Abstract

A rapid and efficient all-optical method for forming propagation invariant shaped beams by exploiting the optical feedback of a laser cavity is presented. The method is based on the modified degenerate cavity laser (MDCL), which is a highly incoherent cavity laser. The MDCL has a very large number of degrees of freedom (320,000 modes in our system) that can be coupled and controlled, and allows direct access to both the real space and Fourier space of the laser beam. By inserting amplitude masks into the cavity, constraints can be imposed on the laser in order to obtain minimal loss solutions that would optimally lead to a superposition of Bessel-Gauss beams forming a desired shaped beam. The resulting beam maintains its transverse intensity distribution for relatively long propagation distances.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
    [Crossref]
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    [Crossref] [PubMed]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  5. A. Sukhorukov, “Nonlinear optics: Diffraction cancellation,” Nat. Photonics 5(1), 4–5 (2011).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  15. R. Chriki, M. Nixon, V. Pal, C. Tradonsky, G. Barach, A. A. Friesem, and N. Davidson, “Manipulating the spatial coherence of a laser source,” Opt. Express 23(10), 12989–12997 (2015).
    [Crossref] [PubMed]
  16. M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7(11), 919–924 (2013).
    [Crossref]
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    [Crossref]
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  19. A. E. Siegman, Lasers (University Science Books, 1986), ch. 14.2.
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    [Crossref] [PubMed]
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2016 (1)

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 2, 1–3 (2016).

2015 (1)

2013 (3)

M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. 38(19), 3858–3861 (2013).
[Crossref] [PubMed]

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref] [PubMed]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7(11), 919–924 (2013).
[Crossref]

2011 (1)

A. Sukhorukov, “Nonlinear optics: Diffraction cancellation,” Nat. Photonics 5(1), 4–5 (2011).
[Crossref]

2010 (1)

2009 (1)

2007 (2)

D. M. Cottrell, J. M. Craven, and J. A. Davis, “Nondiffracting random intensity patterns,” Opt. Lett. 32(3), 298–300 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

2002 (1)

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[Crossref] [PubMed]

1999 (1)

G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science 286(5444), 1518–1523 (1999).
[Crossref] [PubMed]

1998 (1)

1991 (1)

1989 (1)

1987 (2)

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[Crossref] [PubMed]

Barach, G.

Bouchal, Z. K.

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Bromberg, Y.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7(11), 919–924 (2013).
[Crossref]

Cao, H.

Chriki, R.

Christodoulides, D. N.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Cottrell, D. M.

Craven, J. M.

Davidson, N.

R. Chriki, M. Nixon, V. Pal, C. Tradonsky, G. Barach, A. A. Friesem, and N. Davidson, “Manipulating the spatial coherence of a laser source,” Opt. Express 23(10), 12989–12997 (2015).
[Crossref] [PubMed]

M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. 38(19), 3858–3861 (2013).
[Crossref] [PubMed]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7(11), 919–924 (2013).
[Crossref]

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref] [PubMed]

Davis, J. A.

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Dudley, A.

Durnin, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[Crossref] [PubMed]

Forbes, A.

Friberg, A. T.

Friesem, A. A.

R. Chriki, M. Nixon, V. Pal, C. Tradonsky, G. Barach, A. A. Friesem, and N. Davidson, “Manipulating the spatial coherence of a laser source,” Opt. Express 23(10), 12989–12997 (2015).
[Crossref] [PubMed]

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref] [PubMed]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7(11), 919–924 (2013).
[Crossref]

M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. 38(19), 3858–3861 (2013).
[Crossref] [PubMed]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Hamerly, R.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 2, 1–3 (2016).

Helmerson, K.

Inaba, K.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 2, 1–3 (2016).

Inagaki, T.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 2, 1–3 (2016).

Indebetouw, G.

Inoue, K.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 2, 1–3 (2016).

Katz, O.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7(11), 919–924 (2013).
[Crossref]

Kettunen, V.

Khilo, N.

López-Mariscal, C.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[Crossref] [PubMed]

Nixon, M.

