Abstract

We propose phase space distributions, based on an extension of the Wigner distribution function, to describe fields of any state of coherence that contain evanescent components emitted into a half-space. The evanescent components of the field are described in an optical phase space of spatial position and complex-valued angle. Behavior of these distributions upon propagation is also considered, where the rapid decay of the evanescent components is associated with the exponential decay of the associated phase space distributions. To demonstrate the structure and behavior of these distributions, we consider the fields generated from total internal reflection of a Gaussian Schell-model beam at a planar interface.

© 2012 Optical Society of America

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References

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  1. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).
  2. A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).
    [CrossRef]
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    [CrossRef]
  4. L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–562 (1964).
  5. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  6. K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
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  7. M. A. Alonso, “Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions,” J. Opt. Soc. Am. A 18, 902–909 (2001).
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  8. C. J. R. Sheppard and K. G. Larkin, “Wigner function for nonparaxial wave fields,” J. Opt. Soc. Am. A 18, 2486–2490 (2001).
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  9. J. C. Petruccelli and M. A. Alonso, “Propagation of partially coherent fields through planar dielectric boundaries using angle-impact Wigner functions I. Two dimensions,” J. Opt. Soc. Am. A 24, 2590–2603 (2007).
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  10. M. A. Alonso, “Radiometry and wide-angle wave fields III: partial coherence,” J. Opt. Soc. Am. A 18, 2502–2511 (2001).
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  11. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
    [CrossRef]
  12. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
    [CrossRef]
  13. S. B. Oh, J. C. Petruccelli, L. Tian, and G. Barbastathis, “Wigner functions defined with Laplace transform kernels,” Opt. Express 19, 21938–21944 (2011).
    [CrossRef]
  14. L. Q. Wang, L. G. Wang, S. Y. Shu, and M. S. Zubairy, “The influence of spatial coherence on the Goos–Hänchen shift at total internal reflection,” J. Phys. B 41, 055401 (2008).
    [CrossRef]
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 160–176.
  16. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 287–292.
  17. M. A. Alonso, “Measurement of Helmholtz wave fields,” J. Opt. Soc. Am. A 17, 1256–1264 (2000).
    [CrossRef]

2011 (1)

2008 (1)

L. Q. Wang, L. G. Wang, S. Y. Shu, and M. S. Zubairy, “The influence of spatial coherence on the Goos–Hänchen shift at total internal reflection,” J. Phys. B 41, 055401 (2008).
[CrossRef]

2007 (1)

2001 (3)

2000 (2)

M. A. Alonso, “Measurement of Helmholtz wave fields,” J. Opt. Soc. Am. A 17, 1256–1264 (2000).
[CrossRef]

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

1999 (1)

1979 (1)

1968 (1)

1964 (1)

L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–562 (1964).

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Alonso, M. A.

Barbastathis, G.

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

Dolin, L.

L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–562 (1964).

Forbes, G. W.

Friberg, A. T.

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Kwok, C. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Lai, H. M.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Larkin, K. G.

Loo, Y. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 160–176.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 287–292.

Oh, S. B.

Petruccelli, J. C.

Sheppard, C. J. R.

Shu, S. Y.

L. Q. Wang, L. G. Wang, S. Y. Shu, and M. S. Zubairy, “The influence of spatial coherence on the Goos–Hänchen shift at total internal reflection,” J. Phys. B 41, 055401 (2008).
[CrossRef]

Tian, L.

Walther, A.

Wang, L. G.

L. Q. Wang, L. G. Wang, S. Y. Shu, and M. S. Zubairy, “The influence of spatial coherence on the Goos–Hänchen shift at total internal reflection,” J. Phys. B 41, 055401 (2008).
[CrossRef]

Wang, L. Q.

L. Q. Wang, L. G. Wang, S. Y. Shu, and M. S. Zubairy, “The influence of spatial coherence on the Goos–Hänchen shift at total internal reflection,” J. Phys. B 41, 055401 (2008).
[CrossRef]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 160–176.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 287–292.

Wolf, K. B.

Xu, B. Y.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Zubairy, M. S.

