Foundations of Scalar Diffraction Theory(advanced stuff for fun)

The phenomenon known as diffraction plays a role of the utmost importance in the branches of physics and engineering that deal with wave propagation. We now consider some of the foundations of scalar diffraction theory. While the theory discussed here is sufficiently general to be applied in other fields, such as acoustic-wave propagation and radio-wave propagation, the applications of primary concern will be in the realm of physical optics. To fully understand the properties of optical imaging and optical data-processing systems, it is essential thatdiffraction and the limitations it imposes on system performance be appreciated.

1. Historical introduction

The term diffraction has been conveniently defined by Sommerfeld as "any deviation of light rays from rectilinear paths which cannot be interpreted as reflection or refraction". The first accurate report and description of such a phenomenon was made by Grimaldi and was published in the year 1665, shortly after his death. The measurements reported were made with an experimental apparatus similar to that shown in Fig. 11: an aperture in an opaque screen was illuminated by a light source, and the light intensity was observed across a plane some distance behind the screen.

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Figure 11: Apparatus for observing diffraction of light

The corpuscular theory of light propagation, which was the accepted means of explaining optical phenomena at the time, predicted that the shadow behind the screen should be well defined, with sharp borders. Grimaldi's observations indicated, however, that the transition from light to shadow was gradual rather than abrupt. If the quality of his light source had been better, he might have observed even more striking results, such as the presence of light and dark fringes extending far into the geometrical shadow of the screen. Such effects cannot be satisfactorily explained by a corpuscular theory of light, whichrequires rectilinear propagation of light rays in the absence of reflection and refraction.

The initial step in the evolution of a theory that would explain such effects was made by the first proponent of the wave theory of light, Christian Huygens, in the year 1678. Huygens' expressed an intuitive conviction that if each point on the wavefront of a light disturbance were considered to be a new source of a "secondary" spherical disturbance, then the wavefront at any later instant could be found by constructing the "envelope" of the secondary wavelets, as illustrated in Fig. 12.

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Figure 12: Huygens' envelope construction

Huygens' intuitive ideas were greatly improved in 1818 in the famous memoir of Augustin Jean Fresnel, who supplemented Huygens' envelope construction with Young's principle of interference. By making some rather arbitrary assumptions about the effective amplitudes and phases of Huygens' secondary sources, and by allowing the various wavelets to mutually interfere, Fresnel was able to calculate the distribution of light in diffraction patterns with excellent accuracy.The ideas of Huygens and Fresnel were put on a firmer mathematical foundation in 1882 by Gustav Kirchhoff, who succeeded in showing that the amplitudes and phases ascribed to the secondary sources by Fresnel were indeed logical consequences of the wave nature of light. Kirchhoff based his mathematical formulation upon two assumptions about the boundary values of the light incident on the surface of an obstacle placed in the way of the propagation of light.

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These assumptions were later proved inconsistent with each other, by Poincare in 1892 and by Sommerfeld in 1894. As a consequence of these criticisms, Kirchhoff's formulation of the so-called Huygens-Fresnel principle must be regarded as a firstapproximation, although under most conditions it yields results that agree amazingly well with experiment.

The Kirchhoff theory was also modified by Sommerfeld, who eliminated one of the aforementioned assumptions concerning the light at the boundary by making use of the theory of Green's functions. This so-called Rayleiqh-Sommerfeld diffraction theorywill be treated later.

It should be emphasized from the start that the Kirchhoff and Rayleigh-Sommerfeld theories share certain major simplifications and approximations. Most important, light is treated as a scalar phenomenon; i.e., only the scalar amplitude of one transverse component of either the electric or the magnetic field is considered, it being assumed that any other components of interest can be treated independently in a similar fashion. Such an approach entirely neglects the fact that the various components of the electric and magnetic field vectors are coupled through Maxwell's equations and cannot be treated independently. Fortunately, experiments in the microwave region of the spectrum have shown that the scalar theory yields very accurate results if two conditions are met: (1) the diffracting aperture must be large compared with a wavelength, and (2) the diffracted fields must not be observed too close to theaperture. These conditions will be well-satisfied in the problems treated in this class.

The first truly rigorous solution of a diffraction problem was given in 1896 by Sommerfeld who treated the two-dimensional case of a plane wave incident on an infinitesimally thin, perfectly conducting half plane.

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2. Mathematical preliminaries

Before embarking on a treatment of diffraction itself, we first consider a number of mathematical preliminaries that form the basis of the later diffraction-theory derivations. These initial discussions will also serve to introduce the notation to be used throughout.

The Helmholtz equation

Let the light disturbance at position P and time t be represented by the scalar function u(P,t); for the case of linearly polarized waves, we may regard this function as representing either the electric or the magnetic field strength. Attention is restricted to the case of purely monochromatic waves.

