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Fresnel diffraction in phase-space
Fresnel diffraction in phase-space R. Palomino1, 2, M. de Icaza-Herrera2, V. M. Castaño2 1 2
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, A.P. 1152 Puebla, Pue., México Instituto de Física, UNAM, A.P. 1-1010 Querétaro, 76000, México
Abstract: Fresnel Diffraction is worked out in phase space by simple calculations based on the Wigner Distribution Functions for the transfer functions of an array of ideal lenses, which allow the simulation of a composite material and of a continuous medium. Key words: Wigner-distribution-function – fresnel-diffraction
2. Mathematical background 2.1. The Cornu’s integrals Fresnels integrals are defined as: s
pt2 C( s ) = ∫ cos dt 2 0
(1)
s
1. Introduction One of the classical problems of optics is the elegant solution to the diffraction by an aperture derived by Augustin Jean Fresnel, from whose formulation the so-called Fresnel integrals emerge naturally, constituting not only a must in any optics course but also a powerful tool in many areas of Physics. Marie Alfred Cornu, another pioneer of modern optics, devised a rather elegant and ingenuous geometrical description of the Fresnel integrals, which, in addition to representing a tool for calculating numerical values, also stresses out some of the very fundamentals of optics, related to trajectories, diffraction intensities and the like, opening the way to physical optics in a modern sense. With the emerging of computers, graphical methods such as the beautiful spiral discovered by Cornu [1] have lost interest in the eyes of the specialists in optics. The fundamental physics behind, however, deserves further exploration under the light of the new mathematical tools, specially those developed with the aid of phase space functions and group theory, for they have brought exciting possibilities into optics, not only for achieving a deeper understanding of some classical problems, but also for studying modern optical devices, such as fiber optics. Accordingly, this article shows an alternative derivation of Cornu’s integrals and spiral, purely from phase space considerations, which also enables us to recover the wave distribution from the integrals themselves.
Received 19 June 2000; accepted 2 September 2000. Correspondence to: V. M. Castaño Fax: ++52-1-56234165 E-mail: [email protected] Optik 112, No. 1 (2001) 37–39 © 2001 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
pt2 S ( s ) = ∫ sin dt . 2 0
(2)
being customary to define s
B (s) = ∫
ipt2 e 2
dt ≡ C ( s ) + i S ( s ).
(3)
0
As it is well known in optics, the locus of this function is the so-called Cornu’s spiral, which is used in various problems in optics.
2.2. The Wigner distribution function The so-called Wigner Distribution Function (WDF) is a phase-space mathematical tool, originally designed in the 1920s for dealing with quantum mechanics problems [3]. In the last few decades, however, photon, X-ray and electron optics have found interesting applications of this and other related phase-space functions [3–8]. In general, for a pure quantum system, described by the wave function Y ( p) of its momentum p, the Wigner Function (WDF) is defined as: F ( q, p ) = 1 ph
n ∞
∫d
n
yΨ ( q + y ) Ψ * ( q – y )
–∞
2i ⋅ exp – ( p ⋅ y ) . h Where, as usual, h is Planck’s constant divided by 2 p and n the number of freedom’s degrees of the system. The main properties and mathematical characteristics of the WDF, as well as their relation to other functions (Cohens, ambiguity, wavelets, etc.) have been reviewed elsewhere [3–11] being here only relevant to point out that the WDF is indeed a phase-space distribution, since it depends, as observed in the above equation (1) on both spatial and momentum coordinates. 0030-4026/2001/112/1-37 $ 15.00/0
38
R. Palomino et al., Fresnel diffraction in phase-space
It is customary, however, to define the WDF of a function f, which represents a wave function or any other physical quantity, as: Wf ( x, w) =
∞
u f * x – u 2
∫ f x + 2
–∞
⋅ exp {– i w ⋅ u} d 2 u,
(4)
where x and w are the position and momentum vectors. The kernel of the WDF is, of course: H f ( p, u ) = f p + u f * p – u . 2 2
(5)
3. Cornu’s integrals and the Wigner distribution function Let us consider a physical medium formed by an array of tiny lenses, as schematically depicted in figure 1. This model can describe nearly any composite material, that is, a physical system formed by a continuous matrix in which a collection of “lenses”, i.e. particles with different scattering properties as compared to that of the matrix, are embedded. By changing the periodicity, the size of the “lenses” and by introducing “defects” in the array, it is possible to simulate many real situations. It is straightforward to calculate [4], in terms of a “Physical Optics” language, the transfer function fL of an ideal lens (of focal distance f placed in the position qkx, qky of the crystalline structure: ik ⋅ exp – o ( x – qkx ) 2 + ( y – qky ) 2 , (6) 2f where x, y and qkx, qky are the spatial and momentum spaces coordinates respectively, ko the wave vector of the incident wave and f(x, y) is the distribution before the lens. To calculate the whole contribution of a layer of thickness Dz, all one has to do is, according to the principle of superposition, to add over all the lenses contained within that layer:
]
gT ( x , y , z ) = ∑ g ( x , y ) k
ik ⋅ exp – o ( x – qkx ) 2 + ( y – qky ) 2 . (7) 2f By applying the definition of equation (4), the WDF of equation (7) is:
[
Where Wf is the WDF before the layer D z. For the case of having a plane wave incident over that layer, the resulting WDF is: W fTP ( p, w ) 4 p 2 AA *
f L ( x , y; qk ) = f ( x , y )
[
Fig. 1. Schematical representation of a “crystal” characterized by the lattice parameter a.
