Complementary Huygens principle for geometrical

and nongeometrical optics

Alfredo Luis

Departamento de Optica, Facultad de Ciencias Fısicas, Universidad Complutense,

28040 Madrid, Spain

Abstract. We develop a fundamental principle depicting the generalized ray

formulation of optics provided by the Wigner function. This principle is formally

identical to the Huygens-Fresnel principle but in terms of opposite concepts, rays

instead of waves, and incoherent superpositions instead of coherent ones. This ray

picture naturally includes diffraction and interference, and provides a geometrical

picture of the degree of coherence.

PACS numbers: 42.25.-p, 42.15.-i, 42.25.Fx, 42.25.Hz

Published in European Journal of Physics vol. 28, p. 231-240, (2007)

European Journal of Physics c°copyright (2007) IOP Publishing Ltd.

Complementary Huygens principle 2

1. Introduction

The description of light in terms of the Wigner function provides a ray picture of

optics which is exact, complete, and rigorous (at least within paraxial propagation in

homogeneous or weakly inhomogeneous media) covering most of undergraduate optics

[1, 2, 3, 4, 5, 6, 7, 8]. In particular, and in sharp contrast to standard geometrical optics,

this Wigner ray picture encompasses coherent wave phenomena, such as interference,

diffraction, and polarization [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. The price to be

paid is that the Wigner function can take negative values. This is rather disturbing,

since otherwise it would be a first-rate aspirant to represent the energy content of rays

(radiance or specific intensity).

In this work we elaborate the Wigner function approach to optics by formulating

it in terms of a simple and basic principle, formally analogous to the Huygens-

Fresnel principle but with inverted terms: Each point reached by the light becomes a

secondary source of rays, so that the disturbance evolves as the result of the incoherent

superposition of these secondary rays. Given the importance and insight of the Huygens-

Fresnel principle for the wave picture we may expect that this inverted version (let us

call it complementary Huygens principle) might well condensate and illustrate this ray

picture of optics.

Using this complementary Huygens principle as an starting point we derive typical

coherent phenomena such as interference and diffraction. In particular we show that

this approach provides a natural geometrical picture of coherence as an average value

of the phase difference.

This issue has interesting didactical applications in undergraduate optics as an

illustration of coexistence of different theories. On the one hand we have two partially

incompatible theories (standard geometrical optics versus Wigner ray optics) leading to

contradictory predictions in spite of using the same concept of light trajectories. On

the other hand, we have two equivalent theories (Wigner ray optics and wave optics)

providing different explanations for the same predictions using very different concepts.

2. Ray picture of wave optics

For the sake of simplicity we focus on harmonic, scalar, fully coherent waves where the

electric field E is expressed as

E(x, z, t) = Uz(x)ei(nkz−ωt), (1)

where k is the wave number k = 2π/λ, λ is the wavelength in vacuum, n is the refraction

index, ω is the frequency, and throughout we assume that the amplitude Uz(x) dependsexclusively on a single transversal coordinate x orthogonal to the main propagation

direction along axis z.

The associated Wigner function is defined as

Wz(x, p) =k

2π

Z ∞−∞dx0Uz(x− x0/2)U∗z (x+ x0/2)eikpx

0. (2)

Complementary Huygens principle 3

x

W (x,p)z

z

��p/n

Figure 1. Illustration of the ray parameters.

The Wigner function provides complete information about the corresponding wave, as

in can be explicitly demonstrated by inverting (2)

Uz(x) = 1

U∗z (0)Z ∞−∞dpeikpxWz(x/2, p), (3)

where we have assumed without loss of generality that U∗z (0) 6= 0. In particular the fieldintensity Iz(x), which is proportional to the modulus square of the wave amplitude, can

be expressed in terms of Wz(x, p) as

Iz(x) = |Uz(x)|2 =Z ∞−∞dpWz(x, p). (4)

The variables (x, p) can be interpreted as representing a light ray at the transversal

coordinate x in the plane z propagating in a direction forming an angle θ = p/n with

the axis z (see figure 1). Thus, Wz(x, p) is a kind of weight of the (x, p) ray at plane

z. This weight is intimately connected with the field energy as revealed by the above

relation (4) which shows that the field intensity Iz(x) at each point (x, z) is the sum of

the values of the Wigner function associated to all rays passing through (x, z).

