Abstract

This tutorial gives an overview of the use of the Wigner function as a tool for modeling optical field propagation. Particular emphasis is placed on the spatial propagation of stationary fields, as well as on the propagation of pulses through dispersive media. In the first case, the Wigner function gives a representation of the field that is similar to a radiance or weight distribution for all the rays in the system, since its arguments are both position and direction. In cases in which the field is paraxial and where the system is described by a simple linear relation in the ray regime, the Wigner function is constant under propagation along rays. An equivalent property holds for optical pulse propagation in dispersive media under analogous assumptions. Several properties and applications of the Wigner function in these contexts are discussed, as is its connection with other common phase-space distributions like the ambiguity function, the spectrogram, and the Husimi, P, Q, and Kirkwood–Rihaczek functions. Also discussed are modifications to the definition of the Wigner function that allow extending the property of conservation along paths to a wider range of problems, including nonparaxial field propagation and pulse propagation within general transparent dispersive media.

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2010

J. C. Petruccelli and M. A. Alonso, "Phase space distributions tailored for dispersive media," J. Opt. Soc. Am. A 27, 1194‒1201 (2010).
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R. Horstmeyer, S. B. Oh, and R. Raskar, "Iterative aperture mask design in phase space using a rank constraint," Opt. Express 18, 22545‒22555 (2010).
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H. Gao, L. Tian, B. Zhang, and G. Barbastathis, "Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function," Opt. Lett. 35, 4148‒4150 (2010).
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J. C. Petruccelli, N, J. Moore, and M. A. Alonso, "Two methods for modeling the propagation of the coherence and polarization properties of nonparaxial fields," Opt. Commun. 283, 4457‒4466 (2010).
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S. B. Mehta and C. J. R. Sheppard, "Using the phase-space imager to analyze partially coherent imaging systems: bright-field, phase contrast, differential interference contrast, differential phase contrast, and spiral phase contrast," J. Mod. Opt. 57, 718‒739 (2010).
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W. P. Schleich, J. P. Dahl, and S. Varró, "Wigner function for a free particle in two dimensions: a tale of interference," Opt. Commun. 283, 786‒789 (2010).
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2009

2008

2007

2006

2005

A. C. Fannjiang, "White-noise and geometrical optics limits of Wigner–Moyal equation for wave beams in turbulent media," Commun. Math. Phys. 254, 289‒322 (2005).
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C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNulty, "X-ray imaging: a generalized approach using phase-space tomography," J. Opt. Soc. Am. A 22, 1691‒1700 (2005).
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2004

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R. W. Robinett, "Quantum wave packet revivals," Phys. Rep. 392, 1‒119 (2004).
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M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free-space," J. Opt. Soc. Am. A. 21, 2233‒2243 (2004).
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2003

2002

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L. E. Vicent and M. A. Alonso, "Generalized radiometry as a tool for the propagation of partially coherent fields," Opt. Commun. 207, 101‒112 (2002).
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2001

2000

1999

F. Hlawatsch, A. Papandreou-Suppappola, and G. Boudreaux-Bartels, "The power classes-quadratic time–frequency representations with scale covariance and dispersive time-shift covariance," IEEE Trans. Signal Process. 47, 3067‒3083 (1999).
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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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1998

J. Tu and S. Tamura, "Analytic relation for recovering the mutual intensity by means of intensity information," J. Opt. Soc. Am. A 15, 202‒206 (1998).
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A. Wax and J. E. Thomas, "Measurement of smoothed Wigner phase-space distributions for small-angle scattering in a turbid medium," J. Opt. Soc. Am. A 15, 1896‒1908 (1998).
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A. Papandreou-Suppappola, F. Hlawatsch, and G. Boudreaux-Bartels, "Quadratic time–frequency representations with scale covariance and generalized time-shift covariance: A unified framework for the affine, hyperbolic, and power classes," Digital Signal Processing 8, 3‒48 (1998).
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1997

J. Tu and S. Tamura, "Wave field determination using tomography of the ambiguity function," Phys. Rev. E 55, 1946‒1949 (1997).
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M. A. Alonso and G. W. Forbes, "Semigeometrical estimation of Green’s functions and wave propagators in optics," J. Opt. Soc. Am. A 14, 1076‒1086 (1997).
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D. Dragoman, J. P. Meunier, and M. Dragoman, "Beam-propagation method based on the Wigner transform: a new formulation," Opt. Lett. 22, 1050‒1052 (1997).
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K. B. Wolf and A. L. Rivera, "Holographic information in the Wigner function," Opt. Commun. 144, 36‒42 (1997).
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1996

