Abstract

A new class of partially coherent model sources is introduced on the basis of the second-order coherence theory of nonstationary optical fields. These model sources are spatially fully coherent at each frequency but can have broadband spectra and variable spectral coherence properties, which lead to reduced spatiotemporal coherence in the time domain. The source model is motivated by the spectral coherence properties of supercontinuum pulse trains generated in single-spatial-mode optical fibers. We demonstrate that such broadband light is highly (but not completely) spatially coherent, even though the spectral and temporal coherence properties may vary over a wide range. The model sources introduced here are convenient in assessing the spatiotemporal coherence of broadband pulses in optical systems.

© 2014 Optical Society of America

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References

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  1. R. R. Alfano, ed., The Supercontinuum Laser Source: Fundamentals with Updated References, 2nd ed. (Springer, 2006).
  2. J. M. Dudley and J. R. Taylor, eds., Supercontinuum Generation in Optical Fibres (Cambridge University, 2010).
  3. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
    [CrossRef]
  4. G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources,” J. Opt. Soc. Am. B 24, 1771–1785 (2007).
    [CrossRef]
  5. G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Second-order coherence of supercontinuum light,” Opt. Lett. 35, 3057–3059 (2010).
    [CrossRef]
  6. 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]
  7. M. Erkintalo, M. Surakka, J. Turunen, A. T. Friberg, and G. Genty, “Coherent-mode representation of supercontinuum,” Opt. Lett. 37, 169–171 (2012).
    [CrossRef]
  8. A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains,” Opt. Lett. 20, 623–625 (1995).
    [CrossRef]
  9. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  10. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  11. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–927 (1982).
    [CrossRef]
  12. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
    [CrossRef]
  13. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
    [CrossRef]

2012 (1)

2011 (1)

2010 (1)

2007 (1)

2006 (1)

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

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]

1995 (1)

1988 (1)

1982 (2)

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–927 (1982).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Coen, S.

G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources,” J. Opt. Soc. Am. B 24, 1771–1785 (2007).
[CrossRef]

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

Dudley, J. M.

G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources,” J. Opt. Soc. Am. B 24, 1771–1785 (2007).
[CrossRef]

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

Erkintalo, M.

Friberg, A. T.

Genty, G.

Gori, F.

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[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]

Starikov, A.

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Surakka, M.

Turunen, J.

Vahimaa, P.

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

Wolf, E.

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]

J. Opt. Soc. Am. (1)

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

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

Opt. Commun. (3)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

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

Opt. Lett. (3)

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 (2)

R. R. Alfano, ed., The Supercontinuum Laser Source: Fundamentals with Updated References, 2nd ed. (Springer, 2006).

J. M. Dudley and J. R. Taylor, eds., Supercontinuum Generation in Optical Fibres (Cambridge University, 2010).

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Figures (3)

Fig. 1.
Fig. 1.

Spatial variation of the normalized integrated degree of coherence μ¯(ρ,0)/μ¯ given by Eq. (30) with Ω/ω0=0.5 and μ¯4=0 (red), 0.25 (green), 0.5 (blue), and 0.75 (black). The dashed lines illustrate the normalized Gaussian spectral densities S(ρ)/S0 at ω=ω0 (red), ω=ω0Ω (green), and ω=ω0+Ω (blue).

Fig. 2.
Fig. 2.

Overall degree of spectral coherence as a function of the fractional spectral width of the pulse for different states of spectral coherence. Blue solid line: β=0.99. Orange long-dashed line: β=0.75. Green short-dashed line: β=0.5. Blue-dotted line: β=0.1.

Fig. 3.
Fig. 3.

Normalized transverse intensity profile at t=0 for the fully coherent case β=1 if Ω/ω0=0.5, calculated with finite integration limits (solid blue line) and infinite limits (dashed red line).

Equations (55)