For a monochromatic wave, the field may be written explicitly as

u(P,t) =U(P)cos 2πνt + φ(P)[ ] (2-1)φ(P)

νwhere U(P) and are the amplitude and phase, respectively, of the wave at position P, while is the optical frequency. A more compact form of (2-1) is found by using complex notation, writing

u(P,t) = Re Uc (P)exp −2πiνt[ ]⎡⎣ ⎤⎦ (2-2)

where Uc(P) is the following complex function of position (sometimes called a phasor)

Uc (P) =U(P)exp −iφ(P)[ ] (2-3)

and Re is shorthand notation meaning the real part of.5

If the real disturbance U(P,t) is to represent an optical wave, it must satisfy the scalar wave equation

∇2u −1c2

∂2u∂t 2

= 0 (2-4)

∇2at each source-free point, being the Laplacian operator

∇2 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2

But the complex function Uc(P) serves as an adequate description of the disturbance, since the time dependence is known a priori. If (2-2) is substituted in (2-4), it follows that the complex disturbance Uc must obey the time-independent equation

(∇2 + k2 )Uc = 0 (2-5)where k is termed the wave number and is given by

k = 2π νc=2πλ

The relation (2-5) is known as the Helmholtz equation; we may assume in the future that the complex amplitude of any monochromatic optical disturbance propagating through free space must obey such a relation.Green's theorem

Calculation of the complex disturbance Uc at an observation point in space can be accomplished with the help of the mathematical relation known as Green's theorem. This theorem can be stated as follows:

(2-5)

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Green's theorem. Let Uc(P) and Gc(P) be any two complex-valued functions of position, and let S be a closed surface surrounding a volume V. If Uc, Gc, and their first and second partial derivatives are single-valued and continuous within and on S, then we have

Gc∇2Uc −Uc∇

2Gc( )V∫∫∫ dv = Gc

∂Uc

∂n−Uc

∂Gc

∂n⎛⎝⎜

⎞⎠⎟

S∫∫ ds (2-6)

∂ / ∂nwhere signifies a partial derivative in the outward normal direction at each point on S.

This theorem is in many respects the prime foundation of scalar diffraction theory. However, only a prudent choice of a so-called Green's function Gc and a closed surface S will allow its direct application to the diffraction problem. We turn now to the former of these problems, considering Kirchhoff's choice of a Green's function and the consequent integral theorem that follows.The integral theorem of Helmholtz and Kirchhoff

The Kirchhoff formulation of the diffraction problem is based on a certain integral theorem which expresses the solution of the homogeneous wave equation at an arbitrary point in terms of the values of the solution and its first derivative on an arbitrary closed surface surrounding that point. This theorem had been derived previously in acoustics by H. von Helmholtz. We will develop Green functions fully later.

Let the point of observation be denoted P0, and let S denote an arbitrary closed surface surrounding P0, as indicated in Fig. 2-3.

(2-6)

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Figure 2-3: Surface of integration

The problem is, of course, to express the optical disturbance at P0 in terms of its values on the surface S. To solve this problem, we follow Kirchhoff in applying Green's theorem and in choosing as a Green's function G a unit-amplitude spherical wave expanding about the point P0, that is,the so-called free-space Green's function; we will derive this later. Thus, the value of G at an arbitrary point P1 is given by

Gc (P1) =exp(ikr01)

r01(2-7)

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r01where we adopt the notation that r01 is the length of the vector pointing from P0 to P1.

Sε

εSε

SεSε

To be legitimately used in Green's theorem, the function G (as well as its first and second derivatives) must be continuous within the enclosed volume V. Therefore, to exclude the discontinuity at P0, a small spherical surface , of radius , is inserted about the point P0. Green's theorem is then applied, the volume of integration V' being that volume lying between S and , and the surface of integration being the composite surface S' = S + , as indicated in Fig. 2-3. Note that the "outward" normal to the composite surface points outward in the conventional sense on S, but inward (toward P0 ) on .

Within the volume V', the disturbance Gc, being simply an expanding spherical wave, satisfies a Helmholtz equation

(∇2 + k2 )Gc = 0 (2-8)

Substituting the two Helmholtz equations (2-5) and (2-8) in the left-hand side of Green's theorem, we find

Gc∇2Uc −Uc∇

2Gc( )V∫∫∫ dv = − GcUck

2 −UcGck2( )

V∫∫∫ dv = 0

Thus the theorem reduces to Gc∂Uc

∂n−Uc

∂Gc

∂n⎛⎝⎜

⎞⎠⎟

S '∫∫ ds = 0 or

− Gc∂Uc

∂n−Uc

∂Gc

∂n⎛⎝⎜

⎞⎠⎟

Sε∫∫ ds = Gc

∂Uc

∂n−Uc

∂Gc

∂n⎛⎝⎜

⎞⎠⎟

S∫∫ ds (2-9)