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A. W. Lohmann and B. H. Soffer, "Relationships between the Radon–Wigner and fractional Fourier transforms," J. Opt. Soc. Am. A 11, 1798‒1801 (1994).
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1993

1992

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1991

1989

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1988

1987

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1985

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1984

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1983

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1980

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M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264‒267 (1979).
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1978

E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6‒17 (1978).
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M. J. Bastiaans, "Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26‒30 (1978).
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M. V. Berry, "Semi-classical mechanics in phase space: a study of Wigners function," Philos. Trans. R. Soc. Lond. 287, 237‒271 (1977).
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A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256‒1259 (1968).
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1966

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[CrossRef]

G. S. Agarwal and E. Wolf, "Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators," Phys. Rev. D 2, 2161‒2186 (1970).
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G. S. Agarwal and E. Wolf, "Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. III. A generalized Wick theorem and multitime mapping," Phys. Rev. D 2, 2206‒2225 (1970).
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G. S. Agarwal and E. Wolf, "Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. II. Quantum mechanics in phase space," Phys. Rev. D 2, 2187‒2205 (1970).
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Alieva, T.

Alonso, M. A.

S. Cho and M. A. Alonso, "Ambiguity function and phase-space tomography for nonparaxial fields," J. Opt. Soc. Am. A 28, 897‒902 (2011).
[CrossRef]

M. A. Alonso, T. Setälä, and A. T. Friberg, "Optimal pulses for arbitrary dispersive media," J. Eur. Opt. Soc. R.P. 6, 1100 (2011).

J. C. Petruccelli and M. A. Alonso, "Generalized radiometry model for the propagation of light within anisotropic and chiral media," J. Opt. Soc. Am. A 28, 791‒800 (2011).
[CrossRef]

J. C. Petruccelli, N, J. Moore, and M. A. Alonso, "Two methods for modeling the propagation of the coherence and polarization properties of nonparaxial fields," Opt. Commun. 283, 4457‒4466 (2010).
[CrossRef]

J. C. Petruccelli and M. A. Alonso, "Phase space distributions tailored for dispersive media," J. Opt. Soc. Am. A 27, 1194‒1201 (2010).
[CrossRef]

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R. W. Boyd, Radiometry and the Detection of Optical Radiation, Wiley, 1983, pp. 13‒27.

P. Moon and D. E. Spencer, The Photic Field, MIT Press, 1981.

L. Cohen, Time–Frequency Analysis, Prentice Hall, 1995.

W. Mecklenbräuker and F. Hlawatsch, The Wigner Distribution: Theory and Applications in Signal Processing, Elsevier, 1997.

U. Leonhardt, Measuring the Quantum State of Light, Cambridge U. Press, 1997.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Vol. 1, Wiley, 1977, pp. 214‒227.

See Chapter 4 by G. Saavedra and W. Furlan in Ref. [15], pp. 107–164

K. B. Wolf, Integral Transforms in Science and Engineering, Plenum Press, 1979, Ch. 9, 10.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, E. Wolf, ed., "Fractional transformations in optics," Progress in Optics XXXVIII, 1998, pp. 263‒242.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, John Wiley & Sons, 2001.

A. Erdélyi, Asymptotic Expansions, Dover, 1956.

Supplementary Material (24)

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» Media 19: MOV (545 KB)     
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Figures (28)

Figure 1
Figure 1

Illustration of the definition of the spectrogram. The first row shows the real (blue) and imaginary (red) parts of the function in Eqs. (40) for q [ 16 , 16 ] . The second row shows the Gaussian window function w ( q q ) over the same range, for varying q. The third row shows the products of each of the curves in the first row with that in the second row. The fourth row shows the spectrogram for the corresponding value of q, given by the squared modulus of the Fourier transform of the function in the third row. In the movie (Media 1), q varies from 12 to 12.