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V(ρ;t)=0V(ρ;ω)exp(iωt)dω,
W(ρ1,ρ2;ω1,ω2)=V*(ρ1;ω1)V(ρ2,ω2).
S(ρ;ω)=W(ρ,ρ;ω,ω)
μ(ρ1,ρ2;ω1,ω2)=W(ρ1,ρ2;ω1,ω2)S(ρ1;ω1)S(ρ2;ω2)
W(ρ1,ρ2;ω1,ω2)=W(ω1,ω2)U*(ρ1;ω1)U(ρ2;ω2),
μ¯2=0|W(ω1,ω2)|2dω1dω2[0S(ω)dω]2,
μ¯2(ρ1,ρ2)=0|W(ρ1,ρ2;ω1,ω2)|2dω1dω20S(ρ1;ω1)dω10S(ρ2;ω2)dω2,
Γ(ρ1,ρ2;t1,t2)=V*(ρ1;t1)V(ρ2,t2).
Γ(ρ1,ρ2;t1,t2)=0W(ρ1,ρ2;ω1,ω2)×exp[i(ω1t1ω2t2)]dω1dω2.
I(ρ;t)=Γ(ρ,ρ;t,t),
γ(ρ1,ρ2;t1,t2)=Γ(ρ1,ρ2;t1,t2)I(ρ1;t1)I(ρ2;t2),
Γ(ρ1,ρ2;t,t)=0W(ω1,ω2)×U*(ρ1;ω1)U(ρ2;ω2)exp[i(ω1ω2)t]dω1dω2.
Γ(ρ1,ρ2;t,t)=U*(ρ1)U(ρ2)×0W(ω1,ω2)exp[i(ω1ω2)t]dω1dω2,
γ¯(ρ1,ρ2)=Γ¯(ρ1,ρ2)I¯(ρ1)I¯(ρ2),
Γ¯(ρ1,ρ2)=Γ(ρ1,ρ2;t,t)dt
I¯(ρ)=Γ¯(ρ,ρ)=I(ρ;t)dt
Γ¯(ρ1,ρ2)=2π0W(ρ1,ρ2;ω,ω)dω
I¯(ρ)=2π0S(ρ;ω)dω.
U(ρ;ω)=U0exp(aωρ2),
W(ω1,ω2)=[S(ω1)S(ω2)]1/2μ(ω1,ω2),
S(ω)=S0exp[2Ω2(ωω0)2]
μ(ω1,ω2)=exp[(ω1ω2)22Σ2].
U(ρ;ω)=U0exp(ωω0ρ2w2).
μ¯2=02ω¯2ω¯|W(ω¯,Δω)|2dω¯dΔω[0S(ω)dω]2,
W(ω¯,Δω)=S0exp[2Ω2(ω¯ω0)2]exp(T28Δω2),
T24=1Ω2+1Σ2=1Ω2μ¯4,
μ¯=2ΩT=[1+(ΩΣ)2]1/4
W(ρ1,ρ2;ω¯,Δω)=W0exp(ω¯ω0ρ12+ρ22w2)×exp[2Ω2(ω¯ω0)2]exp(T28Δω2)×exp(12Δωω0ρ22ρ12w2),
μ¯(ρ1,ρ2)=μ¯exp[18Ω2(1μ¯4)ω02(ρ12ρ22)2w4].
μ¯(ρ,0)/μ¯=exp[18Ω2(1μ¯4)ω02(ρw)4].
Γ(ρ1,ρ2;t¯,Δt)=02ω¯2ω¯W(ρ1,ρ2;ω¯,Δω)×exp[i(ω¯Δt+Δωt¯)]dω¯dΔω.
Γ(ρ1,ρ2;t¯,Δt)=I0exp(ρ12+ρ22w2)exp(iω0Δt)×exp[Ω28(ρ12+ρ22ω0w2+iΔt)2]×exp[2T2(ρ12ρ222ω0w2it¯)2],
I(ρ;t)=I0exp[2(ρw)2]×exp[12(Ωω0)2(ρw)4]exp(2t2T2).
γ(ρ1,ρ2;t¯,Δt)=exp[18Ω2(1μ¯4)ω02(ρ12ρ22)2w4]×exp(i2Ωμ¯2ω0ρ12ρ22w2t¯T)exp(i2Ωω0μ¯2ρ12+ρ22w2ΔtT)×exp(Δt22Θ2)exp(iω0Δt),
Θ=Tμ¯21μ¯4=TΣΩ
μ¯=[1+(TΘ)2]1/4,
|γ(ρ1,ρ2;t¯,Δt)||γ(ρ1,ρ2;Δt)|=exp[18Ω2(1μ¯4)ω02(ρ12ρ22)2w4]exp(Δt22Θ2),
|γ(ρ,ρ;Δt)|=exp(Δt22Θ2),
|γ(ρ,0;0)|=exp[18Ω2(1μ¯4)ω02(ρw)4].
|γ(ρ,0;0)|=μ¯(ρ,0)/μ¯,
W(ρ1,ρ2,z;ω1,ω2)=W(ω1,ω2)×U*(ρ1,z;ω1)U(ρ2,z;ω2),
U(ρ,z;ω)=ωi2πcBexp(iωL/c)exp(iωD2cBρ2)×U(ρ;ω)exp(iωA2cBρ2)×exp(iωcBρ·ρ)d2ρ
U(ρ;ω)=U0exp(iω2cq0ρ2),
U(ρ,z;ω)=U0q0Aq0+Bexp(iωLc)exp[iω2cq(z)ρ2],
q(z)=Aq0+BCq0+D
iω2cq(z)=ωω0w2(z)+iω2cR(z),
w(z)=wA2+B2/zR2
R(z)=A2zR2+B2ACzR2+BD.
W(ρ1,ρ2,z;ω¯,Δω)=S0|U0|2zR2A2zR2+B2exp(iΔωL)×exp{2Ω2(ω¯ω0)2+i2c[ρ22q(z)ρ12q*(z)]ω¯}×exp{T28Δω2+i4c[ρ12q*(z)+ρ22q(z)]Δω}.
Γ(ρ1,ρ2,z;t¯,Δt)=I0ΩTexp(iω0Δt)×exp{iω02c[ρ22q(z)ρ12q*(z)]}×exp{Ω28[12c(ρ22q(z)ρ12q*(z))Δt]2}×exp{2T2[14c(ρ12q*(z)+ρ22q(z))t¯r]2},
I0=2πS0|U0|2zR2A2zR2+B2,
t¯r=t¯L/c,
β=[1+(ΩΣ)2]1/2=2ΩT,
μ¯2=8βπ1[1+erf(2ω0/Ω)]2×0exp[4(xω0Ω)2]erf(2xβ)dx.
I(ρ,t)=2πΩT|U0|2S0exp(2t2T2)×0exp[2Ωω0(ρw)2x]exp[2(xω0Ω)2]×[erf(i2tT+2xβ)erf(i2tT2xβ)]dx,

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