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Now note that for a general point P1 on S' we have

Gc (P1) =exp(ikr01)

r01(2-10)

and

∂Gc (P1)∂n

= cos(n, r01) ik −1r01

⎛⎝⎜

⎞⎠⎟exp(ikr01)

r01(2-11)

n

r01 Sε cos(n, r01)

where represents the cosine of the angle between the outward normal and the vector joining P0 to P1. For the particular case of P1 on , = -1, and these equations become

cos(n, r01)

Gc (P1) =exp(ikε)

εand

∂Gc (P1)∂n

=exp(ikε)

ε1ε− ik⎛

⎝⎜⎞⎠⎟

(2-12)

εLetting become arbitrarily small, the continuity of Uc (and its derivatives) at P0 allows us to write

Gc∂Uc

∂n−Uc

∂Gc

∂n⎛⎝⎜

⎞⎠⎟

Sε∫∫ ds = 4πε 2 ∂Uc (P0 )

∂nexp(ikε)

ε−Uc (P0 ) exp(ikε)

ε1ε− ik⎛

⎝⎜⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

=ε→0

− 4πUc (P0 )

Substitution of this result in (2-9) yields

Uc (P0 ) =14π

∂Uc

∂nexp(ikr01)

r01−Uc

∂∂n

exp(ikr01)r01

⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟S∫∫ ds (2-13)

P0 is only point as ε → 0

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This result is known as the integral theorem of Helmholtz and Kirchoff; it plays a significant role in the development of the scalar theory of diffraction, for it allows the field at any point P0 to be expressed in terms of the "boundary values" of the wave on any closed surface surrounding that point. As we shall now see, such a relation is instrumental in the development of scalar diffraction equations.

3. The Kirchoff formulation of diffraction by a plane screen

Consider now the problem of diffraction by an aperture in an infinite opaque screen. As illustrated in Fig. 2-4, a wave disturbance is assumed to impinge on the screen and aperture from the left, and the field at the point P0 behind the aperture is to be calculated.

Application of the integral theorem

To find the field at the point P0, we apply the integral theorem of Helmholtz and Kirchhoff, being careful to choose a surface of integration that will allow the calculation to be performed successfully.

Following Kirchhoff, the closed surface S is chosen to consist of two parts, as shown in Fig. 2-4. Let a plane surface, S1 lying directly behind the diffracting screen, be joined and closed by a large spherical cap, S2, of radius R and centered at the observation point P0. The total closed surface S is simply the sum of S1 and S2.

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Figure 2-4: Kirchoff formulation of diffraction by a plane screen

Thus, applying (2-13),Uc (P0 ) =

14π

Gc∂Uc

∂n−Uc

∂Gc

∂n⎛⎝⎜

⎞⎠⎟

S1 +S2∫∫ ds

where, as before,Gc =

exp(ikr01)r01

As R increases, S2 approaches a large hemispherical shell. It is tempting to reason that, since both U and G will fall off as 1/R, the integrand will ultimately vanish yielding a contribution of zero from the surface integral over S2.

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However, the area of integration increases as R2, so this argument is incomplete. It is also tempting to assume that, since the disturbances are propagating with finite velocity c, R will ultimately be so large that the waves have not yet reached S2 and the integrand will be zero on that surface. But this argument is incompatible with our assumption of monochromatic disturbances, which must (by definition) have existed for all time. Evidently a more careful investigation is required before the contribution from S2 can be disposed of.

Examining this problem in more detail, we see that on S2

Gc =exp(ikR)

R

and, from (2-11), ∂Gc

∂n= ik −

1R

⎛⎝⎜

⎞⎠⎟exp(ikR)

R≅ ikGc

where the last approximation is valid for large R. The integral in question can thus be reduced to

Ω RGcwhere is the solid angle subtended by S2 at P0. Now the quantity is uniformly bounded on S2. Therefore, the entire integral over S2 will vanish as R becomes infinite provided the disturbance Uc has the property

uncertainty principle

ΔEΔt ≥ → ΔνΔt ≥ 1Δν → 0⇒ Δt→∞

ds = sinθdθdφ = dΩ

Gc∂Uc

∂n−Uc

∂Gc

∂n⎛⎝⎜

⎞⎠⎟

S2∫∫ ds = Gc

∂Uc

∂n− ikUc

⎛⎝⎜

⎞⎠⎟

Ω∫ R2dΩ

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limR→∞

R∂Uc

∂n− ikUc

⎛⎝⎜

⎞⎠⎟= 0 (2-14)

uniformly in angle. This requirement is known as the Sommerfeld radiation condition and is satisfied if the disturbance Uc vanishes at least as fast as a diverging spherical wave. Since the disturbances that illuminate the aperture will invariably consist of a spherical wave or a linear combination of spherical waves, we can be confident that this requirement will be satisfied in practice, and therefore that the integral over S2 will yield a contribution of precisely zero.