Figure 2
Figure 2

Spectrogram of the superposition of two Gaussians in Eqs. (40) over the range q [ 4 , 10 ] , p [ 4 , 10 ] , for a square window function. Here, black indicates zero, and brighter shades of green indicate higher values. The blue dot indicates the coordinates ( q 0 , p 0 ) . The movie (Media 2) shows, for ( q 0 , p 0 ) = ( 2 , 2 ) , the visible effect of varying the phase difference Φ over a complete cycle. Then, q 0 is varied from 2 to 8, and this is followed by a similar variation in p 0 . Finally, the phase Φ is varied again over a complete cycle, even though the effect of this variation is not noticeable due to the separation of the two components.

Figure 3
Figure 3

Spectrogram of the superposition of two Gaussians in Eqs. (40) over the range q [ 4 , 10 ] , p [ 4 , 10 ] , for a Gaussian window function. In the movie (Media 3), the variation of Φ, q 0 and p 0 is as in Fig. 2. Notice that here the bumps representing the two Gaussian contributions are more symmetric and localized than in Fig. 2.

Figure 4
Figure 4

Illustration of the definition of the Wigner function (Media 4). The first and second rows show plots, respectively, of f ( q + q / 2 ) and f ( q q / 2 ) (real parts in blue, imaginary parts in red) as functions of q [ 16 , 16 ] for different values of q. The third row shows the product of these functions, and the fourth row shows the Fourier transform of this product, equal to the Wigner function for the corresponding value of q.

Figure 5
Figure 5

The effect of a fractional Fourier transformation (with σ = 1 ) is a clockwise rotation of the Wigner function by an angle θ. In the static frame, (a) shows the Wigner function of the original function, and (b) shows the Wigner function following a transformation with θ = π / 3 . The range shown in both cases is [ 4 , 4 ] for both p and q. In Media 5, θ varies from 0 to π.

Figure 6
Figure 6

The effect of a scaling on the Wigner function. In the static frame, (a) shows the Wigner function of the original function, and (b) shows the Wigner function following a transformation with b = 1 . 4 . The range shown in both cases is [ 4 , 4 ] for both p and q. In Media 6, b varies from 1 to 1.4.

Figure 7
Figure 7

The effect of a Fresnel transformation is a horizontal shearing of the Wigner function. In the static frame, (a) shows the Wigner function of the original function, and (b) shows the Wigner function following a transformation with τ = 1 . The range shown in both cases is [ 4 , 4 ] for both p and q. In Media 7, τ varies from 0 to 1.

Figure 8
Figure 8

The effect of chirping is a vertical shearing of the Wigner function. In the static frame, (a) shows the Wigner function of the original function, and (b) shows the Wigner function following a transformation with τ = 1 . The range shown in both cases is [ 4 , 4 ] for both p and q. In Media 8, τ varies from 0 to 1.

Figure 9
Figure 9

The effect of a phase-space shift transformation is precisely to shift the Wigner function in phase space in the specified amounts q 0 , p 0 . In the static frame, (a) shows the Wigner function of the original function, and (b) shows the Wigner function following a transformation with ( q 0 , p 0 ) = ( 1 . 2 , 2 ) . The range shown in both cases is [ 4 , 4 ] for both p and q. Media 9 shows displacements from ( 0 , 0 ) to ( 0 , 3 ) to ( 3 , 3 ) .

Figure 10
Figure 10

Wigner function of the superposition of two Gaussians in Eqs. (40) over the range q [ 4 , 10 ] , p [ 4 , 10 ] . The blue dot indicates the coordinates of ( q 0 , p 0 ) . Media 10 shows, for ( q 0 , p 0 ) = ( 2 , 2 ) , the effect of varying the phase difference Φ over a complete cycle. Then, q 0 is varied from 2 to 8, and this is followed by a similar variation in p 0 . Finally, the phase Φ is varied again over a complete cycle.

Figure 11
Figure 11

Wigner function of (a) the sum of two Gaussians of unit width, given by f ( q ) = π 1 / 4 exp [ ( q + 4 ) 2 / 2 ] + π 1 / 4 exp [ ( q 4 ) 2 / 2 ] , and (b) the sum of three Gaussians of unit width, corresponding to f ( q ) = π 1 / 4 { exp [ ( q 4 ) 2 / 2 ] + exp [ ( q + 4 ) 2 / 2 ] + exp [ ( q 4 ) 2 / 2 ] exp ( i 5 q ) } . Both q and p are shown within the interval [ 8 , 8 ] . The marginal projections given by | f ( q ) | 2 and | f ̃ ( p ) | 2 are shown on the top and left margins, respectively.

