Abstract

We consider spatial shaping of broadband (either stationary or pulsed) spatially coherent light, comparing refractive, standard diffractive, and harmonic diffractive (modulo 2πM) elements. Considering frequency-integrated target profiles we show that, contrary to common belief, standard diffractive (M = 1) elements work reasonably well for, e.g., Gaussian femtosecond pulses and spatially coherent amplified-spontaneous-emission sources such as superluminescent diodes. It is also shown that harmonic elements with M ≥ 5 behave in essentially the same way as refractive elements and clearly outperform standard diffractive elements for highly broadband light.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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2012 (1)

2009 (1)

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

2007 (2)

S. N. Kasarova, Nina Georgieva Sultanova, Christo Dimitrov Ivanov, and Ivan Dechev Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

V. Torres-Company, H. Lajunen, and A.T. Friberg, ”Coherence theory of noise in ultrashort-pulse trains,” J. Opt. Soc. Am. B 24, 1441–1450 (2007).
[Crossref]

2003 (1)

1996 (2)

L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13, 751–760 (1996).
[Crossref]

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[Crossref]

1995 (2)

1992 (1)

1988 (1)

1982 (1)

1974 (1)

1973 (1)

Bryngdahl, O.

Dechev Nikolov, Ivan

S. N. Kasarova, Nina Georgieva Sultanova, Christo Dimitrov Ivanov, and Ivan Dechev Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

DeGama, A.

dePalma, J. J.

Dickey, F.

Dickey, F. M.

L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13, 751–760 (1996).
[Crossref]

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[Crossref]

Dimitrov Ivanov, Christo

S. N. Kasarova, Nina Georgieva Sultanova, Christo Dimitrov Ivanov, and Ivan Dechev Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

du Plessis, A.

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988), Sect. 14.7.

Faklis, D.

Forbes, A.

Friberg, A. T.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

Friberg, A.T.

Georgieva Sultanova, Nina

S. N. Kasarova, Nina Georgieva Sultanova, Christo Dimitrov Ivanov, and Ivan Dechev Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

Goerge, N.

Hoadley, H. O.

Holswade, S. C.

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[Crossref]

Kasarova, S. N.

S. N. Kasarova, Nina Georgieva Sultanova, Christo Dimitrov Ivanov, and Ivan Dechev Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

Kurtz, C. N.

Lajunen, H.

Lee, W.-H.

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, Vol. XVI, E. Wolf, ed. (North-Holland, 1978), pp. 119–223.
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
[Crossref]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988), Sect. 14.7.

Morris, G. M.

Roberts, N. C.

Romero, L. A.

Saastamoinen, K.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

Sommargren, G. E.

Stone, T.

Sweeney, D. W.

Tervo, J.

Torres-Company, V.

Turunen, J.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
[Crossref] [PubMed]

Vahimaa, P.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
[Crossref] [PubMed]

Wolf, E.

Wyrowski, F.

Appl. Opt. (4)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Eng. (1)

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Opt. Mater. (1)

S. N. Kasarova, Nina Georgieva Sultanova, Christo Dimitrov Ivanov, and Ivan Dechev Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

Phys. Rev. A (1)

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

Other (4)

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988), Sect. 14.7.

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, Vol. XVI, E. Wolf, ed. (North-Holland, 1978), pp. 119–223.
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
[Crossref]

J. M. Dudley and J. R. Taylor, eds., Supercontinuum Generation in Optical Fibers (Cambridge University, 2010).
[Crossref]

Supplementary Material (2)

» Media 1: AVI (6896 KB)     
» Media 2: AVI (6896 KB)     

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Figures (5)