The Kirchhoff boundary conditions

Having disposed of the integration over the surface S2, it is now possible to express the disturbance at P0 in terms of the disturbance and its normal derivative over the infinite plane S1 immediately behind the screen, that is,

Uc (P0 ) =14π

Gc∂Uc

∂n−Uc

∂Gc

∂n⎛⎝⎜

⎞⎠⎟

S1∫∫ ds (2-15)

Σ

The screen is opaque, except for the open aperture which will be denoted . It therefore seems intuitively reasonable that the major contribution to the integral (2-15) arises from points of S1 located within the aperture , where we would expect the integrand to be largest. Kirchhoff accordingly adopted the following assumptions:

Σ

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Σ ∂Uc / ∂n1. Across the surface , the field distribution Uc and its derivative are exactly the same as they would be in the absence of the screen.

2. Over the portion of S1 that lies in the geometrical shadow of the screen, the field distribution Uc and its derivative are identically zero.∂Uc / ∂n

These conditions are commonly known as the Kirchhoff boundary conditions. The first allows us to specify the disturbance incident on the aperture by neglecting the presence of the screen. The second allows us to neglect all of the surface of integration except that portion lying directly within the aperture itself. Thus (2-15) is reduced to

Uc (P0 ) =14π

Gc∂Uc

∂n−Uc

∂Gc

∂n⎛⎝⎜

⎞⎠⎟

Σ∫∫ ds (2-16)

Σ

While the Kirchhoff boundary conditions simplify the result considerably, it is important to realize that neither can be exactly true. The presence of the screen will invariably perturb the fields on to some extent, for along the rim of the aperture certain boundary conditions must be met that would not be required in the absence of the screen. In addition, the shadow behind the screen is never perfect, for fields will invariably extend behind the screen for a distance of several wavelengths. However, if the dimensions of the aperture are large compared with a wavelength, these fringing effects can be safely neglected, and the two boundary conditions can be used to yield results that agree very well with experiment.

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The Fresnel-Klrchhoff diffraction formula and the Huygens-Fresnel principle

A further simplification of the expression for Uc(P0) is obtained by noting that the distance r01 from the aperture to the observation point is usually many optical wavelengths, and therefore, since k >> 1/r01, Eq. (2-11) becomes

∂Gc (P1)∂n

= cos(n, r01) ik −1r01

⎛⎝⎜

⎞⎠⎟exp(ikr01)

r01≅ ik cos(n, r01)

exp(ikr01)r01

Substituting this approximation and the expression (2-7) for Gc in Eq. (2-16), we find

Uc (P0 ) =14π

exp(ikr01)r01

∂Uc

∂n− ikUc cos(

n, r01)⎛⎝⎜

⎞⎠⎟

Σ∫∫ ds (2-17)

Now suppose that the aperture is illuminated by a single spherical wave,

Uc (P1) = Aexp(ikr21)

r21arising from a point source at P2, a distance r21 from P1 (see Fig. 2-5).

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Figure 2-5: Point source illumination of a plane screen

If r21 is many optical wavelengths long, then (2-17) can be directly reduced to

Uc (P0 ) =Aiλ

exp(ik(r21 + r01))r21r01

cos(n, r01) − cos(n, r21)

2⎛⎝⎜

⎞⎠⎟

Σ∫∫ ds (2-18)

This result, which applies only for an illumination consisting of a single point source, is commonly known as the Fresnel-Kirchoff diffraction formula.

Finally, we consider an interesting and useful interpretation of the diffraction formula (2-18). Let that equation be rewritten as follows:

Uc (P0 ) = U 'c (P1)exp(ikr01)

r01Σ∫∫ ds (2-19)

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where

U 'c (P1) =1iλAexp(ikr21)

r21

cos(n, r01) − cos(n, r21)

2⎛⎝⎜

⎞⎠⎟

(2-20)

Aexp(ikr21) / r21

Now (2-19) may be interpreted as implying that the field at P0 arises from an infinity of fictitious “secondary” point sources located within the aperture itself. The amplitude Uc'(P1) of the secondary source located at P1 is evidently proportional to the amplitude of the wave incident at P1, but it differs from that amplitude in three respects.

λ−1

(cos(n, r01) − cos(

n, r21)) / 2First, the amplitude differs from the incident amplitude by a factor . Second, that amplitude is also reduced by a so-called obliquity factor which is never greater than unity nor less than zero; in effect, there is a nonisotropic “directivity pattern” associated with each secondary source. Finally, the phase of the secondary source at P1 leads the phase of the incident wave by 90o (the i in 2-20).