Figure 12
Figure 12

(a) Plot of the phase-space curve p = Ω ( q ) (black line) as well as several half-scaled replicas p = Ω ( q 0 ) / 2 Ω ( 2 q q 0 ) / 2 (green, red, and purple lines), for Ω ( q ) = q 5 4 q 3 and different values of q 0 . The anchor point of the scaling (indicated by small circles in the case of the red and purple curves) is the point along the curve [ q 0 , Ω ( q 0 ) ] where the replica and the curve coincide. This construction is also illustrated in Media 11. (b) Wigner function for f ( q ) = A ( q ) exp [ i K Ω ( q ) ] , with A ( q ) = exp [ ( q / 4 ) 4 ] and K = 2 . Notice the correlation between the areas presenting oscillations and those occupied by the scaled replicas in (a). In both figures, the ranges shown are q [ 2 , 2 ] and p [ 10 , 10 ] . Also shown in part (a) is a straight line segment joining two anchor points (for the red and purple lines), whose midpoint coincides with the intersection of the two corresponding replicas. At this point, also shown in part (b) as a yellow dot, the fringes of the Wigner function are approximately parallel to this line segment, and the phase of the oscillations is proportional to K times the phase-space area enclosed by this segment and the phase-space curve p = Ω ( q ) . In the region below the phase-space curve between the two minima, the pattern is more complicated, since each point is a half-point for two pairs of points, i.e., four replicas cross each point. Therefore the pattern is formed by the interference of two sets of fringes.

Figure 13
Figure 13

(a) Plot of the phase-space curve q 2 + p 2 = 2 n + 1 (black line) as well as several half-scaled replicas of this curve (green lines). (b) Wigner function in Eq. (63) for n = 8 , with the curve q 2 + p 2 = 2 n + 1 overlaid (white line).

Figure 14
Figure 14

Frames from Media 12. (a) Wigner function of the n th coherent mode, given in Eq. (86) for σ 1 = σ 2 1 = 4 , i.e.,  ν = 1 / 4 . (b) Partial sum up to n of these Wigner functions, weighted by the factor in Eq. (84). The sum converges to the Wigner function in Eq. (83). In both figures, the range shown for both q and p is [ 8 , 8 ] . Notice that the partial superpositions are free of negative regions.

Figure 15
Figure 15

(a) Wigner function, (b) Husimi function with a = 1 , (c) Husimi function with a = 2 , (d) absolute value of the Kirkwood–Rihaczek function, and (e) real part of the Kirkwood–Rihaczek function, for f ( q ) = exp ( i q 3 / 12 ) and K = 1 . The plot ranges are q [ 8 , 8 ] and p [ 4 , 12 ] .

Figure 16
Figure 16

Frames from movies illustrating (a) Media 13 the Wigner function, (b) (Media 14) the Q function (i.e., the Husimi function with a = 1 ), (c) (Media 15) the Husimi function with a = 2 , and (d) (Media 16) the real part of the Kirkwood–Rihaczek function, for the fractional Fourier transform of f ( q ) = π 1 / 4 { exp [ ( q 4 ) 2 / 2 ] + exp [ ( q + 4 ) 2 / 2 ] } , with σ = K = 1 . The plot ranges are q [ 8 , 8 ] and p [ 8 , 8 ] . In the animations, the degree of the fractional Fourier transform varies from 0 to π .

Figure 17
Figure 17

At a plane of constant z, a ray (red line) is specified by the coordinates x = ( x , y ) of its intersection with this plane, and by the optical momentum p = ( p x , p y ) given by the transverse components of a tangent vector (green arrow) whose norm is the refractive index, n.

Figure 18
Figure 18

(a), (b) Wigner function for a plane wave incident on a slit of width a, and marginal projection in p, corresponding to the intensity (green line on top), for (a) z = 0 and (b) z = z peak , where the vertical alignment of positive regions leads to a peak in the intensity on axis. The horizontal axis corresponds to x [ a , a ] , while the vertical axis is p [ 20 / ( k a ) , 20 / ( k a ) ] . The intensity distribution is shown in (c) for x [ a , a ] and z [ 0 , 2 z peak ] . The red line indicates z = z peak . In Media 17, z varies between 0 and 2 z peak .

