Abstract

A general formulation is presented for spatial shaping of single-spatial-mode broadband pulsed beams with partial spectral and temporal coherence properties. The model is based on the second-order coherence theory of non-stationary fields, and examples are presented on Gaussian to flat-top transformations. Spatiotemporal intensity distributions are evaluated in the target plane of the shaping element for idealized Gaussian Schell-model pulse trains and for realistic supercontinuum pulses.

© 2015 Optical Society of America

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References

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    [Crossref]
  2. H. Aagedal, F. Wyrowski, and M. Schmid, “Paraxial beam splitting and shaping,” J. Turunen and F. Wyrowski, eds. Diffractive Optics for Industrial and Commercial Applications (Akademie–Verlag, 1997), Chapt. 6.
  3. A. Forbes, F. Dickey, A. DeGama, and A. du Plessis, “Wavelength tunable laser beam shaping,” Opt. Lett. 37, 49–51 (2012).
    [Crossref] [PubMed]
  4. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [Crossref]
  5. M. Singh, J. Tervo, and J. Turunen, “Elementary-field analysis of partially coherent beam shaping,” J. Opt. Soc. Am. A 30, 2611–2617 (2013).
    [Crossref]
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  7. S. Zhang, Y. Ren, and G. Lüpke, “Spatial beam shaping of ultrashort laser pulses: theory and experiment,” Appl. Opt. 44, 5818–5823 (2005).
    [Crossref] [PubMed]
  8. M. Singh, J. Tervo, and J. Turunen, “Broadband beam shaping with harmonic diffractive optics,” Opt. Express 22, 22680–22688 (2014).
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  9. B. Alonso, Í. J. Sola, Ó Varela, J. Hernández-Toro, C. Méndez, J. San Román, A. Zaïr, and L. Roso, “Spatiotemporal amplitude-and-phase reconstruction by Fourier-transform of interference spectra of high-complex-beams,” J. Opt. Soc. Am. B 27, 933–940 (2010).
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  10. H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
    [Crossref]
  11. H. Lajunen, J. Tervo, and P. Vahimaa, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1544 (2005).
    [Crossref]
  12. J. Turunen, “Low coherence laser beams,” A. Forbes, ed., Laser Beam Propagation: Generation and Propagation of Customized Light (CRC, 2014), Chapt. 10.
    [Crossref]
  13. J. M. Dudley and J. R. Taylor, eds., Supercontinuum Generation in Optical Fibers (Cambridge University, 2010).
    [Crossref]
  14. G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Second-order coherence of supercontinuum light,” Opt. Lett. 35, 3057–3059 (2010).
    [Crossref] [PubMed]
  15. G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Complete characterization of supercontinuum coherence,” J. Opt. Soc. Am. B 28, 2301–2309 (2011).
    [Crossref]
  16. R. Dutta, M. Korhonen, A. T. Friberg, G. Genty, and J. Turunen, “Broadband spatiotemporal Gaussian Schell-model pulse trains”, J. Opt. Soc. Am. A 31, 637–643 (2014).
    [Crossref]
  17. M. Erkintalo, M. Surakka, J. Turunen, A. T. Friberg, and G. Genty, “Coherent-mode representation of supercon-tinuum,” Opt. Lett. 37, 169–171 (2012).
    [Crossref] [PubMed]
  18. O. Bryngdahl and F. Wyrowski, “Digital holography — Computer-generated holograms,” Progr. Opt.XXVIII, E. Wolf, ed. (Elsevier, 1990), 1–86.
    [Crossref]
  19. I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–77 (1986).
    [Crossref]
  20. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
    [Crossref]
  21. P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).
  22. S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
    [Crossref]
  23. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
    [Crossref]

2014 (2)

2013 (1)

2012 (2)

2011 (1)

2010 (2)

2007 (1)

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

2005 (2)

2004 (1)

2003 (1)

2002 (1)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[Crossref]

1996 (1)

1986 (1)

I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–77 (1986).
[Crossref]

1974 (1)

Aagedal, H.