These curious properties of the secondary sources are significant in a historical sense, for in the earlier work of Fresnel, the combination of Huygens' envelope construction and Young's principle of interference was found to yield accurate predictions of diffraction patterns only if properties such as these could be ascribed to the secondary sources. In order to obtain accurate results, Fresnel assumed that these properties were true. Kirchhoff's mathematical development demonstrated that they were in fact natural consequences of the wave nature of light.

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Note that the above derivation of the so-called Huygens-Fresnel principle has been restricted to the case of an aperture illumination consisting of a single expanding spherical wave. However, we shall return to the Huygens-Fresnel principle shortly to demonstrate that it is indeed more general than our first examination has implied.

3. The RAYLEIGH-SOMMERFELD formulation of diffraction by a plane screen

The Kirchhoff theory has been found experimentally to yield remarkably accurate results and is widely used in practice. However, there are certain internal inconsistencies in the theory which have motivated a search for more satisfactory mathematical developments. The difficulties of the Kirchhoff theory stem from the fact that boundary conditions must be imposed on both the field strength and its normal derivative. In particular, it is a well-known theorem of potential theory that if a two-dimensional potential function and its normal derivative vanish together along any finite curve segment, then that potential function must vanish over the entire plane. Similarly, if a solution of the three-dimensional wave equation vanishes on any finite surface element, it must vanish in a11 space. Thus the two Kirchhoff boundary' conditions together imply that the field is identically zero everywhere behind the aperture, a result which contradicts the known physical situation. A further indication of these inconsistencies is the fact that the Fresnel-Kirchhoff diffraction formula can be shown to fail to reproduce the assumed boundary conditions as the observation point approaches the screen or aperture. In view of these contradictions, it is indeed remarkable that the Kirchhoff theory yields such accurate results.

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The inconsistencies of the Kirchhoff theory were removed by Sommerfeld, who eliminatedthe necessity of imposing boundary values on both the disturbance and its normal derivative simultaneously. We discuss this so-called Rayleigh-Sommerfeld theory now.

Choice of alternative Green's functions

Consider again Eq. (2-15) for the observed field strength in terms of the incident field and its normal derivative across the entire screen:

Uc (P0 ) =14π

Gc∂Uc

∂n−Uc

∂Gc

∂n⎛⎝⎜

⎞⎠⎟

S1∫∫ ds (2-15)∂Gc / ∂n

∂Uc / ∂n

P0

Suppose that the Green's function Gc of the Kirchhoff theory were modified in such a way that, while the development leading to the above equation remains valid, in addition either Gc or vanishes over the entire surface S1. In either case, the necessity of imposing simultaneous boundary conditions on Uc and would be removed, and the inconsistencies of the Kirchhoff theory would be eliminated.

Sommerfeld pointed out that Green's functions with the required properties do indeed exist. Suppose that Gc is generated not only by a point source located at P0, but also by a second point source located at a position which is the mirror image of P0 on the opposite side of the screen (see Fig. 2-6).

∂Gc / ∂n

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Figure 2-6: Rayleigh-Sommerfeld formulation of diffraction by a plane screen

P0 λLet the source at be of the same wavelength as the source at P0, and suppose

that the two sources are oscillating with a 180o phase difference. The Green's function is in this case given by

G− (P1) =exp(ikr01)

r01−exp(ikr01)r01

(2-21)

r01

P0where is the distance from to P1. The corresponding normal derivative of G- is

∂G−

∂n= cos(n, r01) ik −

1r01

⎛⎝⎜

⎞⎠⎟exp(ikr01)

r01− cos(n, r01) ik −

1r01

⎛⎝⎜

⎞⎠⎟exp(ikr01)r01

(2-22)

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The fact that one theory is self-consistent while the other is not does not necessarily mean that the former is more accurate than the latter. The relative accuracy of the Kirchhoff and Rayleigh-Sommerfeld formulations is still a subject of active research.

Now for P1 on S1, we have

r01 = r01 , cos(n, r01) = − cos(n, r01)

and therefore on that surface

G− (P1) = 0

∂G− (P1)∂n

= 2cos(n, r01) ik −1r01

⎛⎝⎜

⎞⎠⎟exp(ikr01)

r01

(2-23)

Thus the function G- vanishes over the entire surface S1.

