Abstract

We demonstrate operation of a cladding-pumped hybrid ytterbium-doped HOM fiber amplifier and reconversion of the HOM output to Gaussian-like beam by using an axicon based reconversion system. The amplifier was constructed by concatenating single-mode and HOM ytterbium-doped double clad fibers, and was excited by a common multimode pump source. A continuous wave (cw) input signal of 97mW was amplified to 100W at the amplifier output, which yielded a gain of more than 30dB. The LP0,10 output of the HOM amplifier could be converted to a Gaussian-like beam with 67% conversion efficiency. We present, both analytically and numerically, the effects of scaling the beam size on axicon’s apex angle, and how shape imperfections affect the mode converter’s performance.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
    [Crossref] [PubMed]
  2. J. W. Nicholson, J. M. Fini, J. Phillips, A. DeSantolo, X. Liu, K. Feder, V. R. Supradeepa, P. Westbrook, E. Monberg, F. DiMarcello, C. Headley, and D. J. DiGiovanni, “Nanosecond pulse amplification in a 6000 μm2 effective area higher-order mode erbium-doped fiber amplifier,” in Conference on Lasers and Electro-Optics 2012, OSA Technical Digest (online) (Optical Society of America, 2012), paper JTh1I.2.
    [Crossref]
  3. R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
    [Crossref] [PubMed]
  4. J. W. Nicholson, C. Headley, J. Phillips, A. Desantolo, E. Gonzalos, S. Ghalmi, M. F. Yan, P. W. Wisk, D. Trevor, J. Fleming, E. Monberg, F. Dimarcello, R. S. Windeler, J. M. Fini, D. J. DiGiovanni, and S. Ramachandran, “Higher-order-mode fiber amplifiers,” in Lasers, Sources and Related Photonic Devices,” OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper LSWD1.
  5. S. Ramachandran, “Multimode nonlinear fiber optics: New opportunities & challenges,” Tutorial, in Conference on Lasers and Electro-Optics (CLEO) (2018).
  6. K. S. Abedin, R. Ahmad, A. M. DeSantolo, J. W. Nicholson, C. E. Headley, P. S. Westbrook, and D. J. DiGiovanni, “Cladding pumped Yb-doped HOM power amplifier with high gain”, Proc. SPIE 10512, Fiber Lasers XV: Technology and Systems, 105121E (2018).
    [Crossref]
  7. N. Lindlein, G. Leuchs, and S. Ramachandran, “Achieving Gaussian outputs from large-mode-area higher-order-mode fibers,” Appl. Opt. 46(22), 5147–5157 (2007).
    [Crossref] [PubMed]
  8. J. Demas, L. Rishøj, and S. Ramachandran, “Free-space beam shaping for precise control and conversion of modes in optical fiber,” Opt. Express 23(22), 28531–28545 (2015).
    [Crossref] [PubMed]
  9. K. S. Abedin, R. Ahmad, A. M. DeSantolo, J. W. Nicholson, P. S. Westbrook, C. Headley, and D. J. DiGiovanni, “Cladding-pumped hybrid single- and higher-order-mode (HOM) amplifier,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SW4K.4.
    [Crossref]
  10. K. S. Abedin, R. Ahmad, A. M. DeSantolo, J. W. Nicholson, P. S. Westbrook, C. Headley, and D. J. DiGiovanni, “Reconversion of higher-order-mode (HOM) output from a cladding-pumped hybrid Yb:HOM amplifier,” in Laser Congress 2018 (ASSL), OSA Technical Digest (Optical Society of America, 2018), paper ATh2A.2.
    [Crossref]
  11. J. W. Nicholson, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “Axicons for mode conversion in high peak power, higher-order mode, fiber amplifiers,” Opt. Express 23(26), 33849–33860 (2015).
    [Crossref] [PubMed]
  12. S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002).
    [Crossref] [PubMed]
  13. O. Brzobohatý, T. Cizmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16(17), 12688–12700 (2008).
    [Crossref] [PubMed]
  14. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

2017 (1)

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

2015 (2)

J. Demas, L. Rishøj, and S. Ramachandran, “Free-space beam shaping for precise control and conversion of modes in optical fiber,” Opt. Express 23(22), 28531–28545 (2015).
[Crossref] [PubMed]

J. W. Nicholson, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “Axicons for mode conversion in high peak power, higher-order mode, fiber amplifiers,” Opt. Express 23(26), 33849–33860 (2015).
[Crossref] [PubMed]

2008 (1)

O. Brzobohatý, T. Cizmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16(17), 12688–12700 (2008).
[Crossref] [PubMed]

2007 (1)

N. Lindlein, G. Leuchs, and S. Ramachandran, “Achieving Gaussian outputs from large-mode-area higher-order-mode fibers,” Appl. Opt. 46(22), 5147–5157 (2007).
[Crossref] [PubMed]

2006 (1)

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

2002 (1)

S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002).
[Crossref] [PubMed]

Abedin, K. S.

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

Ahmad, R.

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

Brzobohatý, O.