L. Q. Wang, L. G. Wang, S. Y. Shu, and M. S. Zubairy, “The influence of spatial coherence on the Goos–Hänchen shift at total internal reflection,” J. Phys. B 41, 055401 (2008).
[CrossRef]

Ann. Phys. (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–562 (1964).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

J. Phys. B (1)

L. Q. Wang, L. G. Wang, S. Y. Shu, and M. S. Zubairy, “The influence of spatial coherence on the Goos–Hänchen shift at total internal reflection,” J. Phys. B 41, 055401 (2008).
[CrossRef]

Opt. Express (1)

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. E (1)

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Other (3)

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 160–176.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 287–292.

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Figures (10)

Fig. 1.
Fig. 1.

Contour of integration specifying plane-wave components of the field. Points on the horizontal line segment represent homogeneous, propagating plane waves, while points on the vertical segments represent evanescent waves whose phase advances along the x direction and that decay along the z axis.

Fig. 2.
Fig. 2.

Illustration of the changes of variables associated with Blm(0) where plane-wave components are illustrated by points on the integration contour. (a) The change of variables associated with B(0) is to centroid and difference in the parameter η, which describes the complex component of the angle representing evanescent components of the field. [The change of variables for B++(0) is similar.] (b) Change of variables for B+(0) and B+(0), also to centroid and difference in η, the parameter representing the complex portion of the angle representing evanescent waves. (c) Change of variables for Bh(0), given by Eq. (20). [The change of variables for Bh(0), Bh+(0), and B+h(0) are similar.]

Fig. 3.
Fig. 3.

Illustrating the effect of interface between two media for which n>n¯ on plane-wave components of the incident beam. (a) A propagating plane-wave component incident from the glass which makes an angle θ less than the critical angle θc with respect to the interface normal is refracted into a propagating plane-wave component after the interface, where the red arrow denotes the direction in which the phase accumulates and phase fronts are indicated by the red line segments. (b), (c) Propagating plane-wave components that make an angle with respect to the interface normal of greater than the critical angle become evanescent after the interface. Phase is accumulated along the x axis, increasing in the same direction as the transverse component of the incident plane-wave direction. The amplitude of the field decays exponentially along the z axis. Shading of the incident field indicates which plane-wave directions correspond, after transmission past the interface, to which segments of the contour illustrated in Fig. 1.

Fig. 4.
Fig. 4.

Radiant intensity, |A(θ)|2, of an incident Gaussian Schell-model beam that is (a) angularly wide (σ=2.0) and normally incident on the interface (θ0=0) and (b) angularly narrow (σ=0.1) and incident beyond the critical angle θ0=θc/2+π/4.

Fig. 5.
Fig. 5.

Bhh(0)(x,0,θ¯) and 2Re{Bh+(0)(x,0,θ¯)+Bh(0)(x,0,θ¯)} plotted for a Gaussian Schell-model beam propagating from glass to air that is highly globally incoherent (μg=0.011), partially coherent (μg=0.57), and coherent (μg=1.0). Figures are normalized so that the radiant intensity has a maximal value of unity.

Fig. 6.
Fig. 6.

B(0)(x,0,η¯), B++(0)(x,0,η¯), and 2Re{B+(0)(x,0,θ¯)} plotted for a Gaussian Schell-model field propagating from glass to air that is highly globally incoherent (μg=0.011), partially coherent (μg=0.57), and coherent (μg=1.0). Figures are normalized so that the radiant intensity has a maximal value of unity.

Fig. 7.
Fig. 7.

Bhh(0)(x,z,θ¯) and 2Re{Bh+(0)(x,z,θ¯)+Bh(0)(x,z,θ¯)} plotted for a partially coherent (μg=0.57) Gaussian Schell-model beam propagating from glass to air at distances kz=0, 10, and 20 from the interface. Figures are normalized so that the radiant intensity has a maximal value of unity.

Fig. 8.
Fig. 8.

B(0)(x,0,η¯), B++(0)(x,0,η¯), and 2Re{B+(0)(x,0,θ¯)} plotted for a partially coherent (μg=0.57) Gaussian Schell-model beam propagating from glass to air at distances kz=0, 10, and 20 from the interface. Figures are normalized so that the radiant intensity has a maximal value of unity.

Fig. 9.
Fig. 9.