For completeness we mention three remarks:

(i) When the wave is not fully coherent, the definition of the Wigner function is

modified by replacing Uz(x1)U∗z (x2) by the cross-spectral density function Γ(x1, x2) =hUz(x1)U∗z (x2)i, where the brackets denote ensemble average.

(ii) The inversion formula (3) holds provided that the wave in fully coherent, i. e.

Γ(x1, x2) = U(x1)U∗(x2). Otherwise, the inversion provides the cross-spectral densityfunction Γ(x1, x2) =

R∞−∞ dpe

ikp(x1−x2)W [(x1 + x2)/2, p].(iii) Not all functions are admissible as Wigner functions, since the associated

Γ(x1, x2) must be always positive definiteR∞−∞ dx1dx2f

∗(x1)Γ(x1, x2)f(x2) ≥ 0 for everyf(x).

Complementary Huygens principle 4

2.1. Free wave propagation

Free propagation of an electromagnetic wave in an homogeneous medium is usually

described in undergraduate optics via the Huygens-Fresnel principle: (i) Each point

reached by the light becomes a secondary source of waves whose amplitudes are

proportional to the amplitude of the incoming wave at that point, (ii) The disturbance

evolves as the result of the coherent superposition of the secondary waves. The

mathematical expression of this principle is of the form

Uz(r) = nk

2πi

Z ∞−∞d2r0U0(r0) cos θe

inkR

R, (5)

where we have considered the most general case in which two transversal coordinates

are required to describe both the observation plane z > 0 with r = (x, y), and the

source plane z = 0 with r0 = (x0, y0). The angle θ is the one between the vector

R = (x− x0, y − y0, z) and the z axis, and R =q(r− r0)2 + z2 is the distance between

the source point r0 at z = 0 and the observation point r at z > 0.We appreciate in (5) that each secondary-source point at r0 produces a spherical

wave einkR/R with amplitude U0(r0). The integration represents the coherent

superposition of these secondary waves. We recall that coherent superposition occurs

when the complex amplitude of the superposition is the sum of the complex amplitudes

of the incoming waves. On the other hand, the superposition is incoherent when the

intensity of the superposition is the sum of the intensities of the incoming waves.

Within paraxial approximation |r − r0| << z the mathematical expression of the

Huygens-Fresnel principle becomes the Fresnel diffraction formula

Uz(r) = nk

2πizeinkz

Z ∞−∞d2r0U0(r0)eink(r−r0)2/(2z). (6)

Furthermore, for far enough observation planes (z → ∞) the Fraunhofer diffractionformula holds

Uz(r) = nk

2πizeinkzeinkr

2/(2z)Z ∞−∞d2r0U0(r0)e−inkr·r0/z. (7)

For the sake of simplicity we adapt the above expressions (6), (7) to the simpler

one-dimensional dependence of the transversal amplitude [19]. For Fresnel diffraction

we have

Uz(x) =snk

2πizeinkz

Z ∞−∞dx0U0(x0)eink(x−x0)2/(2z), (8)

while for Fraunhofer diffraction

Uz(x) =snk

2πizeinkzeinkx

2/(2z)Z ∞−∞dx0U0(x0)e−inkxx0/z, (9)

which is equivalent to

Uz(x) =rn

izeinkzeinkx

2/(2z)U0(nx/z), (10)

Complementary Huygens principle 5

= 0z z��p/n

x-zp/n

x

p

p

W (x-zp/n,p)0

W (x,p)z

Figure 2. Transport of the value of the Wigner function along a ray.

where U(p) is the Fourier transform of U(x),

U(p) =sk

2π

Z ∞−∞dxU(x)e−ipkx. (11)

The modulus square of the Fourier transform |U(p)|2 is often referred to as angularspectrum (or angular distribution) and can be obtained directly as the spatial integration

of the Wigner function as¯U(p)

¯2=Z ∞−∞dxW (x, p). (12)

2.2. Free ray propagation

By combining (2) and (8) we can compute the Wigner functionWz(x, p) within paraxial

approximation at any plane z > 0, relating it to the Wigner function W0(x, p) at z = 0.