O. Brzobohatý, T. Cizmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16(17), 12688–12700 (2008).
[Crossref] [PubMed]

Cizmár, T.

O. Brzobohatý, T. Cizmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16(17), 12688–12700 (2008).
[Crossref] [PubMed]

Demas, J.

J. Demas, L. Rishøj, and S. Ramachandran, “Free-space beam shaping for precise control and conversion of modes in optical fiber,” Opt. Express 23(22), 28531–28545 (2015).
[Crossref] [PubMed]

DeSantolo, A.

J. W. Nicholson, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “Axicons for mode conversion in high peak power, higher-order mode, fiber amplifiers,” Opt. Express 23(26), 33849–33860 (2015).
[Crossref] [PubMed]

DiGiovanni, D. J.

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

J. W. Nicholson, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “Axicons for mode conversion in high peak power, higher-order mode, fiber amplifiers,” Opt. Express 23(26), 33849–33860 (2015).
[Crossref] [PubMed]

Dimarcello, F. V.

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

Ghalmi, S.

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

Headley, C.

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

J. W. Nicholson, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “Axicons for mode conversion in high peak power, higher-order mode, fiber amplifiers,” Opt. Express 23(26), 33849–33860 (2015).
[Crossref] [PubMed]

Kremp, T.

J. W. Nicholson, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “Axicons for mode conversion in high peak power, higher-order mode, fiber amplifiers,” Opt. Express 23(26), 33849–33860 (2015).
[Crossref] [PubMed]

Leuchs, G.

N. Lindlein, G. Leuchs, and S. Ramachandran, “Achieving Gaussian outputs from large-mode-area higher-order-mode fibers,” Appl. Opt. 46(22), 5147–5157 (2007).
[Crossref] [PubMed]

Lindlein, N.

N. Lindlein, G. Leuchs, and S. Ramachandran, “Achieving Gaussian outputs from large-mode-area higher-order-mode fibers,” Appl. Opt. 46(22), 5147–5157 (2007).
[Crossref] [PubMed]

Monberg, E.

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

Monberg, E. M.

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

Nicholson, J. W.

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

J. W. Nicholson, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “Axicons for mode conversion in high peak power, higher-order mode, fiber amplifiers,” Opt. Express 23(26), 33849–33860 (2015).
[Crossref] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

Ramachandran, S.

J. Demas, L. Rishøj, and S. Ramachandran, “Free-space beam shaping for precise control and conversion of modes in optical fiber,” Opt. Express 23(22), 28531–28545 (2015).
[Crossref] [PubMed]

N. Lindlein, G. Leuchs, and S. Ramachandran, “Achieving Gaussian outputs from large-mode-area higher-order-mode fibers,” Appl. Opt. 46(22), 5147–5157 (2007).
[Crossref] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002).
[Crossref] [PubMed]

S. Ramachandran, “Multimode nonlinear fiber optics: New opportunities & challenges,” Tutorial, in Conference on Lasers and Electro-Optics (CLEO) (2018).

Rishøj, L.

J. Demas, L. Rishøj, and S. Ramachandran, “Free-space beam shaping for precise control and conversion of modes in optical fiber,” Opt. Express 23(22), 28531–28545 (2015).
[Crossref] [PubMed]

Wang, Z.

S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002).
[Crossref] [PubMed]

Westbrook, P. S.

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

J. W. Nicholson, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “Axicons for mode conversion in high peak power, higher-order mode, fiber amplifiers,” Opt. Express 23(26), 33849–33860 (2015).
[Crossref] [PubMed]

Windeler, R. S.

J. W. Nicholson, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “Axicons for mode conversion in high peak power, higher-order mode, fiber amplifiers,” Opt. Express 23(26), 33849–33860 (2015).
[Crossref] [PubMed]

Wisk, P.

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

Wisk, P. W.

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

Yan, M.

S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002).
[Crossref] [PubMed]

Yan, M. F.

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

Zemánek, P.

O. Brzobohatý, T. Cizmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16(17), 12688–12700 (2008).
[Crossref] [PubMed]

Appl. Opt. (1)

N. Lindlein, G. Leuchs, and S. Ramachandran, “Achieving Gaussian outputs from large-mode-area higher-order-mode fibers,” Appl. Opt. 46(22), 5147–5157 (2007).
[Crossref] [PubMed]

Opt. Express (3)

J. Demas, L. Rishøj, and S. Ramachandran, “Free-space beam shaping for precise control and conversion of modes in optical fiber,” Opt. Express 23(22), 28531–28545 (2015).
[Crossref] [PubMed]

J. W. Nicholson, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “Axicons for mode conversion in high peak power, higher-order mode, fiber amplifiers,” Opt. Express 23(26), 33849–33860 (2015).
[Crossref] [PubMed]

O. Brzobohatý, T. Cizmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16(17), 12688–12700 (2008).
[Crossref] [PubMed]

Opt. Lett. (3)

S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002).
[Crossref] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

R. Ahmad, M. F. Yan, J. W. Nicholson, K. S. Abedin, P. S. Westbrook, C. Headley, P. W. Wisk, E. M. Monberg, and D. J. DiGiovanni, “Polarization-maintaining, large-effective-area, higher-order-mode fiber,” Opt. Lett. 42(13), 2591–2594 (2017).
[Crossref] [PubMed]

Other (7)

J. W. Nicholson, C. Headley, J. Phillips, A. Desantolo, E. Gonzalos, S. Ghalmi, M. F. Yan, P. W. Wisk, D. Trevor, J. Fleming, E. Monberg, F. Dimarcello, R. S. Windeler, J. M. Fini, D. J. DiGiovanni, and S. Ramachandran, “Higher-order-mode fiber amplifiers,” in Lasers, Sources and Related Photonic Devices,” OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper LSWD1.