H. Aagedal, F. Wyrowski, and M. Schmid, “Paraxial beam splitting and shaping,” J. Turunen and F. Wyrowski, eds. Diffractive Optics for Industrial and Commercial Applications (Akademie–Verlag, 1997), Chapt. 6.

Alonso, B.

Bryngdahl, O.

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
[Crossref]

O. Bryngdahl and F. Wyrowski, “Digital holography — Computer-generated holograms,” Progr. Opt.XXVIII, E. Wolf, ed. (Elsevier, 1990), 1–86.
[Crossref]

Christov, I. P.

I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–77 (1986).
[Crossref]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

DeGama, A.

Dickey, F.

Dickey, F. M.

du Plessis, A.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Dutta, R.

Eberly, J. H.

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

Erkintalo, M.

Forbes, A.

Friberg, A. T.

Genty, G.

Hernández-Toro, J.

Ivanov, C. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

Kasarova, S. N.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

Korhonen, M.

Lajunen, H.

Lüpke, G.

Méndez, C.

Milonni, P. W.

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

Nikolov, I. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[Crossref]

Ren, Y.

Román, J. San

Romero, L. A.

Roso, L.

Schmid, M.

H. Aagedal, F. Wyrowski, and M. Schmid, “Paraxial beam splitting and shaping,” J. Turunen and F. Wyrowski, eds. Diffractive Optics for Industrial and Commercial Applications (Akademie–Verlag, 1997), Chapt. 6.

Singh, M.

Sola, Í. J.

Sultanova, N. G.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

Surakka, M.

Tervo, J.

Turunen, J.

Vahimaa, P.

Varela, Ó

Wyrowski, F.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[Crossref]

O. Bryngdahl and F. Wyrowski, “Digital holography — Computer-generated holograms,” Progr. Opt.XXVIII, E. Wolf, ed. (Elsevier, 1990), 1–86.
[Crossref]

H. Aagedal, F. Wyrowski, and M. Schmid, “Paraxial beam splitting and shaping,” J. Turunen and F. Wyrowski, eds. Diffractive Optics for Industrial and Commercial Applications (Akademie–Verlag, 1997), Chapt. 6.

Zaïr, A.

Zhang, S.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

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

Opt. Acta (1)

I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–77 (1986).
[Crossref]

Opt. Commun. (1)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Opt. Mater. (1)

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[Crossref]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Other (5)

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

H. Aagedal, F. Wyrowski, and M. Schmid, “Paraxial beam splitting and shaping,” J. Turunen and F. Wyrowski, eds. Diffractive Optics for Industrial and Commercial Applications (Akademie–Verlag, 1997), Chapt. 6.

J. Turunen, “Low coherence laser beams,” A. Forbes, ed., Laser Beam Propagation: Generation and Propagation of Customized Light (CRC, 2014), Chapt. 10.
[Crossref]

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

O. Bryngdahl and F. Wyrowski, “Digital holography — Computer-generated holograms,” Progr. Opt.XXVIII, E. Wolf, ed. (Elsevier, 1990), 1–86.
[Crossref]

Supplementary Material (4)

» Media 1: MP4 (240 KB)     
» Media 2: MP4 (169 KB)     
» Media 3: MP4 (135 KB)     
» Media 4: MP4 (192 KB)     

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Figures (7)