Figure 19
Figure 19

(a) Intensity distribution over the ( x , z ) plane for a plane wave diffracted by a slit of width a at z = 0 . Note that the z direction is significantly compressed with respect to the x direction. (b) Superposition of a dense sampling of lines representing rays, over a grey background. The lines are black or white depending on whether the Wigner function for the corresponding ray is positive or negative, and each line has an opacity proportional to the magnitude of the Wigner function. Media 18 shows the gradual accumulation of the rays in this figure.

Figure 20
Figure 20

(a), (b) Wigner function for an Airy beam, given by Eq. (147), over the phase-space region x [ 5 a , 5 a ] , p [ 3 / k a , 3 / k a ] , for (a) z = 0 and (b) z = 2 . 5 k a 2 . In both cases, the marginal in p, equal to the spatial intensity distribution, is shown as a green curve at the top. (c) Intensity profile for this Airy beam, described by Eq. (145), over the same interval in x and for z [ 0 , 5 k a 2 ] . The red line indicates z = 2 . 5 k a 2 , corresponding to the Wigner function in part (b). Media 19 shows the evolution of this Wigner function and its marginal under propagation in z, where the value of z is indicated by the red line.

Figure 21
Figure 21

(a) Intensity profile of a section of a field that is periodic in x with period Λ, for z from the initial plane z = 0 to the first Talbot image, at z 1 . (b) Wigner function for this field. Note that the Wigner function is discrete and delta-like in p. The green line on top shows the marginal projection in p, corresponding to the intensity. Media 20 shows the change of these quantities under propagation in z between 0 and z 1 (the first Talbot image). Note that at z = z 1 / 2 the intensity is a perfect image of the initial intensity except for a lateral shift of Λ / 2 .

Figure 22
Figure 22

(a) Representation of three stacked sampled replicas of the Wigner function, whose spacing in p, given by 2 π / k Λ , must be greater than the support in p for each replica, 2 p M , so they do not overlap. (b) After propagation in p, the Wigner function is sheared, and so is each replica following sampling, resulting in more effective separation between the replicas. (c) The sheared replicas can be packed more densely without leading to significant overlapping.

Figure 23
Figure 23

Representation, in the phase space composed of the two quadratures of the electric field, of (a) a classical monochromatic field, (b) a quantum coherent state, and (c) a quantum squeezed state. Note that the state in (c) is squeezed such that it has a more confined phase spread (corresponding to the angular spread of the distribution) than the coherent state in (b). Media 21 shows the evolution of these states and their marginals in p over one optical cycle.

Figure 24
Figure 24

Illustration of the geometric interpretation of the change of variables needed for the construction of a Wigner function that is conserved under propagation or evolution. The values of η 1 and η 2 correspond to two intersections of the curves with a straight line normal to u ( ξ ) , and α controls the displacement of this straight line.

Figure 25
Figure 25

(a) Illustration of the change of variables when the curve is the unit circle, for which α can be chosen as the angular separation of the two points, which varies from 0 to π, and θ indicates the direction of the bisector. (b) For nonparaxial fields in three dimensions, the manifold spanned by g is a unit sphere. The change of variables here is from u 1 and u 2 (two points over this sphere) to u, the direction of the bisector of the two points, α, the angular separation between the points, and ϕ, which controls the orientation of the line joining the two points for fixed u and α.

Figure 26
Figure 26

(a), (b) Nonparaxial Wigner function in terms of x and τ for a truncated cylindrical wave with numerical aperture of 1 / 2 (half-angle of π / 4 ), at (a) z = 0 and (b) z = 10 / k . In both cases, the marginal in τ, plotted at the top, equals the intensity profile at the corresponding propagation distance. (c) Intensity profile over the region x [ 10 / k , 10 / k ] , z [ 0 , 20 / k ] . The red line indicates z = 10 / k , corresponding to the Wigner function in (b). See Media 22.