Fig. 1
Fig. 1 The 2F geometry for transformation of an incident field with spatial distribution V(x, ω) and spectrum S0(ω) into a target field V(u, ω) in the Fourier plane of an achromatic lens with focal length F, using a beam shaping element with complex-amplitude transmission function t(x, ω) located at the plane z = 0. We stress that the distance between the beam shaping element and the lens is not critical; they could, e.g., be in contact.
Fig. 2
Fig. 2 Fourier-plane spatial distributions S(U, ω) generated by (a) modulo 2π(M = 1) and (b) modulo 10π(M = 5) diffractive elements at the design frequency ω = ω0 (green), ω = 1.05ω0 (red), and ω = 0.95ω0 (blue). The black lines represent frequency-integrated profiles S(U). Here the expansion factor is Q = 20.
Fig. 3
Fig. 3 The design-frequency phase functions of purely diffractive (red) and modulo M = 5 diffractive (blue) beam shaping elements with Q = 20. The video ( Media 1) shows the evolution of the phase profile when M is increased.
Fig. 4
Fig. 4 Target-plane amplitude profiles |V(U, Ω)| produced individually by the most significant generalized diffraction orders. (a) Standard diffractive element: orders m = 0, m = 1, and m = 2. (b) Harmonic element with with M = 5: orders m = 4, m = 5, and m = 6. In both cases we have considered a non-resonant frequency ω = 0.99ω0 and Q = 20.
Fig. 5
Fig. 5 Target-plane profiles generated by red (R), green (G), and blue (B) light sources, and the frequency-integrated result for (a) standard diffractive and (b) modulo M = 5 diffractive elements. The frequency-integrated results (black curves) are scaled down for clarity. Also shown are true-color profiles as seen by the human eye. Again Q = 20. The video ( Media 2) shows the evolution of the true-color profile when M is increased. (c) The result with no element in place (note the different lateral scale).

Equations (24)

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V ( u , ω ) = ω i 2 π c F V ( x , ω ) t ( x , ω ) exp ( i ω c F u π ) d x .
S ( u , ω ) = S 0 ( ω ) | V ( u , ω ) | 2 ,
S ( u ) = 0 S ( u , ω ) d ω .
I ( u ) = 2 π S ( u ) ,
I ( u ) = 1 2 π I ( u , t ) d t ,
S 0 ( ω ) = S 0 exp [ 2 ( ω ω 0 ) 2 ω S 2 ] ,
V ( x , ω ) = ( 2 π ω ω 0 ) 1 / 4 1 w 0 exp ( ω ω 0 x 2 w 0 2 ) ,
ϕ ( x , ω ) = ω ω 0 n ( ω ) 1 n ( ω 0 ) 1 ϕ ( x , ω 0 ) ,
t ( X , Ω ) = exp [ i Ω D ( Ω ) ϕ ( X , ω 0 ) ]
V ( U , Ω ) = ( 2 / π ) 1 / 4 w 0 w F Ω 3 / 4 exp ( Ω X 2 ) exp [ i Ω D ( Ω ) ϕ ( X , ω 0 ) ] exp [ i 2 Ω U ( X ) ] d X .
S ( U , Ω ) = S 0 exp [ 2 ( Ω 1 ) 2 / Ω S 2 ] | V ( U , Ω ) | 2 ,
t ( x ) = exp [ i α ϕ ( x ) ] ,
t ( x ) = m = G m exp [ i m ϕ ( x ) / M ] ,
G m = 1 2 π M 0 2 π M t ( x ) exp [ i m ϕ ( x ) / M ] d ϕ ( x ) = sinc ( M α m ) exp [ i π ( M α m ) ] .
G m ( Ω ) = sinc [ M Ω D ( Ω ) m ] exp { i π [ M Ω D ( Ω ) m ] } ,
η m ( Ω ) = | G m ( Ω ) | 2 = sinc 2 [ M Ω D ( Ω ) m ] ,
V ( U , Ω ) = ( 2 / π ) 1 / 4 w 0 w F Ω 3 / 4 m = G m ( Ω ) × exp ( Ω X 2 ) exp [ i ( m / M ) ϕ ( X , ω 0 ) ] exp ( i 2 Ω U X ) d X
M Ω D ( Ω ) = m = integer ,
ϕ ( X , ω 0 ) = 2 Q { 1 2 π [ exp ( 2 X 2 ) 1 ] + X erf ( 2 X ) } .
S ( u ) = 2 π I ( u )
E ( u , ω ) = 1 2 π E ( u , t ) exp ( i ω t ) d t .
W ( u 1 , u 2 , ω 1 , ω 2 ) = 1 ( 2 π ) 2 Γ ( u 1 , u 2 , t 1 , t 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d t 1 d t 2 .
S ( u , ω ) = 1 ( 2 π ) 2 Γ ( u , u , t 1 , t 2 ) exp [ i ω ( t 1 t 2 ) ] d t 1 d t 2 .
S ( u ) = 1 2 π Γ ( u , u , t 1 , t 1 ) d t 1

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