The Rayleigh-Sommerfeld diffraction formula

Let the Green's function G- be substituted for Gc in Eq. (2-15). Using (2-23), it follows directly that

Uc (P0 ) =1iλ

Uc (P1)exp(ikr01)

r01S1∫∫ cos(n, r01) (2-24)

r01 λwhere it has been assumed that . The, Kirchhoff boundary conditions may now be

applied to Uc alone, yielding the very general result

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Uc (P0 ) =1iλ

Uc (P1)exp(ikr01)

r01Σ∫∫ cos(n, r01)ds (2-25)

∂Uc / ∂nSince no boundary conditions need be applied to , the inconsistencies of the Kirchhoff theory have evidently been removed.

For the purpose of comparing these results with the predictions of the Kirchhoff theory, let the aperture be illuminated by a spherical wave diverging from a point source at position P2 (see Fig. 2-5 again). Thus

Uc (P1) = Aexp(ikr21)

r21and

Uc (P0 ) =Aiλ

exp(ik(r21 + r01))r21r01

cos(n, r01)Σ∫∫ ds (2-26)

This latter expression is known as the Rayleigh-Sommerfeld diffraction formula, and it should be compared with the corresponding result of the Kirchhoff theory, Eq. (2-18). Note that the only difference between the two expressions occurs in the obliquity factor.

III. Fresnel and Frauenhofer Diffraction

In the discussion, the results of scalar diffraction theory were presented in their most general forms. Attention is now turned to certain approximations to the general theory-approximations that will allow diffraction-pattern calculations to be reduced to more simple mathematical manipulations. These approximations, which are commonly made in many fields that deal with wave propagation, will be referred to as the Fresnel and Fraunhofer approximations.

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This treatment will differ from more common treatments of Fresnel and Fraunhofer diffraction in that, in accordance with the view of the propagation phenomenon as a “system”, we will attempt to find approximations that are valid for any of a wide class of “input” field distributions.

Σ

1. Approximations to the Huygens-Fresnel principle

Consider again diffraction of monochromatic light by a finite aperture in an infinite opaque screen. As indicated in Fig. 3-1, the screen is assumed to be planar, with a rectangular coordinate system (x1,y1) attached. In addition, the region of observation is assumed to be a plane, standing parallel with the plane of the screen at a normal distance z. A coordinate system (x0,y0) is attached to the plane of observation, with the coordinate axes parallel with those in the x1y1 plane.

Figure 3-1: Diffraction geometry

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Initial approximations

Using the mathematical expression of the Huygens-Fresnel principle, in particular Eqs. (2-27) and (2-28), the field amplitude at point (x0,y0) is readily written(dropping c subscript) as

U(x0 , y0 ) = h(x0 , y0 , x1, y1)U(x1, y1)Σ∫∫ dx1dy1 (3-1)

where

h(x0 , y0 , x1, y1) =1iλexp(ikr01)

r01cos(n, r01) (3-2)

U(x1, y1)Σ

In the future, this superposition integral will be written with infinite limits, it being understood that, in accordance with the Kirchhoff boundary conditions, is identically zero outside the aperture . Thus

U(x0 , y0 ) = h(x0 , y0 , x1, y1)U(x1, y1)−∞

∞

∫−∞

∞

∫ dx1dy1 (3-3)

Our approximations will be based on the assumption that the distance z between aperture and observation plane is very much greater than the maximum linear dimension of the aperture . In addition, we shall assume that in the plane of observation only a finite region about the z-axis is of interest, and that the distance z is much greater than the maximum linear dimension of this region. With these assumptions, the obliquity factor is readily approximated by

Σ

25

cos(n, r01) ≅ 1

(n, r01)where the accuracy is to within 5 percent if the angle does not exceed 18o.

Under similar conditions the quantity r01 in the denominator of (3-2) will not differ significantly from z, allowing the weighting function to be approximated as

h(x0 , y0 , x1, y1) ≅1iλz

exp(ikr01) (3-4)

2π

Note that the quantity r01 in the exponent cannot be replaced simply by z, for the resulting errors will be multiplied by a very large number k, and consequently can generate phase errors much greater than radians.

The Fresnel approximations

Further simplification can be accomplished only by adopting certain approximations to the quantity r01 in the exponent of Eq. (3-4). From Fig. 3-1, the distance r01 is given exactly by

r01 = z2 + (x0 − x1)2 + (y0 − y1)2

= z 1+ x0 − x1

z⎛⎝⎜

⎞⎠⎟

2

+y0 − y1

z⎛⎝⎜

⎞⎠⎟

2 (3-5)

A convenient means of approximation is offered by a binomial expansion of the square root,

1+ b = 1+ 12b −

18b2 + ........ b < 1 (3-6)