]
W fTP ( p, w ) = ∑ k, l
i k q 2 – qk2 ⋅ exp o l + p – Dz w ⋅ ( qk – ql ) × k0 f 2 q – ql k × W f p – Dz w , w + o p – Dz w – k . k0 f k0 2 (8)
=∑ k, l
i k q 2 – qk2 ⋅ exp o l + p – Dz w ⋅ ( qk – ql ) × f 2 k 0 k q – ql ×d o p – k + 1 – Dz w . (9) f { 2 Obviously, what matters, from a practical standpoint, is to calculate the original function, either as a function f of the spatial coordinates, f(x) or as its Fourier transform F(w). To do this, let us consider the kernel of the WDF and its relation to the original WDF [12]: H f ( x,w ) = f x + w f * x – w 2 2 = 12 4p
∞
∫
W f ( x, K ) exp {i w ⋅ K} dK .
(10)
–∞
By evaluating Hf in x = w/2, one gets: g ( u ) g * ( 0 ) = Hg u , u , 2
(11)
since f (0) f * (0) = Hf (0,0), then f * ( 0 ) = H f ( 0, 0 ) and thus f ( 0 ) = ( H f ( 0, 0 )) exp iδ . Substituting in equation (11): Hg u , u 2 iδ e . g (u ) = (12) Hg ( 0 , 0 )
R. Palomino et al., Fresnel diffraction in phase-space
In the case of the layer D z, one can repeat the above inversion procedure to obtain the function, which yields: 2 i k0 H f ( x, w ) q – x – w = ∑ exp l AA * 2 l 2 ( f – Dz 2 i k0 q – x + w × ∑ exp – k 2 l 2 ( f – Dz (13)
k0 , P+ = x + w , P– = x – w ( f – Dz ) 2 2 the exponential functions can be re-written as:
a=
iα ( q – P ) 2 l –
∑e
e2
l
– iα ( qk – P+ ) 2 2 .
(14)
(15)
k
By taking the limit when the lattice parameter a of the array (fig. 1) is very small as compared to the wavelength of the incoming wave, that is, in the case of having a continuous instead of a composite medium, the summations of equation (10) become integrals:
∑e
– iα ( qk – P+ ) 2 2
∆q x ∆q y a2
k
≈1 a
⋅1 a
Na 2
∫
e
– iα ( q x – P+ x ) 2 2 dq
– Na 2 Na 2 – iα ( q y – P+ y ) 2 dq e 2
1 a
∫
e
α (l – u ) y p α (– l – u ) y p
.
(19)
∫
References (16)
y.
– Na 2
– iα [ q x – P– x ]2 2 dq
x
– Na 2
α Na – P –x π 2
=1 a
p α
=1 a
p B (s) α
∫
iπ t2 dt e 2
α – Na – P –x p 2 α p α p
Na – P –x 2 – Na – P –x 2
.
(17)
One can re-write the integrals as:
a (qx – P– x)2 = p t 2.