Figure 27
Figure 27

(a) Dispersion curve in Eq. (208) with β = 0 . 4 ω R corresponding to a transparent Lorentz-model medium with one resonance. The green rectangle indicates the pulse’s spectral amplitude, U 0 ( ω ) . (b), (c) Generalized Wigner function in Eq. (210) for this pulse, in terms of z [ 0 . 2 c t 20 ω R / c , 0 . 2 c t + 20 ω R / c ] and v [ 0 , 0 . 4 c ] , for (b) t = 0 and (c) t = 100 / ω R . The marginal in v, plotted at the top, equals the intensity profile at the corresponding propagation time. Media 25 shows the horizontal shearing around the line v = 0 . 2 c of the Wigner function and the evolution of the spatial intensity profile for t between 0 and 100 / ω R . Notice that we chose to shear around v = 0 . 2 c instead of around v = 0 in order to move with the pulse, i.e., to factor out the pulse’s spatial displacement.

Figure 28
Figure 28

MEDIA 24 (a) Dispersion curve for a hollow metallic waveguide, given in Eq. (211). The green rectangle indicates the pulse’s spectral amplitude, U 0 ( ω ) . (b), (c) Generalized Wigner function in Eq. (213) for this pulse, in terms of z [ 0 . 6 c t 40 ω co / c , 0 . 6 c t + 40 ω co / c ] and v [ 0 . 4 c , 0 . 8 c ] , for (b) t = 0 and (c) t = 200 / ω co . The marginal in v, plotted at the top, equals the intensity profile at the corresponding propagation time. Media 24 shows the horizontal shearing around the line v = 0 . 6 c of the Wigner function and the evolution of the spatial intensity profile for t between 0 and 200 / ω co . Notice that shearing around v = 0 . 6 c instead of around v = 0 factors out the spatial displacement of the pulse.

Tables (2)

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Table 1. Kernels and Blurs for Several Phase-Space Sistributions of the Cohen Class

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Table 2. Transformation Properties a of Several Cohen-Class Phase-Space Distributions b under Linear Unitary Transformations c

Equations (233)

