.

S. Jonas, A. Kuczewski, and C. Thorn, “An improved phase retrieval algorithm for coherent beams,” in Conference on Lasers and Electro-Optics (CLEO 2001), Technical Digest (Optical Society of America, 2001), paper CThL35.

Kudryashov, A. V.

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack–Hartmann wavefront sensor and M2-sensor: Comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[CrossRef]

Kundikova, N.

A. Dooghin, N. Kundikova, and B. Zel’dovich, “Phase retrieval from laser beam intensity profiles,” Opt. Commun. 91, 193–196 (1992).
[CrossRef]

Kunert, B.

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

Liu, J.

J. Liu, A. Caley, and M. Taghizadeh, “Symmetrical iterative Fourier-transform algorithm using both phase and amplitude freedoms,” Opt. Commun. 267, 347–355 (2006).
[CrossRef]

Lofdahl, M. G.

M. G. Lofdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II telescope,” Proc. SPIE 3356, 1190–1201 (1998).
[CrossRef]

Maas, D. J. H. C.

Maclean, A.

P. Roth, A. Maclean, A. Kemp, S. Calvez, M. Dawson, and D. Burns, “Efficiency and beam quality analysis of a semiconductor disk laser,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies 2008 (CLEO 2008), Technical Digest (Optical Society of America, 2008), paper CWD3.
[PubMed]

Matsuoka, S.

S. Matsuoka and K. Yamakawa, “Wave-front measurements of terawatt-class ultrashort laser pulses by the Fresnel phase-retrieval method,” J. Opt. Soc. Am. B 17, 663–667 (2000).
[CrossRef]

S. Matsuoka and K. Yamakawa, “Wavefront reconstruction from intensity measurements using Fresnel phase retrieval method,” Jpn. J. Appl. Phys. 38, L1183–L1185 (1999).
[CrossRef]

Mitchell, K. E.

M. G. Lofdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II telescope,” Proc. SPIE 3356, 1190–1201 (1998).
[CrossRef]

Moloney, J. V.

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

Paxman, R. G.

M. G. Lofdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II telescope,” Proc. SPIE 3356, 1190–1201 (1998).
[CrossRef]

Reed, M.

J. Chilla, Q.-Z. Shu, H. Zhou, E. Weiss, M. Reed, and L. Spinelli, “Recent advances in optically pumped semiconductor lasers,” Proc. SPIE 6451, 645109 (2007).
[CrossRef]

Roth, P.

P. Roth, A. Maclean, A. Kemp, S. Calvez, M. Dawson, and D. Burns, “Efficiency and beam quality analysis of a semiconductor disk laser,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies 2008 (CLEO 2008), Technical Digest (Optical Society of America, 2008), paper CWD3.
[PubMed]

Rudin, B.

Rutz, A.

Rydberg, C.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Sandusky, J.

J. Sandusky and S. Brueck, “A CW external-cavity surface-emitting laser,” IEEE Photonics Technol. Lett. 8, 313–315(1996).
[CrossRef]

Saxton, W.

R. Gerchberg and W. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Seldin, J. H.

M. G. Lofdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II telescope,” Proc. SPIE 3356, 1190–1201 (1998).
[CrossRef]

Sheldakova, J. V.

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack–Hartmann wavefront sensor and M2-sensor: Comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[CrossRef]

Shu, Q.-Z.

J. Chilla, Q.-Z. Shu, H. Zhou, E. Weiss, M. Reed, and L. Spinelli, “Recent advances in optically pumped semiconductor lasers,” Proc. SPIE 6451, 645109 (2007).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” Proc. SPIE 1868, 2 (1993).
[CrossRef]

Spinelli, L.

J. Chilla, Q.-Z. Shu, H. Zhou, E. Weiss, M. Reed, and L. Spinelli, “Recent advances in optically pumped semiconductor lasers,” Proc. SPIE 6451, 645109 (2007).
[CrossRef]

Stolz, W.

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

Südmeyer, T.

Taghizadeh, M.

J. Liu, A. Caley, and M. Taghizadeh, “Symmetrical iterative Fourier-transform algorithm using both phase and amplitude freedoms,” Opt. Commun. 267, 347–355 (2006).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Thorn, C.