B++(0) plotted for narrow (σ=0.1) Gaussian Schell-model beams incident at a central angle θ0=θc+π/4 for (a) varying states of coherence μg=0.027, 0.36, and 1.0 and (b) propagation away from the interface at distances kz=6, 8, 10. (c) Relative error between Eq. (15) and the lowest-order approximation of propagation given by Eq. (17). Figures are normalized so that the radiant intensity has a maximal value of unity.

Fig. 10.
Fig. 10.

B++(1)·x^, B++(1)·z^, and B++(2) plotted for a narrow (σ=0.1) Gaussian Schell-model beam incident at a central angle θ0=θc+π/4 with a global degree of coherence given by μg=0.36. Values of B++(p) are shown (a) at the interface z=0 and (b) after propagation away from the interface by a small amount kz=2. The lowest-order approximation in a Taylor series expansion of propagation is used to estimate the propagated Bs at kz=2, and the relative error, according to Eq. (35), is illustrated in (c). Figures are normalized so that the radiant intensity has a maximal value of unity.

Equations (57)

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B(r,u)=d2ΦdΩudAn^·u,
u·B(r,u)=0.
I(r)=4πB(r,u)n^·udΩu,
(2+k2n2)U(r)=0,
W(r1,r2)=U*(r1)U(r2),
U(r)=exp[iknu(ψ)·r],
U(r)=exp[iknu(θ)·rcoshη]exp[knu(θ)·rsinhη],
U(r)=kn2πCA(ψ)exp[iknu(ψ)·r]dψ,
Uh(r)=kn2ππ/2π/2Ah(θ)exp[ikn(xsinθ+zcosθ)]dθ,
U(r)=kn2π0ηMA(η)exp(iknxcoshη)exp(knzsinhη)dη,
U+(r)=kn2πηM0A+(η)exp(iknxcoshη)exp(knzsinhη)dη,
U(r)=l=,h,+Ul(r).
S(r)=l,m=+,,hUl*(r)Um(r)=kn2πl,m=+,,hAl*(ψ1)Am(ψ2)×exp{ikn[u(ψ2)u*(ψ1)]·r}dψ1dψ2,
S(r)=l,m=+,,hBl,m(0)(r,ψ)dψl,m,
Shh(r)=kn2ππ/2π/2Ah*(θ1)Ah(θ2)exp{ikn[u(θ2)u(θ1)]·r}dθ1dθ2.
Shh(r)=π/2π/2Bhh(0)(r,θ)dθ.
u(θ)·Bhh(0)(r,θ)=0.
Bhh(0)(r,θ)=kn2ππ2|θ|π2|θ|A*(θα2)A(θ+α2)exp[2iknr·u(θ)sinα2]dα.
Bll(0)(x,z,η)=kn2πβMβMAl*(ηβ2)Al(η+β2)exp(2iknxsinhηsinhβ2)×exp(2knz|sinhη|coshβ2)dβ,
Bll(0)(x,z,η)=exp(2knz|sinhη|114k2sinh2η2x2)Bll(0)(x,0,η),
Bll(0)(x,z,η)exp(2kz|sinhη|)Bll(0)(x,0,η),
B+(0)(x,z,η)=kn2πηMηMA*(η+β2)A+(ηβ2)exp(2ikxcoshηcoshβ2)×exp(2kzcoshηsinhβ2)dβ,