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f ̃ ( p ) = { F ˆ f } ( p ) = | K | 2 π N / 2 f ( q ) exp ( i K q p ) d N q ,
f ( q ) = { F ˆ 1 f ̃ } ( q ) = | K | 2 π N / 2 f ̃ ( p ) exp ( i K q p ) d N p .
f ( q ) g ( q ) d N q = f ̃ ( p ) g ̃ ( p ) d N p .
Δ q i Δ p i 1 2 | K | ,
Δ q i 2 = ( q i q ̄ i ) 2 | f ( q ) | 2 d N q | f ( q ) | 2 d N q , q ̄ i = q i | f ( q ) | 2 d N q | f ( q ) | 2 d N q ,
Δ p i 2 = ( p i p ̄ i ) 2 | f ̃ ( p ) | 2 d N p | f ̃ ( p ) | 2 d N p , p ̄ i = p i | f ̃ ( p ) | 2 d N p | f ̃ ( p ) | 2 d N p .
f θ ( s ) = { F ˆ θ f } ( s ) = | K | ( tan θ i ) 2 π σ tan θ N / 2 f ( q ) exp i K ( q 2 + σ 2 s 2 ) cos θ 2 σ q s 2 σ 2 sin θ d N q ,
f θ ( s ) g θ ( s ) d N s = f ( q ) g ( q ) d N q .
f 0 ( s ) = σ N / 2 f ( σ s ) ,
f π / 2 ( s ) = σ N / 2 f ̃ ( σ 1 s ) .
{ C ˆ f } ( q ) = a f ( q ) exp i K v ( q , q ) d N q ,
v ( q , q ) v 0 + v q + v q + q V q + 2 q V q + q V q 2 ,
{ C ˆ f 1 } ( q ) { C ˆ f 2 } ( q ) d N q = | a | 2 2 π K N 1 Det ( V ) f 1 ( q ) f 2 ( q ) d N q ,
a = exp ( i ϕ 0 ) K 2 π N / 2 Det ( V ) ,
v ( q , q ) = q B 1 A q 2 q B 1 q + q D B 1 q 2 ,
B 1 A = ( B 1 A ) T , D B 1 = ( D B 1 ) T ,
A B T = B A T , B T D = D T B .
{ C ˆ ( S ) f } ( q ) = i K 2 π N / 2 1 Det ( B ) f ( q ) × exp i K q B 1 A q 2 q B 1 q + q D B 1 q 2 d N q ,
S = A B C D .
O I I O S T O I I O = S 1 ,
D A T C B T = I ,
C = D B 1 A ( B 1 ) T = ( D A T I ) ( B 1 ) T ,
Det ( S ) = 1 .
{ C ˆ ( S 2 ) C ˆ ( S 1 ) f } ( q ) = { C ˆ ( S 2 S 1 ) f } ( q ) ,
S FrFT ( θ , σ ) = σ 1 I cos θ σ I sin θ σ 1 I sin θ σ I cos θ .
{ S ˆ ( b ) f } ( q ) = b N / 2 f ( q / b ) ,
S Sc ( b ) = b I O O b 1 I .
{ H ˆ ( τ ) f } ( q ) = i K 2 π τ N / 2 f ( q ) exp i K | q q | 2 2 τ d N q ,
S Fresnel ( τ ) = I τ I O I .
{ V ˆ ( τ ) f } ( q ) = exp i K τ | q | 2 2 f ( q ) .
S Chirp ( τ ) = I O τ I I .
S Chirp ( τ ) = lim ϵ 0 I ϵ I τ I I + ϵ τ I .
{ T ˆ ( q 0 , 0 ) f } ( q ) = f ( q q 0 ) .
{ T ˆ ( 0 , p 0 ) f } ( q ) = exp ( i K q p 0 ) f ( q ) ,
{ T ˆ ( 0 , p 0 ) T ˆ ( q 0 , 0 ) f } ( q ) = exp ( i K q 0 p 0 ) { T ˆ ( q 0 , 0 ) T ˆ ( 0 , p 0 ) f } ( q ) .
{ T ˆ ( q 0 , p 0 ) f } ( q ) = exp i K q 0 p 0 2 exp ( i K q p 0 ) f ( q q 0 ) .
f ̃ w ( q , p ) = exp i K q p 2 | K | 2 π N / 2 f ( q ) w ( q q ) exp ( i K q p ) d N q ,
f ̃ w ( q , p ) = exp i K q p 2 | K | 2 π N / 2 f ̃ ( p ) w ̃ ( p p ) exp ( i K q p ) d N p .
f ̃ w ( q , p ) = | K | 2 π N / 2 { T ˆ ( q , p ) w } ( q ) f ( q ) d N q ,
S f , w ( q , p ) = | f ̃ w ( q , p ) | 2 = | K | 2 π N f ( q ) w ( q q ) exp ( i K q p ) d N q 2 .
f ( q ) = f 1 ( q ) + f 2 ( q ) ,
f 1 ( q ) = π 1 / 4 exp q 2 2 ,
f 2 ( q ) = exp ( i Φ ) { T ˆ ( q 0 , p 0 ) f 1 } ( q ) .
W f ( q , p ) = | K | 2 π N f q q 2 f q + q 2 exp ( i K q p ) d N q .
W f ( q , p ) = | K | 2 π N f ̃ p p 2 f ̃ p + p 2 exp ( i K q p ) d N p .