Fig. 1
Fig. 1 The beam shaping geometry, where x and u denote the transverse coordinates in the input and output planes, separated by an optical system with a response function K(u, x; ω). A thin beam-shaping element with complex-amplitude transmission function t(x; ω) transforms the incident field V0(x; ω) into a spatially shaped output field V(u; ω).
Fig. 2
Fig. 2 Target-plane space–frequency profile for a Gaussian Schell-model pulse train (Top). Space–time profiles of pulse trains with the same spectral width but different spectral coherence widths (Bottom):Σ/Ω = 3 (left), Σ/Ω = 0.5 (center), and Σ/Ω = 0.2 (right) (the horizontal axis represents the retarded time tr = t − Δz/c). Media 1 illustrates the space–time profile when the ratio Σ/Ω is reduced.
Fig. 3
Fig. 3 Spatial intensity distribution in the target at different instants instants of time: tr = 20 fs (red), tr = 0 (green), and tr = −10 fs (blue). The black line is the final time-integrated spatial intensity profile. Media 2 shows the temporal evolution of the both the integrated (top) and the instantaneous (bottom) target-plane intensity profile within the range −60 fs ≤ t ≤ 60 fs.
Fig. 4
Fig. 4 Space–time profiles of Gaussian Schell-model pulse trains in the target plane when the spectral width and the spectral coherence width are kept constants (Σ/Ω = 0.5) but the target flat-top beam width is (i) Q = 20. (ii) Q = 10, and (iii) Q = 5. Media 3 illustrates the evolution of these profiles when Q is varied.
Fig. 5
Fig. 5 Spatial intensity profiles of a pulse with Q = 20 and Σ/Ω = 0.5 when the central maximum of the pulse is at z0 = 0.1m (left), z0 = 0.3m (center), and z0 = 0.5m (right). Media 4 shows the evolution of the shape upon propagation from the source plane z0 = 0 to the target plane z0 = 0.5m.
Fig. 6
Fig. 6 (a) The spectral density and (b) the temporal intensity of the incident supercontinuum pulse train. The blue lines show the profiles for the entire pulse train, whereas the green and red lines illustrate the quasi-coherent and quasi-stationary contributions, respectively.
Fig. 7
Fig. 7 (a) The target-plane space–frequency profiles of the supercontinuum pulse train. Top: the entire pulse train. Middle: the the quasi-coherent contribution. Bottom: the quasi-stationary contribution. (b) The corresponding space–time profiles. The profiles are scaled to their maximum values in each case.

Equations (37)