Sh(x,z)=kn2πAh*(θ1)A(η2)exp{ik[coshη2x^+u(θ1)·r]}×exp(ksinhη2z)dθ1dη2.
η2(θ,α)=±arccosh[cos(α2)cscθ],
Bh(0)(x,z,θ)=σ=1σα1α2Ah*(θ+σα2)A[arccosh(cosα2cscθ)]×exp{ikncos(θ+σα2)cscθu(θ)·r}exp(knzcos2α2csc2θ1)×cos(θ+σα2)cscθ[cos2αsin2θ]1/2dα,
α1=2Re[arccosh(coshηM|sinθ|)],
α2=2arccos(|sinθ|),
Bh+(0)(x,z,θ)=σ=1σα1α2Ah*(θ+σα2)A+[arccosh(cosα2cscθ)]×exp{ikncos(θ+σα2)cscθu(θ)·r}exp(knzcos2α2csc2θ1)×cos(θ+σα2)cscθ[cos2αsin2θ]1/2dα,
Bre(0)(x,z,θ)={Bhh(0)(x,z,θ)+2Re{Bh(0)(x,z,θ)}ifπ/2θ<0Bhh(0)(x,z,0)ifθ=0Bhh(0)(x,z,θ)+2Re{Bh+(0)(x,z,θ)}if0<θπ/2.
Bim(0)(x,z,η)={B++(0)(x,z,η)ifηMη<ηM/2B++(0)(x,z,η)+2Re{B+(0)(x,z,η)}ifηM/2η<02Re{B+(0)(x,z,η)}ifη=0B(0)(x,z,η)+2Re{B+(0)(x,z,η)}if0<ηηM/2B(0)(x,z,η)ifηM/2η<ηM.
S(x,z)=π/2π/2Bim(0)(x,z,θ)dθ+ηMηMBim(0)(x,z,η)dη.
μg=2πk|A(θ1,θ2)|2dθ1dθ2[A(θ,θ)dθ]2,
θ(ϕ¯)=arcsin(n¯nsinϕ¯).
τ(ϕ¯)=2ncosθ(ϕ¯)ncosθ(ϕ¯)+n¯cosϕ¯.
Tl(ϕ¯)={τ(ϕ¯)cosϕ¯cosθϕ¯ifl=hτ(ϕ¯)ifl=.
A¯*(ϕ¯1)A¯(ϕ¯2)=Tl*(ϕ1¯)Tm(ϕ2¯)A*[θ(ϕ¯1)]A¯[θ(ϕ¯2)],
A*(θ1)A(θ2)=exp{[p(θ2)p(θ0)]2+[p(θ1)p(θ0)]22σ2}×exp{[p(θ2)p(θ1)]22ϵ2}cosθ1cosθ2,
ΔB++(0)(x,z,η¯)=B++(0)(x,z,η¯)B++(0)(x,0,η¯)exp(2kzsinhη¯)Max{B++(0)(x,z,η¯)},
B++(1)(x,z,η)=kn2πsign(η)coshηβMβMA+*(ηβ2)A+(η+β2)u(iβ2)exp(2iknxsinhηsinhβ2)exp(2knz|sinhη|coshβ2)dβ,
B++(2)(x,z,η)=nccosh2ηB++(0)(x,z,η),
ΔB++(p)(x,z,η¯)=B++(p)(x,z,η¯)B++(p)(x,0,η¯)exp(2kzsinhη¯)Max{B++(p)(x,z,η¯)}.
u(ψ)·Blm(0)(x,z,ψ)=0
F(r)=i2knU*(r)U(r)U(r)U*(r)
H(r)=n2cU*(r)U(r)+1n2k2U*(r)·U(r)
I(p)(r)=l,m=+,,hIlm(p)(r)=kn2πl,m=+,,hV(p)(ψ1,ψ2)Al*(ψ1)Am(ψ2)×exp{ikn[u(ψ2)u*(ψ1)]·r}dψ1dψ2,
V(0)(θ1,θ2)=1,
V(1)(θ1,θ2)=u*(ψ1)+u(ψ2)2,
V(2)(θ1,θ2)=n2c[1+u*(ψ1)·u(ψ2)].
I(p)(x,z)=π/2π/2Bre(p)(x,z,θ)dθ+ηMηMBim(p)(x,z,η)dη,
vhh(1)(θ,α)=u(θ)cosα2,
vhh(2)(θ,α)=nccos2α2,
vll(1)(η,β)=sign(η)u(iβ2),
vll(2)(η,β)=nccosh2η,
v+(1)(η,β)=isinhηu(iβ2)=v+(1)(η,β),
v+(2)(η,β)=ncsinh2η,
vh(1)(θ,σα)=vh(1)*(θ,σα)=12[u(θ+σα2)(cscθcosα2,0)i(0,cos2α2csc2θ1)],
vh(2)(θ,σα)=vh(2)*(θ,σα)=14[3+cosα+σcotθsinα2icos(θ+σα2)cos2α2csc2θ1].

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