W f ( q , p ) d N p = | f ( q ) | 2 ,
W f ( q , p ) d N q = | f ̃ ( p ) | 2 .
W f [ σ ( s cos θ t sin θ ) , σ 1 ( t cos θ + s sin θ ) ] d N t = | f θ ( s ) | 2 ,
W f [ σ ( s cos θ t sin θ ) , σ 1 ( t cos θ + s sin θ ) ] = | K | 2 π N f θ s s 2 f θ s + s 2 exp ( i K s t ) d N s .
W f ( q , p ) d N p d N q = | f ( q ) | 2 d N q .
W C ˆ ( S ) f ( q , p ) = W f ( A q + B p , C q + D p ) ,
W f ( q , p ) = W C ˆ ( S ) f ( A q + B p , C q + D p ) .
f ( q ) = 2 π 1 / 4 exp ( 2 q 2 ) ,
W f ( q , p ) = exp 4 q 2 p 2 4 .
W f θ ( s , t ) = W f [ σ ( s cos θ t sin θ ) , σ 1 ( t cos θ + s sin θ ) ] ,
W f ( q , p ) = W f θ ( σ 1 q cos θ + σ p sin θ , σ p cos θ σ 1 q sin θ ) .
W S ˆ b f ( q , p ) = W f ( q / b , b p ) .
W H ˆ τ f ( q , p ) = W f ( q τ p , p ) .
W V ˆ τ f ( q , p ) = W f ( q , p τ q ) .
W T ˆ q 0 , p 0 f ( q , p ) = W f ( q q 0 , p p 0 ) .
W f ( q , p ) W g ( q , p ) d N q d N p = | K | 2 π N g ( q ) f ( q ) d N q 2 .
W f 2 ( q , p ) d N q d N p = | K | 2 π N | f ( q ) | 2 d N q 2 .
W f 1 + f 2 ( q , p ) = W f 1 ( q , p ) + 2 [ W f 1 , f 2 ( q , p ) ] + W f 2 ( q , p ) ,
W f 1 , f 2 ( q , p ) = W f 2 , f 1 ( q , p ) = | K | 2 π N f 1 q q 2 f 2 q + q 2 exp ( i K q p ) d N q .
W f 1 ( q , p ) = exp ( q 2 p 2 ) ,
W f 2 ( q , p ) = exp [ ( q q 0 ) 2 ( p p 0 ) 2 ] ,
2 [ W f 1 , f 2 ( q , p ) ] = 2 exp q q 0 2 2 p p 0 2 2 cos q p 0 q 0 p + Φ .
p = Ω q .
f ( q ) = 1 π 1 / 4 2 n n ! H n ( q ) exp q 2 2 ,
q 2 + p 2 = 2 n + 1 .
W f ( q , p ) = ( 1 ) n π L n [ 2 ( q 2 + p 2 ) ] exp ( q 2 p 2 ) ,
F ( q 1 , q 2 ) = f ( q 1 ) f ( q 2 ) ,
f n ( q ) F ( q , q ) d N q = Λ n f n ( q ) .
f n ( q ) f n ( q ) d N q = δ n , n ,
n f n ( q ) f n ( q ) = δ ( q q ) .
F ( q 1 , q 2 ) = n Λ n f n ( q 1 ) f n ( q 2 ) .
W F ( q , p ) = | K | 2 π N f q q 2 f q + q 2 exp ( i K q p ) d N q = | K | 2 π N F q q 2 , q + q 2 exp ( i K q p ) d N q .
S F , w ( q , p ) = | K | 2 π N F ( q 1 , q 2 ) w ( q 1 q ) w ( q 2 q ) × exp [ i K ( q 2 q 1 ) p ] d N q 1 d N q 2 ,
W F ( q , p ) = n Λ n W f n ( q , p ) ,
S F , w ( q , p ) = n Λ n S f n , w ( q , p ) .
ν = n Λ n 2 1 / 2 n Λ n = | F ( q 1 , q 2 ) | 2 d N q 1 d N q 2 1 / 2 F ( q , q ) d N q .
W F 2 ( q , p ) d N q d N p = | K | 2 π N | F ( q 1 , q 2 ) | 2 d N q 1 d N q 2 ,
ν = 2 π | K | N / 2 W F 2 ( q , p ) d N q d N p 1 / 2 W F ( q , p ) d N q d N p .
F QH ( q 1 , q 2 ) = F 0 q 1 + q 2 2 μ ( q 2 q 1 ) ,
| μ ( q ) | μ ( 0 ) = 1 , μ ( q ) = μ ( q ) .
ν = F 0 2 ( q ) d N q | μ ( q ) | 2 d N q 1 / 2 F 0 ( q ) d N q ,
W F QH ( q , p ) = | K | 2 π N / 2 F 0 ( q ) μ ̃ ( p ) ,
W f n ( q , p ) d N q d N p = 1 .
W f n ( q , p ) W f n ( q , p ) d N q d N p = | K | 2 π N f n ( q ) f n ( q ) d N q 2 = 0 .
F ( q 1 , q 2 ) = π 1 / 2 σ 2 exp ( q 1 + q 2 ) 2 4 σ 1 2 exp ( q 2 q 1 ) 2 4 σ 2 2 ,
W F ( q , p ) = exp q 2 σ 1 2 σ 2 2 p 2 .
Λ n = 2 π 1 / 2 σ 1 σ 2 σ 1 + σ 2 σ 1 σ