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W 0 ( x 1 , x 2 ; ω 1 , ω 2 ) = U 0 * ( x 1 ; ω 1 ) U 0 ( x 2 ; ω 2 ) = W 0 ( ω 1 , ω 2 ) V 0 * ( x 1 ; ω 1 ) V 0 ( x 2 ; ω 2 ) ,
t ( x , ω ) = exp [ i ω ω 0 D ( ω ) ϕ ( x ) ] ,
W ( x 1 , x 2 ; ω 1 , ω 2 ) = t * ( x 1 ; ω 1 ) t ( x 2 ; ω 2 ) W 0 ( x 1 , x 2 ; ω 1 , ω 2 ) = W 0 ( ω 1 , ω 2 ) t * ( x 1 ; ω 1 ) t ( x 2 ; ω 2 ) V 0 * ( x 1 ; ω 1 ) V 0 ( x 2 ; ω 2 ) .
V ( u ; ω ) = t ( x ; ω ) V 0 ( x ; ω ) K ( u , x ; ω ) d x .
W ( u 1 , u 2 ; ω 1 , ω 2 ) = U * ( u 1 ; ω 1 ) U ( u 2 ; ω 2 ) = W 0 ( ω 1 , ω 2 ) V * ( u 1 ; ω 1 ) V ( u 2 ; ω 2 )
S ( u ; ω ) = S 0 ( ω ) | V ( u ; ω ) | 2 ,
K ( u , x ; ω ) = ω i 2 π c F exp ( i ω 2 F / c ) exp ( i ω c F u x ) ,
K ( u , x ; ω ) = ω i 2 π c Δ z exp ( i ω Δ z / c ) exp [ i ω 2 c Δ z ( u x ) 2 ]
Γ ( u 1 , u 2 ; t 1 , t 2 ) = 0 W ( u 1 , u 2 ; ω 1 , ω 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d ω 1 d ω 2 .
W 0 ( ω 1 , ω 2 ) = m = 0 α m ψ m * ( ω 1 ) ψ m ( ω 2 ) ,
0 W 0 ( ω 1 , ω 2 ) ψ m ( ω 1 ) d ω 1 = α m ψ m ( ω 2 ) .
W ( u 1 , u 2 ; ω 1 , ω 2 ) = m = 0 α m ψ m * ( ω 1 ) ψ m ( ω 2 ) V * ( u 1 ; ω 1 ) V ( u 2 ; ω 2 ) ,
Γ ( u 1 , u 2 ; t 1 , t 2 ) = m = 0 α m V m * ( u 1 ; t 1 ) V m ( u 2 ; t 2 ) ,
V m ( u ; t ) = 0 ψ m ( ω ) V ( u ; ω ) exp ( i ω t ) d ω .
I ( u ; t ) = m = 0 α m | V m ( u ; t ) | 2 .
W ( ω 1 , ω 2 ) = [ S ( ω 1 ) S ( ω 2 ) ] 1 / 2 μ ( ω 1 , ω 2 ) ,
S ( ω ) = S 0 exp [ 2 ( ω ω 0 ) 2 Ω 2 ]
μ ( ω 1 , ω 2 ) = exp [ ( ω 1 ω 2 ) 2 2 Σ 2 ]
ψ m ( ω ) = ( 2 π ) 1 / 4 ( 2 m m ! Ω β ) 1 / 2 H m [ 2 ( ω ω 0 ) Ω β ] exp [ ( ω ω 0 ) 2 Ω 2 β ] ,
α m = S 0 2 π 1 + 1 / β ( 1 β 1 + β ) m
β = [ 1 + ( Ω / Σ ) 2 ] 1 / 2 .
V 0 ( x ; ω ) = V 0 exp ( ω ω 0 x 2 w 0 2 ) ,
ϕ ( x ) = ω 0 x 2 2 c Δ z + ω 0 a c Δ z { w 0 2 π [ exp ( 2 x 2 w 0 2 ) 1 ] + x erf ( 2 x w 0 ) } .
w = 2 c Δ z ω 0 w 0 = w 0 Δ z z R
Q = a w = z R Δ z a w 0 ,
t ( X , ω ) = exp [ i ω ω 0 D ( ω ) z R Δ z X 2 ] exp [ i ω ω 0 D ( ω ) ϕ S ( X ) ] ,
ϕ S ( X ) = 2 Q { 1 2 π [ exp ( 2 X 2 ) 1 ] + X erf ( 2 X ) } .
V ( U ; ω ˜ ) = V 0 ω ˜ i π z R Δ z exp [ i ( ω 0 Δ z / c ) ω ˜ ] exp ( i Δ z z R ω ˜ U 2 ) exp ( ω ˜ X 2 ) × exp [ i D ( ω ) ω ˜ ϕ S ( X ) ] exp { i z R Δ z [ 1 D ( ω ) ] ω ˜ X 2 } exp ( i 2 ω ˜ X U ) d X .
I ( int ) ( U , t r ) = t r I ( U ; t ) d t .
W ( ω 1 , ω 2 ) = A * ( ω 1 ) A ( ω 2 ) = 1 N n = 1 N A n * ( ω 1 ) A n ( ω 2 )
A n ( u ; t ) = 0 A n ( ω ) V ( u ; ω ) exp ( i ω t ) d ω .
I ( u ; t ) = | A n ( u ; t ) | 2 = 1 N n = 1 N | A n ( u ; t ) | 2 .
V m ( u ; t r ) = ω i 2 π c F ϕ m ( ω ) exp ( i ω t r ) 0 t ( x ; ω ) V 0 ( x ; ω ) exp ( i ω c F u x ) d x d ω .
t ( x ; ω ) V 0 ( x ; ω ) t * ( x ; ω ) V 0 * ( x ; ω ) .
V m ( u ; t r ) i V m * ( u ; t r ) .
Γ ( u 1 , u 2 ; t r 1 , t r 2 ) Γ * ( u 1 , u 2 ; t r 1 ; t r 2 ) .
I ( u ; t r ) I ( u , t r ) .

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