S. Ramachandran, “Multimode nonlinear fiber optics: New opportunities & challenges,” Tutorial, in Conference on Lasers and Electro-Optics (CLEO) (2018).

K. S. Abedin, R. Ahmad, A. M. DeSantolo, J. W. Nicholson, C. E. Headley, P. S. Westbrook, and D. J. DiGiovanni, “Cladding pumped Yb-doped HOM power amplifier with high gain”, Proc. SPIE 10512, Fiber Lasers XV: Technology and Systems, 105121E (2018).
[Crossref]

K. S. Abedin, R. Ahmad, A. M. DeSantolo, J. W. Nicholson, P. S. Westbrook, C. Headley, and D. J. DiGiovanni, “Cladding-pumped hybrid single- and higher-order-mode (HOM) amplifier,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SW4K.4.
[Crossref]

K. S. Abedin, R. Ahmad, A. M. DeSantolo, J. W. Nicholson, P. S. Westbrook, C. Headley, and D. J. DiGiovanni, “Reconversion of higher-order-mode (HOM) output from a cladding-pumped hybrid Yb:HOM amplifier,” in Laser Congress 2018 (ASSL), OSA Technical Digest (Optical Society of America, 2018), paper ATh2A.2.
[Crossref]

J. W. Nicholson, J. M. Fini, J. Phillips, A. DeSantolo, X. Liu, K. Feder, V. R. Supradeepa, P. Westbrook, E. Monberg, F. DiMarcello, C. Headley, and D. J. DiGiovanni, “Nanosecond pulse amplification in a 6000 μm2 effective area higher-order mode erbium-doped fiber amplifier,” in Conference on Lasers and Electro-Optics 2012, OSA Technical Digest (online) (Optical Society of America, 2012), paper JTh1I.2.
[Crossref]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (11)

Fig. 1
Fig. 1 (a) Cross-sectional view of the Yb:HOM fiber used in the amplifier, (b) The normalized electric field amplitude distribution of the LP0,1 and LP0,10 modes plotted in relation to the refractive index profile, RIP (not drawn to scale).
Fig. 2
Fig. 2 (a) Schematic of a cladding-pumped hybrid Yb:HOM amplifier employing single-mode and higher-order mode gain fiber, (b) Beam profile of the HOM mode, LP0,10, at the amplifier output.
Fig. 3
Fig. 3 CW amplification characteristics of the cladding-pumped 0.5-m long single mode gain fiber, studied experimentally (a) Signal output power vs pump power, (b) Residual pump vs incident pump power.
Fig. 4
Fig. 4 Amplification characteristics of the hybrid amplifier.
Fig. 5
Fig. 5 Schematic diagram of HOM reconversion using axicon.
Fig. 6
Fig. 6 Electric field amplitude and phase distribution of HOM calculated at different locations within the reconversion system. (a) Exit of the HOM fiber, (b) right before the axicon.
Fig. 7
Fig. 7 Optimum apex angle and corresponding overlap integral (OI) plotted as a function of the position of the axicon.
Fig. 8
Fig. 8 (a) Schematic of reconversion of magnified HOM beam using axicon, (b) Optimum apex angle and position of axicon plotted as a function of beam magnification, calculated for LP0,10 mode of HOM fiber used in the experiment.
Fig. 9
Fig. 9 (a) Radial distribution of phase delay due to ideal and experimental 1-deg silica-glass axicon, (b) OI and optimum axicon location as a function of beam magnification.
Fig. 10
Fig. 10 Experimental setup of reconversion of HOM amplifier output using axicon. For beam profiling, camera is placed at the location of aperture.
Fig. 11
Fig. 11 Reconversion of HOM amplifier output using axicon. (a) Far-field pattern of the reconverted beam, (b) M2 as a function of aperture transmission.

Equations (4)

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u( x,y,z )= e jkz jkz u( ξ,η ) exp[ j k 2z { ( xξ ) 2 + ( yη ) 2 } ]dξdη
u M ( xM,yM,z M 2 )= e jkz M 2 jkz M 2 u( ξ M , η M ) exp[ j k 2z M 2 { ( xMξ ) 2 + ( yMη ) 2 } ]dξdη = e jkz( M 2 1 ) u( x,y,z )
M.tan α M =tan α 0
Z M = M 2 . Z 0

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