S. Jonas, A. Kuczewski, and C. Thorn, “An improved phase retrieval algorithm for coherent beams,” in Conference on Lasers and Electro-Optics (CLEO 2001), Technical Digest (Optical Society of America, 2001), paper CThL35.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics, 3rd ed. (Prentice Hall, 1995).

Wang, T.-L.

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

Weiss, E.

J. Chilla, Q.-Z. Shu, H. Zhou, E. Weiss, M. Reed, and L. Spinelli, “Recent advances in optically pumped semiconductor lasers,” Proc. SPIE 6451, 645109 (2007).
[CrossRef]

Wyrowski, F.

Yamakawa, K.

S. Matsuoka and K. Yamakawa, “Wave-front measurements of terawatt-class ultrashort laser pulses by the Fresnel phase-retrieval method,” J. Opt. Soc. Am. B 17, 663–667 (2000).
[CrossRef]

S. Matsuoka and K. Yamakawa, “Wavefront reconstruction from intensity measurements using Fresnel phase retrieval method,” Jpn. J. Appl. Phys. 38, L1183–L1185 (1999).
[CrossRef]

Yarborough, J. M.

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

Zavalova, V. Y.

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack–Hartmann wavefront sensor and M2-sensor: Comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[CrossRef]

Zel’dovich, B.

A. Dooghin, N. Kundikova, and B. Zel’dovich, “Phase retrieval from laser beam intensity profiles,” Opt. Commun. 91, 193–196 (1992).
[CrossRef]

Zhao, F.

Zhou, H.

J. Chilla, Q.-Z. Shu, H. Zhou, E. Weiss, M. Reed, and L. Spinelli, “Recent advances in optically pumped semiconductor lasers,” Proc. SPIE 6451, 645109 (2007).
[CrossRef]

Appl. Opt.

IEEE Photonics Technol. Lett.

J. Sandusky and S. Brueck, “A CW external-cavity surface-emitting laser,” IEEE Photonics Technol. Lett. 8, 313–315(1996).
[CrossRef]

T.-L. Wang, Y. Kaneda, J. M. Yarborough, J. Hader, J. V. Moloney, A. Chernikov, S. Chatterjee, S. W. Koch, B. Kunert, and W. Stolz, “High-power optically pumped semiconductor laser at 1040 nm,” IEEE Photonics Technol. Lett. 22, 661–663 (2010).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

S. Matsuoka and K. Yamakawa, “Wavefront reconstruction from intensity measurements using Fresnel phase retrieval method,” Jpn. J. Appl. Phys. 38, L1183–L1185 (1999).
[CrossRef]

Opt. Commun.

A. Dooghin, N. Kundikova, and B. Zel’dovich, “Phase retrieval from laser beam intensity profiles,” Opt. Commun. 91, 193–196 (1992).
[CrossRef]

J. Liu, A. Caley, and M. Taghizadeh, “Symmetrical iterative Fourier-transform algorithm using both phase and amplitude freedoms,” Opt. Commun. 267, 347–355 (2006).
[CrossRef]

Opt. Lett.

Optik

R. Gerchberg and W. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Proc. SPIE

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack–Hartmann wavefront sensor and M2-sensor: Comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[CrossRef]

J. Chilla, Q.-Z. Shu, H. Zhou, E. Weiss, M. Reed, and L. Spinelli, “Recent advances in optically pumped semiconductor lasers,” Proc. SPIE 6451, 645109 (2007).
[CrossRef]

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” Proc. SPIE 1868, 2 (1993).
[CrossRef]

L. Bruel, “Numerical phase retrieval from beam intensity measurements in three planes,” Proc. SPIE 4932, 590–598(2003).
[CrossRef]

M. G. Lofdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II telescope,” Proc. SPIE 3356, 1190–1201 (1998).
[CrossRef]

Other

J. T. Verdeyen, Laser Electronics, 3rd ed. (Prentice Hall, 1995).

S. Jonas, A. Kuczewski, and C. Thorn, “An improved phase retrieval algorithm for coherent beams,” in Conference on Lasers and Electro-Optics (CLEO 2001), Technical Digest (Optical Society of America, 2001), paper CThL35.