26

We therefore assume that the square root in Eq. (3-5) is adequately approximated by the first two terms of its expansion, giving

z 1+ x0 − x1z

⎛⎝⎜

⎞⎠⎟2

+y0 − y1z

⎛⎝⎜

⎞⎠⎟2

≅ z 1+ 12

x0 − x1z

⎛⎝⎜

⎞⎠⎟2

+12

y0 − y1z

⎛⎝⎜

⎞⎠⎟2⎡

⎣⎢⎢

⎤

⎦⎥⎥

This assumption, which will be referred to as the Fresnel approximation, allows the weighting function to be rewritten as

h(x0 , y0 , x1, y1) ≅exp(ikz)iλz

exp ik2z

x0 − x1( )2 + y0 − y1( )2⎡⎣

⎤⎦

⎡⎣⎢

⎤⎦⎥

(3-8)

When the distance z is sufficiently large for this approximation to be an accurate one, the observer is said to be in the region of Fresnel diffraction.

Accepting the validity of the Fresnel approximation, the superposition integral can be expressed in either of two equivalent forms. First, U(x0,y0) may be regarded as a convolution of U(x1,y1) with h; that is,

U(x0 , y0 ) =exp(ikz)iλz

U(x1, y1)−∞

∞

∫−∞

∞

∫ exp ik2z

x0 − x1( )2 + y0 − y1( )2⎡⎣

⎤⎦

⎡⎣⎢

⎤⎦⎥dx1dy1

(3-7)

(3-9)

Alternatively, the quadratic terms in the exponent may be expanded to yield

27

U(x0 , y0 ) = exp(ikz)iλz

exp ik2z

x02 + y0

2⎡⎣ ⎤⎦⎡⎣⎢

⎤⎦⎥

U(x1, y1){−∞

∞

∫−∞

∞

∫

exp ik2z

x12 + y1

2⎡⎣ ⎤⎦⎡⎣⎢

⎤⎦⎥⎫⎬⎭

exp −i2πλz

x0x1 + y0y1[ ]⎡⎣⎢

⎤⎦⎥dx1dy1

(3-10)

Thus, aside from multiplicative amplitude and phase factors that are independent of (x1,y1), the function U(x0,y0) may be found from a Fourier transform of

U(x1, y1)exp ik2z

x12 + y1

2⎡⎣ ⎤⎦⎡⎣⎢

⎤⎦⎥

(kx = x0 / λz,ky = y0 / λz)where the transform must be evaluated at frequencies to assure the correct space scaling in the observation plane.

exp i(k / 2z) x12 + y1

2⎡⎣ ⎤⎦⎡⎣ ⎤⎦

The Fraunhofer approximation

Diffraction-pattern calculations can be further simplified if restrictions more stringent than those used in the Fresnel approximations are adopted. In particular, it was seen earlier that in the region of Fresnel diffraction the observed field strength U(x1,y1) can be found from a Fourier transform of the product of the aperture distribution U(x1,y1) with a quadratic phase function

If, in addition the stronger(Fraunhofer) assumption

(3-11)

28

zk2x12 + y1

2⎡⎣ ⎤⎦max (3-12)

is adopted, then the quadratic phase factor is approximately unity over the entire aperture, and the observed field distribution can be found directly from a Fourier transform of the aperture distribution itself. Thus, in the region of Fraunhofer diffraction,

U(x0 , y0 ) = exp(ikz)iλz

exp ik2z

x02 + y0

2⎡⎣ ⎤⎦⎡⎣⎢

⎤⎦⎥

U(x1, y1){−∞

∞

∫−∞

∞

∫ exp −i2πλz

x0x1 + y0y1[ ]⎡⎣⎢

⎤⎦⎥dx1dy1

(3-13)

(kx = x0 / λz,ky = y0 / λz)

Aside from the multiplicative factors preceding the integral, this expression is simply the Fourier transform of the aperture distribution, evaluated at frequencies

6 ×10−7meter

At optical frequencies, the conditions required for the validity of the Fraunhofer diffraction equation can indeed be severe ones. For example, at a wavelength of (red light) and an aperture width of 2.5 cm (1 inch), the observation distance z must satisfy z >> 1,600 meters. Hence, the lens in earlier discussions.Nonetheless, the required conditions are met in a number of important problems. In addition, Fraunhofer diffraction patterns can be observed at distances closer than implied by (3-12), provided the aperture is illuminated by a spherical wave converging toward the observer or if a converging lens is properly situated between the observer and the aperture (more about this later).

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I = U(P) 2

2. Examples of Frauenhofer diffraction patterns

We consider next a number of examples of Fraunhofer diffraction calculations. The results of the preceding discussion can be applied directly to find the complex field distribution across the Fraunhofer diffraction pattern of any given aperture. However, it should be noted that real radiation detectors, including the eye, respond to optical intensity( ) rather than field amplitude. The final description of the diffraction patterns will therefore be a distribution of intensity.