The result is

Wz(x, p) = W0(x− zp/n, p). (13)

Among other consequences this relation means that the Wigner function remains

constant along straight rays (see figure 2). As a matter of fact this constancy along

rays actually holds for every optical system within paraxial approximation.

2.3. Complementary Huygens principle

The propagation law (13) is the cornerstone of the ray picture of optics offered by the

Wigner function and can be taken as the basis of the complementary Huygens principle

encapsulating this picture of optics. We enunciate it in two steps: (i) Each point reached

by the light becomes a secondary source of rays, (ii) The disturbance evolves as the result

of the incoherent superposition of the secondary rays.

The first statement (i) derives from the interpretation of (x, p) as ray parameters.

Thus, from each point x emerge a continuous distribution of rays propagating along

different directions specified by p (see figure 3(a)), where W (x, p) plays the role of a

kind of weight or relative importance of each ray. According to the transport law (13)

Complementary Huygens principle 6

x

x

= 0z z

(a) (b)

= 0z

Figure 3. Illustration of the complementary Huygens principle.

this weight remains invariant along the ray. This is a clear departure from standard

geometrical optics, where rays are often defined as normal to the wavefronts, so that

for a fully coherent wave U(x) there would be single ray at each point (leaving asidecaustics and singularities).

The second statement (ii) derives from the formula (4) for the field intensity Iz(x)

at each point

Iz(x) =Z ∞−∞dpWz(x, p) =

Z ∞−∞dpW0(x− zp/n, p). (14)

This means that Iz(x) is the sum of the ray ”intensities” Wz(x, p) associated to all rays

p passing through the same point x. This superposition of rays is incoherent since it

deals with the addition of ”intensities” instead of complex amplitudes. The last equality

in (14) means that each observation point is reached by a ray from each point of the

source plane parametrized by the ray slope p (see figure 3(b)). We stress that the

waves U(x) we are considering are fully coherent, so that the incoherent character ofthe superposition is a universal key feature of the formalism, independent of the actual

coherence properties of the wave.

2.4. Spatial filtering

In order to complete the picture we can consider the effect of spatial filters such as the

pass of the wave through diffracting apertures. In the wave picture this is described by

amplitude-transmission coefficients t(x) (inhomogeneous in general) transforming the

input wave amplitude U(x) intoU(x)→ t(x)U(x). (15)

In the ray picture, the corresponding transformation of the Wigner function is

Wtu(x, p) =Z ∞−∞dp0Wt(x, p− p0)Wu(x, p

0), (16)

Complementary Huygens principle 7

where Wtu, Wu are the Wigner functions immediately after and before, respectively, of

the spatial filter, and

Wt(x, p) =k

2π

Z ∞−∞dx0t(x− x0/2)t∗(x+ x0/2)eikpx0, (17)

is the Wigner function of the filter. This is to say that the output Wigner function

arises from the convolution of the input Wigner function with the Wigner function of

the filter.

2.5. Negativity

As we have mentioned in the Introduction, a complete radiometric interpretation of W

is prevented by the fact that it can take negative values. This is the price to be paid

to include rigorously wave coherence phenomena within a ray picture. Addition of rays

with positive weights corresponds exclusively to incoherent wave mixing, so that the

inclusion of coherent wave phenomena demands the appearance of rays with negative

weights. More specifically, we may say that this formalism includes two classes of weird

rays, which might be termed fictitious and dark rays.

Fictitious rays are those emerging from regions where the wave amplitude vanishes

Uz(x) = 0. The paradigmatic example is the Young interferometer which be elaboratedin some detail below.

Dark rays are those associated with negative values of the Wigner function

[8, 9, 10, 15, 17, 18]. Waves with dark rays might be termed nongeometrical in the same

sense that quantum states with negative Wigner functions are considered as nonclassical.