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Figures (10)

Fig. 1
Fig. 1

Sketch of the setup for the simultaneous capture of the focus and farfield of the beam from an OP-SDL. The tilt angle of L2 is greatly exaggerated, and the little mirror used to redirect the light reflected from L2 toward the camera is omitted for clarity. The two branches of the setup are shown unfolded in Fig. 2.

Fig. 2
Fig. 2

The unfolded branches 1 and 2 of the setup in Fig. 1. The reflection in the curved surface of L2 has been drawn as equivalent to propagation through a positive lens.

Fig. 3
Fig. 3

The required beam waist radius ω 0 as a function of the distance s 4 in the setup. The intersection point indicated with a circle is the common solution to Eqs. (2, 3, 4) to achieve simultaneous focus and farfield capture with optimal field diameters on the camera.

Fig. 4
Fig. 4

Outline of the different numerical steps in one iteration of the phase retrieval algorithm.

Fig. 5
Fig. 5

Images on the camera for the case when lens L1 has been correctly positioned. In this case the partial obscuration of the beam leads to a corresponding obscuration in the farfield image on the camera, while there is a global change in the focus image, typical for a field related to the obscured field by a Fourier transform. The OP-SDL was intentionally made to lase in a higher order mode to enhance the difference between the fields when the beam is partially obscured.

Fig. 6
Fig. 6

Convergence of the phase retrieval algorithm. The plotted quantity is the average phase change in a sample position from one iteration to the next, as defined in Eq. (5), for the example shown in Fig. 7. Inset, evolution of the reconstruction error, as defined in Eq. (6), with the number of iterations for the 130 trials used in the Monte Carlo simulation of the accuracy of the M 2 determination.

Fig. 7
Fig. 7

Captured camera image and the intensity distributions for the focus and farfield, together with the intensity distributions calculated with the retrieved phase distribution. The retrieved phase distribution in the farfield is also shown, where the rapidly varying spherical, deterministic part of the starting approximation has been subtracted. Overlayed are contours for the measured (white) and retrieved (black) intensity distributions for 50%, 10%, and 2% of the peak intensity.

Fig. 8
Fig. 8

Same as Fig. 7 but for the case when the external mirror of the OP-SDL was misaligned to favor lasing on a mode with a more complex field distribution.

Fig. 9
Fig. 9

Measured and retrieved intensity distributions, as in Fig. 7 but for the case when a cylindrical lens was inserted before L1 to introduce astigmatism.

Fig. 10
Fig. 10

Left, retrieved phase distribution, as in Fig. 7 but for the case when a cylindrical lens was inserted before L1 to introduce astigmatism. Overlayed are contours for the measured (white) and retrieved (black, partly obscured by the white) intensity distributions for 50%, 10%, and 2% of the peak intensity. Right, phase distribution in the farfield on the camera as calculated with a numerical propagation from the gain element of the OP-SDL through the entire optical system, assuming the field in the cavity of the OP-SDL to be an ideal Gaussian beam. Overlayed are black contour lines for the calculated intensity distribution for 50%, 10%, and 2% of the peak intensity.

Equations (8)

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f r = f 2 ( n 2 1 ) 2 n 2 , laser ,
D 1 = 2 ω 0 1 + ( s 4 + s 5 ) 2 λ 2 π 2 ω 0 4 ,
D 2 = 2 ω 0 s 5 s 4 .
1 f r = 1 s 4 + 1 s 5 .
Δ ϕ q ¯ = m , n ( I q ( m , n ) { arg [ E q ( m , n ) E q 1 ( m , n ) ] } 2 ) m , n I q ( m , n ) ,
ε = m , n | I m ( m , n ) c min I ( m , n ) | 2 m , n | I m ( m , n ) | 2 ,
D 4 σ , x = 4 m , n { I ( m , n ) [ x ( m , n ) x ¯ ] 2 } m , n I ( m , n ) ,
M x 2 = min ( D 4 σ , x ) θ f f , x 4 λ / π .

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