Rectangular aperture

Consider first a rectangular aperture with an amplitude transmittance given by

t(x1, y1) = rectx1 X

⎛⎝⎜

⎞⎠⎟rect

y1Y

⎛⎝⎜

⎞⎠⎟

X

Yx1 y1

The constants and are simply the respective widths of the aperture in the and directions. If the aperture is normally illuminated by a unit-amplitude, monochromatic plane wave, then the field distribution across the aperture is equal to the transmittance function t. Thus using Eq. (3-13), the Fraunhofer diffraction pattern of the aperture is seen to be

U(x0 , y0 ) =exp(ikz)iλz

exp ik2z

x02 + y0

2⎡⎣ ⎤⎦⎡⎣⎢

⎤⎦⎥ℑ U(x1, y1){ } kx = x0 /λz

ky = y0 /λz

Noting that

30

ℑ U(x1, y1){ } = XYsinc Xkx( )sinc Y ky( )

we find

U(x0 , y0 ) =exp(ikz)iλz

exp ik2z

x02 + y0

2⎡⎣ ⎤⎦⎡⎣⎢

⎤⎦⎥ XYsinc

X x0λz

⎛⎝⎜

⎞⎠⎟sinc Y y0

λz⎛⎝⎜

⎞⎠⎟

and

I(x0 , y0 ) = U(x0 , y0 )2 = X2Y2

λ2z2sinc2 X x0

λz⎛⎝⎜

⎞⎠⎟sinc2 Y y0

λz⎛⎝⎜

⎞⎠⎟

(3-14)x0

Figure 3-2 shows a cross section of the Fraunhofer diffraction pattern along the axis. Note that the width between the first two zeros (the width of the main lobe) is

x0

Δx0 = 2λz X

(3-15)

Figure 3-2: Cross section of the Fraunhofer diffraction pattern of a rectangular aperture

31

X / Y =2

Figure 3-3 shows a photograph of the diffraction pattern produced by a rectangular aperture with a width ratio of .

X

Y

Figure 3-3: The Fraunhofer diffraction pattern of a rectangular aperture( / =2)

32

Circular aperture

Consider a diffracting aperture that is circular rather than rectangular, and let the diameter of the aperture be . Thus, if r1 is a radius coordinate in the plane of the aperture, then

t(r1) = circr1 / 2

⎛⎝⎜

⎞⎠⎟

The circular symmetry of the problem suggests that the Fourier transform in Eq. (3-13) be rewritten as a Fourier-Bessel transform. Thus if r0 is the radius coordinate in the observation plane, we have

U(r0 ) =exp(ikz)iλz

exp ikr02

2z⎛⎝⎜

⎞⎠⎟B U(r1){ }

ρ= r0 /λz(3-16)

U(r1) = t(r1)For unit-amplitude, normally incident plane-wave illumination, we have ; in addition,

B circr1 / 2

⎛⎝⎜

⎞⎠⎟

⎧⎨⎩

⎫⎬⎭=

2⎛⎝⎜

⎞⎠⎟2 J1(πρ)ρ / 2

Thus the amplitude distribution in the Fraunhofer diffraction pattern is seen to be

U(r0 ) = exp(ikz)iλz

exp ikr02

2z⎛⎝⎜

⎞⎠⎟

2⎛⎝⎜

⎞⎠⎟

2 J1(πr0 / λz)r0 / 2λz

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= exp(ikz)exp ikr02

2z⎛⎝⎜

⎞⎠⎟k2

i8z2 J1(kr0 / 2z)

kr0 / 2z⎡

⎣⎢

⎤

⎦⎥

33

and the intensity distribution can be written

I(r0 ) =k2

i8z⎛⎝⎜

⎞⎠⎟

2

2 J1(kr0 / 2z)kr0 / 2z

⎡

⎣⎢

⎤

⎦⎥

2

(3-17)

This intensity distribution is generally referred to as the Airy pattern, after G. B. Airy who first derived it. Table 3-1 shows the values of the Airy pattern at successive maxima and minima, from which it can be seen that the radius to the first zero is given by

Δr0 = 1.22λz

(3-18)

Remember earlier discussions; This is the Airy disk corresponding to the Rayleigh limitand the the angular resolution of a telescope.

x 2 J1(π x)π x

⎡⎣⎢

⎤⎦⎥

2 maxormin

0 1 max1.220 0 min1.635 0.0175 max2.233 0 min2.679 0.0042 max3.238 0 min3.699 0.0016 max

Table 3-1

34

Figure 3-4 shows a cross section of the Airy pattern, while Fig. 3-5 is a photograph of the Fraunhofer diffraction pattern of a circular aperture.

Figure 3-4: Cross section of the Airy pattern Figure 3-5: Fraunhofer diffraction pattern of a circular aperture

35