The approach described here allows to deduce the Cornu’s integrals from fundamental considerations in phase space, enhancing the physics involved, for the trajectories in phase space can be regarded as a generalized “geometrical” optics, as has been demonstrated previously [6]. It is also important to point out the possibility of recovering the wave function f (x) or its Fourier transform F(u) from the Cornu’s integrals in a very simple way, as shown in equation (16). The method also allows to calculate the optical properties of composite materials as well as those of a continuous medium, by conveniently choosing the parameters which define the lattice of fig. 1. This, along with computational advantages of the WDF, which have been described elsewhere [4], open interesting opportunities for explaining, and even predicting, the optical behavior of complex media.
x
We have basically deduced Cornu’s integrals, since, by making the following change of variables: Na 2
⋅ B (s)
α (l – u ) x p α (– l – u ) x p
4. Concluding remarks
By defining:
∑
AA * e iδ p B ( s ) α a2
f (u ) =
39
(18)
Finally, by using the method described before to calculate f (u) from the Kernel of the WDF, one obtains:
[1] Hecht E, Zajak A: Optics. pp. 378–382. Addison-Wesley Publishing Co, Tokyo 1974 [2] Wigner EP: On the Quantum Correction for Thermodynamic Equilibrium. Phys. Rev. 40 (1932) 749–759 [3] Castaño VM: A generalized display of the Contrast Transfer Function in Transmission Electron Microscopy. Optik 81 (1988) 35–37 [4] Gutiérrez-Castrejón R, Castaño VM: Analysis of optical systems through intermediate functions. Optik 91 (1992) 24–28 [5] Vázquez G, Acosta D, Castaño VM: Two dimensional analysis of the contrast transfer function in high resolution transmission electron microscopy. Optik 89 (1992) 181–183 [6] Castaño VM, Santiago P, Gómez A: Image displacements in transmission electron microscopy: a phase space approach. Optik 95 (1993) 31–33 [7] Gómez A, Santiago P, Castaño VM: The Van Cittert-Zernike theorem revisited: alternative approaches in the small angle approximation for electron diffraction. Optik 95 (1993) 59–62 [8] Reyes J, Rodríguez R, Cotorogea M, Castaño VM: Application of Wigner Distribution Function to Fiber Optics. Optik 110 (1999) 305–308 [9] Castaño VM, Gómez A: A phase space approach to X-ray optics. J. X-Ray Sci. Technol. 4 (1993) 67–76 [10] Cohen L: Generalized Phase-Space Distribution Functions. J. Math. Phys. 7 (1966) 781–786 [11] Papoulis A: Ambiguity function in Fourier optics. J. Opt. Soc. Am. 64 (1974) 779–788 [12] Palomino R: Aplicaciones de las técnicas de espacio fase a la caracterización ultrasónica de materiales. M.Sc. thesis (Physics), pp 45–53, U.N.A.M., Mexico City 1998
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, A.P. 1152 Puebla, Pue., México Instituto de Física, UNAM, A.P. 1-1010 Querétaro, 76000, México
Abstract: Fresnel Diffraction is worked out in phase space by simple calculations based on the Wigner Distribution Functions for the transfer functions of an array of ideal lenses, which allow the simulation of a composite material and of a continuous medium. Key words: Wigner-distribution-function – fresnel-diffraction
2. Mathematical background 2.1. The Cornu’s integrals Fresnels integrals are defined as: s
pt2 C( s ) = ∫ cos dt 2 0
(1)
s
1. Introduction One of the classical problems of optics is the elegant solution to the diffraction by an aperture derived by Augustin Jean Fresnel, from whose formulation the so-called Fresnel integrals emerge naturally, constituting not only a must in any optics course but also a powerful tool in many areas of Physics. Marie Alfred Cornu, another pioneer of modern optics, devised a rather elegant and ingenuous geometrical description of the Fresnel integrals, which, in addition to representing a tool for calculating numerical values, also stresses out some of the very fundamentals of optics, related to trajectories, diffraction intensities and the like, opening the way to physical optics in a modern sense. With the emerging of computers, graphical methods such as the beautiful spiral discovered by Cornu [1] have lost interest in the eyes of the specialists in optics. The fundamental physics behind, however, deserves further exploration under the light of the new mathematical tools, specially those developed with the aid of phase space functions and group theory, for they have brought exciting possibilities into optics, not only for achieving a deeper understanding of some classical problems, but also for studying modern optical devices, such as fiber optics. Accordingly, this article shows an alternative derivation of Cornu’s integrals and spiral, purely from phase space considerations, which also enables us to recover the wave distribution from the integrals themselves.
Received 19 June 2000; accepted 2 September 2000. Correspondence to: V. M. Castaño Fax: ++52-1-56234165 E-mail: [email protected] Optik 112, No. 1 (2001) 37–39 © 2001 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
pt2 S ( s ) = ∫ sin dt . 2 0
(2)
being customary to define s
B (s) = ∫
ipt2 e 2
dt ≡ C ( s ) + i S ( s ).