The amount of negativity can be measured for example in terms of the distance between

W (x, p) and its modulus [18, 20]

D =Z ∞−∞dpZ ∞−∞dx [W (x, p)− |W (x, p)|]2 . (18)

3. Examples

Let us illustrate the general approach with some simple but meaningful examples. For a

proper comparison we will always consider normalized waves carrying the same intensityZ ∞−∞dx

Z ∞−∞dpW (x, p) =

Z ∞−∞dx |U(x)|2 = 1. (19)

3.1. Gaussian

The Gaussian

U0(x) = 1qσ√2πe−x

2/(4σ2), (20)

provides a paradigmatic example of geometrical wave with positive definite Wigner

function

W0(x, p) =k

πe−x

2/(2σ2)e−2k2σ2p2 , (21)

Complementary Huygens principle 8

so that D = 0 irrespectively of the its spatial width σ. Actually, this is the only coherent

wave with positive definite Wigner function [21].

3.2. Slit

The simplest example of diffraction is provided by a homogenous slit of width a

U0(x) =(

1√a, for |x| � a

2,

0, for |x| > a2,, (22)

leading to

W0(x, p) =

( sin[kp(a−2|x|)]πap

, for |x| � a2

0, for |x| > a2,, (23)

which takes negative values with

D ' 0.25

λ, (24)

that does not depend on the width of the slit a.

3.3. Exponential

For the exponential

U0(x) = 1√be−|x|/b, (25)

we have

W0(x, p) =k

πe−2|x|/b

(1

1 + (kbp)2[cos(2kp|x|)

− kpb sin(2kp|x|)] + sin(2kp|x|)kpb

), (26)

that takes negative values with

D ' 0.016

λ, (27)

which does not depend on b.

4. Coherence phenomena

As we have mentioned in the Introduction the geometrical ray picture provided by

the Wigner formalism is complete and properly includes coherence phenomena such as

diffraction and interference.

Complementary Huygens principle 9

4.1. Diffraction

The complementary Huygens principle developed above indicates the existence of

diffraction in much the same way the standard Huygens principle does. This is via

the superposition of the emissions of the secondary sources in directions different from

the input one. The only difference is the magnitude which is added. This is the coherent

superposition of amplitudes in the wave picture versus the incoherent addition of the

ray ”intensities” provided by the Wigner function.

We can show that diffraction is included within the Wigner ray formalism by

deriving the Fresnel propagation formula (8) from the ray propagation law (13). From

Eqs. (3) and (13) we can express the field amplitude Uz(x) at any plane z > 0 in termsof the Wigner function at z = 0 as

Uz(x) = 1

U∗z (0)Z ∞−∞dpeikpxW0(x/2− zp/n, p). (28)

Using (2), after a suitable change of integration variables we get

Uz(x) = kn

2πz

1

U∗z (0)Z ∞−∞dx00U∗0 (x00)e−iknx

002/(2z)

×Z ∞−∞dx0U0(x0)eink(x−x0)2/(2z). (29)

Evaluating this expression at x = 0 we get that the first integral is proportional to U∗z (0)Z ∞−∞dx00U∗0 (x00)e−inkx

002/(2z) =

s2πz

kneiϕU∗z (0), (30)

where ϕ is a global phase. Therefore, Eq. (29) reproduces the Fresnel diffraction formula

(8).

We can also show explicitly the complete quantitative equivalence of the Wigner

ray approach with Fraunhofer diffraction. To this end we derive the intensity of the

Fraunhofer diffraction at the far field z →∞ directly from the complementary Huygens

principle (13). Performing the p integration in (13) and after a suitable change of

integration variables we get

Iz(x) =Z ∞−∞dpW0(x− zp/n, p) = n

z

Z ∞−∞dx0W0 [x

0, (x− x0)n/z] , (31)

where x0 represent the points in the plane of the aperture (see figure 4). Considering thatthe size of the aperture is finite, when z →∞ we have for finite x/z that p = n(x−x0)/ztends to be independent of the integration variable x0 so that p→ px = xn/z. Therefore

I∞(x) =n

z

Z ∞−∞dx0W0(x

0, px = xn/z) =n

z

¯U0(nx/z)

¯2, (32)

in full agreement with (10).