R. Chriki, M. Nixon, V. Pal, C. Tradonsky, G. Barach, A. A. Friesem, and N. Davidson, “Manipulating the spatial coherence of a laser source,” Opt. Express 23(10), 12989–12997 (2015).
[Crossref] [PubMed]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7(11), 919–924 (2013).
[Crossref]

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref] [PubMed]

M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. 38(19), 3858–3861 (2013).
[Crossref] [PubMed]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Pal, V.

Redding, B.

Ronen, E.

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref] [PubMed]

Segev, M.

G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science 286(5444), 1518–1523 (1999).
[Crossref] [PubMed]

Silberberg, Y.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7(11), 919–924 (2013).
[Crossref]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[Crossref] [PubMed]

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Small, E.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7(11), 919–924 (2013).
[Crossref]

Stegeman, G. I.

G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science 286(5444), 1518–1523 (1999).
[Crossref] [PubMed]

Sukhorukov, A.

A. Sukhorukov, “Nonlinear optics: Diffraction cancellation,” Nat. Photonics 5(1), 4–5 (2011).
[Crossref]

Takesue, H.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 2, 1–3 (2016).

Tradonsky, C.

Turunen, J.

Vasara, A.

Vasilyeu, R.

Yamamoto, Y.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 2, 1–3 (2016).

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Photonics (3)

A. Sukhorukov, “Nonlinear optics: Diffraction cancellation,” Nat. Photonics 5(1), 4–5 (2011).
[Crossref]

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 2, 1–3 (2016).

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7(11), 919–924 (2013).
[Crossref]

Opt. Commun. (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. Lett. (4)

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[Crossref] [PubMed]

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Science (1)

G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science 286(5444), 1518–1523 (1999).
[Crossref] [PubMed]

Other (2)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University Press, 2007), ch. 19.5.2.

A. E. Siegman, Lasers (University Science Books, 1986), ch. 14.2.

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Figures (4)

Fig. 1
Fig. 1

Experimental arrangement and results. (a) Experimental arrangement of the modified degenerate cavity. (b) Experimental intensity distributions of the MDCL at the Fourier plane for circular or annular apertures. The inset shows a radial crossection of the intensity distribution along the dashed white line. (c) Experimental intensity distributions at the output plane of the MDCL for circular or annular apertures.

Fig. 2
Fig. 2

Experimental and simulation results for diffraction of the MDCL’s output beam as a function of propagation distance. (a) Detected intensity distributions at the output, and after propagation distances of 10mm, 20mm and 30mm, for circular and annular apertures. (b) Experimental and simulation results for the overlap integral of the desired and laser output intensity distributions as a function of propagation distance. Solid lines denote calculated results for a circular (blue) and annular (red) apertures, Blue squares denote measured results obtained with a circular aperture, and red circles denote measured results obtained with an annular aperture.

Fig. 3
Fig. 3

Tradeoff between resolution and maximal distance for propagation invariance. (a) Maximal resolution as a function of propagation distance zmax, for annular apertures with mean diameters of 1.8mm (maximal diameter of 2.1mm, blue curve with circles) and 2.8mm (maximal diameter of 3.1mm, red curve with squares). (b) The desired intensity distribution (first from the left), obtained by illuminating a resolution target outside the cavity, and the intensity distribution obtained in the MDCL laser output (z = 0), when placing at the Fourier plane a large annular aperture (second from the left), a large annular aperture (second from the right) and a small annular aperture (first from the right).

Fig. 4
Fig. 4

Calculated output intensity distributions from the MDCL using Gerchberg-Saxton (GS) and modified Fox-Li simulations. (a, d) Intensity distribution of 1 and 100 realizations of a GS simulation. (b, e) Intensity distribution of 1 and 100 realizations of a cold cavity simulation. (c, f) Intensity distribution of 1 and 100 realizations of an active-cavity simulation.

Equations (3)

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E(r,θ,z=0)=exp( r 2 w 0 ) n A n J n ( k r r ) exp( inθ ),
C( z )= I m ( x,y;z ) I d ( x,y;z )dxdy I m ( x,y;z ) 2 dxdy I d ( x,y;z ) 2 dxdy ,
z max = k z w 0 k r .