(3)
0
As it is well known in optics, the locus of this function is the so-called Cornu’s spiral, which is used in various problems in optics.
2.2. The Wigner distribution function The so-called Wigner Distribution Function (WDF) is a phase-space mathematical tool, originally designed in the 1920s for dealing with quantum mechanics problems [3]. In the last few decades, however, photon, X-ray and electron optics have found interesting applications of this and other related phase-space functions [3–8]. In general, for a pure quantum system, described by the wave function Y ( p) of its momentum p, the Wigner Function (WDF) is defined as: F ( q, p ) = 1 ph
n ∞
∫d
n
yΨ ( q + y ) Ψ * ( q – y )
–∞
2i ⋅ exp – ( p ⋅ y ) . h Where, as usual, h is Planck’s constant divided by 2 p and n the number of freedom’s degrees of the system. The main properties and mathematical characteristics of the WDF, as well as their relation to other functions (Cohens, ambiguity, wavelets, etc.) have been reviewed elsewhere [3–11] being here only relevant to point out that the WDF is indeed a phase-space distribution, since it depends, as observed in the above equation (1) on both spatial and momentum coordinates. 0030-4026/2001/112/1-37 $ 15.00/0
38
R. Palomino et al., Fresnel diffraction in phase-space
It is customary, however, to define the WDF of a function f, which represents a wave function or any other physical quantity, as: Wf ( x, w) =
∞
u f * x – u 2
∫ f x + 2
–∞
⋅ exp {– i w ⋅ u} d 2 u,
(4)
where x and w are the position and momentum vectors. The kernel of the WDF is, of course: H f ( p, u ) = f p + u f * p – u . 2 2
(5)
3. Cornu’s integrals and the Wigner distribution function Let us consider a physical medium formed by an array of tiny lenses, as schematically depicted in figure 1. This model can describe nearly any composite material, that is, a physical system formed by a continuous matrix in which a collection of “lenses”, i.e. particles with different scattering properties as compared to that of the matrix, are embedded. By changing the periodicity, the size of the “lenses” and by introducing “defects” in the array, it is possible to simulate many real situations. It is straightforward to calculate [4], in terms of a “Physical Optics” language, the transfer function fL of an ideal lens (of focal distance f placed in the position qkx, qky of the crystalline structure: ik ⋅ exp – o ( x – qkx ) 2 + ( y – qky ) 2 , (6) 2f where x, y and qkx, qky are the spatial and momentum spaces coordinates respectively, ko the wave vector of the incident wave and f(x, y) is the distribution before the lens. To calculate the whole contribution of a layer of thickness Dz, all one has to do is, according to the principle of superposition, to add over all the lenses contained within that layer:
]
gT ( x , y , z ) = ∑ g ( x , y ) k
ik ⋅ exp – o ( x – qkx ) 2 + ( y – qky ) 2 . (7) 2f By applying the definition of equation (4), the WDF of equation (7) is:
[
Where Wf is the WDF before the layer D z. For the case of having a plane wave incident over that layer, the resulting WDF is: W fTP ( p, w ) 4 p 2 AA *
f L ( x , y; qk ) = f ( x , y )
[
Fig. 1. Schematical representation of a “crystal” characterized by the lattice parameter a.
]
W fTP ( p, w ) = ∑ k, l
i k q 2 – qk2 ⋅ exp o l + p – Dz w ⋅ ( qk – ql ) × k0 f 2 q – ql k × W f p – Dz w , w + o p – Dz w – k . k0 f k0 2 (8)
=∑ k, l
i k q 2 – qk2 ⋅ exp o l + p – Dz w ⋅ ( qk – ql ) × f 2 k 0 k q – ql ×d o p – k + 1 – Dz w . (9) f { 2 Obviously, what matters, from a practical standpoint, is to calculate the original function, either as a function f of the spatial coordinates, f(x) or as its Fourier transform F(w). To do this, let us consider the kernel of the WDF and its relation to the original WDF [12]: H f ( x,w ) = f x + w f * x – w 2 2 = 12 4p
∞
∫
W f ( x, K ) exp {i w ⋅ K} dK .