4.2. Double slit

Concerning the ray picture of interference let us consider the paradigmatic Young

interferometer with two slits located at coordinates x± = ±a in a plane z = 0 illuminated

Complementary Huygens principle 10

x

z

�= (x-x )/z x/z

x

U (x )0U (x)Z

Figure 4. Fraunhofer diffraction.

by a light field of Wigner function Wu(x, p). For definiteness, the two apertures are

assumed to be narrow enough to be described by delta functions so that the amplitude

transmission coefficient is of the form

t(x) = δ(x− a) + δ(x+ a). (33)

From (17) the Wigner function associated to t(x) is

Wt(x, p) =k

2π[δ(x− a) + δ(x+ a) + 2 cos(2kpa)δ(x)] , (34)

and then, from (16), the field at z = 0 immediately after the slits is described by the

Wigner function

Wtu(x, p) =k

2π

h|U0(a)|2 δ(x− a) + |U0(−a)|2 δ(x+ a)

+³e−ikp2aµ+ eikp2aµ∗

´δ(x)

i, (35)

where

µ =Z ∞−∞dp0eikp

02aWu(0, p0) = U0(a)U∗0 (−a), (36)

and U0(±a) is the complex amplitude of the illuminating field at the slits. We canappreciate that Wtu(x, p) 6= 0 only at x = ±a, 0 so that the double slit produces threesecondary point sources of generalized rays, instead of the two sources at x = ±a ofthe Huygens principle (see figure 5). The third source of fictitious rays located at the

midpoint between the slits x = 0 contains all the coherence properties of the input field

required to reproduce exactly the complete interferometric pattern [8, 9, 6, 17].

To see this let us consider that the observation of the interference is performed at

the far plane z → ∞ so that the rays reaching a given observation point x propagate

parallel along a given direction p = xn/z. According to Eqs. (12) and (32) when z →∞the intensity distribution Iz(x = pz/n) is proportional to the angular distribution at

z = 0

I∞(x = pz/n) ∝Z ∞−∞dx0Wtu(x

0, p) ∝¯U0(a) + U0(−a)eiφ

¯2, (37)

where φ = kp2a is the phase difference acquired by the rays between the slits and the

observation plane (see figure 6).

Complementary Huygens principle 11

(a) ( b)

Figure 5. Secondary sources in the Young interferometer: (a) Wave picture (b) Ray

picture.

p

p

W (0,p )u

�=kp2a

�=p/n

�=p/n

�=kp2a

Figure 6. Sketch of the Young interferometer.

This is exactly the same result provided by the wave approach via the Huygens

principle, which leads to the right-hand side of Eq. (37) via the superposition of the

complex amplitudes of the field at the slits U0(±a) after acquiring the phase differenceφ during propagation from the slits to the observation point.

Moreover, the formula for the coherence or visibility factor µ in (36) has a very

simple geometrical meaning. The coherence factor µ turns out to be the average value

of the phase difference φ0 = kp02a at the slits illuminated by a set of plane waves specifiedby the propagation direction p0, where the weight of each plane wave is Wu(0, p

0)

µ = heiφ0iW =Z ∞−∞dp0eikp

02aWu(0, p0) = Γ(a,−a), (38)

where Γ(x1, x2) = hU(x1)U∗(x2)i is the cross-spectral density function.

Complementary Huygens principle 12

5. Conclusions

We have developed a complementary Huygens principle depicting the Wigner ray

formulation of optics. This principle is supported by the transport law satisfied by

the Wigner function and is formally identical to the Huygens-Fresnel principle but in

terms of opposite concepts, i. e., rays instead of waves, and incoherent superpositions

instead of coherent ones. A key feature is that this geometrical ray representation is

complete including all coherent phenomena, such as diffraction and interference, as far

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