(10)
–∞
By evaluating Hf in x = w/2, one gets: g ( u ) g * ( 0 ) = Hg u , u , 2
(11)
since f (0) f * (0) = Hf (0,0), then f * ( 0 ) = H f ( 0, 0 ) and thus f ( 0 ) = ( H f ( 0, 0 )) exp iδ . Substituting in equation (11): Hg u , u 2 iδ e . g (u ) = (12) Hg ( 0 , 0 )
R. Palomino et al., Fresnel diffraction in phase-space
In the case of the layer D z, one can repeat the above inversion procedure to obtain the function, which yields: 2 i k0 H f ( x, w ) q – x – w = ∑ exp l AA * 2 l 2 ( f – Dz 2 i k0 q – x + w × ∑ exp – k 2 l 2 ( f – Dz (13)
k0 , P+ = x + w , P– = x – w ( f – Dz ) 2 2 the exponential functions can be re-written as:
a=
iα ( q – P ) 2 l –
∑e
e2
l
– iα ( qk – P+ ) 2 2 .
(14)
(15)
k
By taking the limit when the lattice parameter a of the array (fig. 1) is very small as compared to the wavelength of the incoming wave, that is, in the case of having a continuous instead of a composite medium, the summations of equation (10) become integrals:
∑e
– iα ( qk – P+ ) 2 2
∆q x ∆q y a2
k
≈1 a
⋅1 a
Na 2
∫
e
– iα ( q x – P+ x ) 2 2 dq
– Na 2 Na 2 – iα ( q y – P+ y ) 2 dq e 2
1 a
∫
e
α (l – u ) y p α (– l – u ) y p
.
(19)
∫
References (16)
y.
– Na 2
– iα [ q x – P– x ]2 2 dq
x
– Na 2
α Na – P –x π 2
=1 a
p α
=1 a
p B (s) α
∫
iπ t2 dt e 2
α – Na – P –x p 2 α p α p
Na – P –x 2 – Na – P –x 2
.
(17)
One can re-write the integrals as:
a (qx – P– x)2 = p t 2.
The approach described here allows to deduce the Cornu’s integrals from fundamental considerations in phase space, enhancing the physics involved, for the trajectories in phase space can be regarded as a generalized “geometrical” optics, as has been demonstrated previously [6]. It is also important to point out the possibility of recovering the wave function f (x) or its Fourier transform F(u) from the Cornu’s integrals in a very simple way, as shown in equation (16). The method also allows to calculate the optical properties of composite materials as well as those of a continuous medium, by conveniently choosing the parameters which define the lattice of fig. 1. This, along with computational advantages of the WDF, which have been described elsewhere [4], open interesting opportunities for explaining, and even predicting, the optical behavior of complex media.
x
We have basically deduced Cornu’s integrals, since, by making the following change of variables: Na 2
⋅ B (s)
α (l – u ) x p α (– l – u ) x p
4. Concluding remarks
By defining:
∑
AA * e iδ p B ( s ) α a2
f (u ) =
39
(18)
Finally, by using the method described before to calculate f (u) from the Kernel of the WDF, one obtains:
[1] Hecht E, Zajak A: Optics. pp. 378–382. Addison-Wesley Publishing Co, Tokyo 1974 [2] Wigner EP: On the Quantum Correction for Thermodynamic Equilibrium. Phys. Rev. 40 (1932) 749–759 [3] Castaño VM: A generalized display of the Contrast Transfer Function in Transmission Electron Microscopy. Optik 81 (1988) 35–37 [4] Gutiérrez-Castrejón R, Castaño VM: Analysis of optical systems through intermediate functions. Optik 91 (1992) 24–28 [5] Vázquez G, Acosta D, Castaño VM: Two dimensional analysis of the contrast transfer function in high resolution transmission electron microscopy. Optik 89 (1992) 181–183 [6] Castaño VM, Santiago P, Gómez A: Image displacements in transmission electron microscopy: a phase space approach. Optik 95 (1993) 31–33 [7] Gómez A, Santiago P, Castaño VM: The Van Cittert-Zernike theorem revisited: alternative approaches in the small angle approximation for electron diffraction. Optik 95 (1993) 59–62 [8] Reyes J, Rodríguez R, Cotorogea M, Castaño VM: Application of Wigner Distribution Function to Fiber Optics. Optik 110 (1999) 305–308 [9] Castaño VM, Gómez A: A phase space approach to X-ray optics. J. X-Ray Sci. Technol. 4 (1993) 67–76 [10] Cohen L: Generalized Phase-Space Distribution Functions. J. Math. Phys. 7 (1966) 781–786 [11] Papoulis A: Ambiguity function in Fourier optics. J. Opt. Soc. Am. 64 (1974) 779–788 [12] Palomino R: Aplicaciones de las técnicas de espacio fase a la caracterización ultrasónica de materiales. M.Sc. thesis (Physics), pp 45–53, U.N.A.M., Mexico City 1998