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Fraunhofer Diffraction
Adrian Down
April 19, 2006
1 Review
Last time, we derived the Fresnel-Kirchhoff Integral Theorem,
U (P ) =
−ıkU 0 4 π
ap
da
eık(r+r ′)
rr′^
(ˆr · nˆ − ˆr′^ · nˆ) (1)
where the optical disturbance UP is defined such that |UP |^2 = I. The physical optical disturbance at the point P is given by
Uphys(P ) = <(UP e−ıωt)
The setup is as follows:
- The axis of the apparatus passes through the center of the slit and is oriented horizontally.
- The width of the slit is δ.
- The observer is located at point P , and the source is located at point S.
- The off-axis distance between the point P and the bottom of the slit is h, and the distance from the plane of P to the plane of the aperture is d.
- The off-axis distance between the point S and the bottom of the slit is h′, and the distance from the plane of S to the plane of the aperture is d′.
- r′^ points from the source to the center of the slit, and r points from the observer to the center of the slit.
- The angle between the axis and r is θ. The angle between the axis and r′^ is θ′.
- nˆ points away from the observer towards the side of slit on which the source is located.
2 Motivation
Our strategy is to make approximations to the integral in (1) until we can solve it analytically. The approximations that we make define the regime known as Fraunhofer diffraction. We will see that the integral will become a Fourier transform.
Note. Fresnel diffraction is a special case of Fraunhofer diffraction in which both the source and the observer are on the axis of the diffraction apparatus.
3 Approximations
3.1 Paraxial approximation
Assume the paraxial approximation, so that the source and observer can be treated as close to the axis of the apparatus. This approximation implies,
paraxial approximation ⇒
θ θ′^
In this approximation, the directional factor in (1) simplifies,
paraxial approximaiton ⇒ ˆr · nˆ − ˆr′^ · nˆ ≈ 2
3.2 Negligible wavefront curvature
Assume the curvature of the wavefront across the area of the aperture is negligible. In particular, if s is the distance from the center of the slit to the wavefront passing through the slit, called the sagitta, we require that ks 1.
The second term, on the other hand, vanishes as d and d′^ approach ∞ for any values of θ and θ′. To ensure negligible curvature, it is this term that must be λ. We require then that
δ^2
2 λ 1 d′^ +^
1 d
k is very small for electromagnetic radiation, so we can impose a more strin- gent condition by multiplying our previous result by k and requiring that it remain small. Assuming that the term in δ^2 is negligible,
k [(r + r′)top − (r + r′)bot] ≈ k
h′ d′^
δ + k
h d
δ + (negligable)
We can ensure that the first term is negligible if we assume normal incidence, so that,
θ′^ = 0 ⇒ sin θ′^ =
h′ d′^
This assumption is equivalent to assuming that the source is on the axis. With this approximation, only the variation in eıkr^ is significant over the
area of the aperture. Taking the factors e −ıkr′ rr′^ to be constant,
UP = C
ap
eıkrda
4 Relation to the Fourier Transform
4.1 Setup
We consider the Fraunhofer integral in a particular context, from which we will be able to derive useful results. Imagine a source of paraxial radiation, such as a laser beam that has been spread and defocused to make a paraxial beam. There is an aperture plane perpendicular to the paraxial beam. Let the vertical position of a beam in the aperture plane be denoted by x. Consider a “transform plane”, denoted by P , located at some distance from the aperture plane. Let the vertical position of the beam at this plane be denoted by X. Finally, consider a planar screen, denoted by I, at some distance from the transform plane,
such that the distance between the aperture and transform planes is the same as that between the transform and image planes. Let the vertical position of a beam at this plane be denoted by x′. Now, place a mirror of height b and focal length f between each pair of planes. Place the mirrors halfway between the planes, and choose the mirrors such that the focal length is equal to half of the distance between successive planes. The set up is then a source, a distance f to the first mirror, a distance f to the aperture plane, a distance f to the second mirror, a distance f to the transform plane, and so on.
Note. The purpose of the lenses is that they allow us to consider an appa- ratus of finite size. The results we obtain in considering the apparatus with the lenses in place are the same as those obtained with no lenses and the transform plane located at infinity. Essentially, the lenses allow us to focus beams that are initially parallel to an intersection point in a finite distance.
4.2 Calculation
We use the Fraunhofer integral to relate the optical disturbance at a point on the transform plane to the radiation entering the aperture. Considering a beam entering the aperture at some height x. One possible path that it can take to reach the transform plane is to travel to the tip of the curved mirror and be bent back downward towards the axis. The total distance traveled by this beam is the r that we will use in applying the Fraunhofer integral.
r =
(b − x)^2 + f 2
)^12
(b − X)^2 + f 2
)^12
Taylor expand in
b f
r = f
b − x f
b − X f
= f
(b^2 − 2 bx + x^2 ) + (b^2 − 2 bX + X^2 ) 2 f 2
= f
2 b^2 − 2 b(x + X) + x^2 + X^2 2 f 2
the terms in (2) are constant over the aperture and appear as proportionality constants. Also, use the definition of the aperture function to extend the integration over all space,
UP (x) ∝
−∞
dx g(x)e−ık^
Xx f
By analogy, in the two-dimensional case,
UP (X, Y ) ∝
−∞
dxdy g(x, y)e−ık^
(Xx+Y y) f
To simplify the integral, define the spatial frequencies,
μ =
kX f
ν =
kY f
Note. • The spatial frequencies have units of inverse meters.
- Notice the close analogy with the Fourier Transform, in which a func- tion of position is integrated over a frequency range. With this substitution,
UP (μ, ν) ∝
−∞
dxdy g(x, y)e−ı(μx+νy)
Thus the optical disturbance on the transform plane is the Fourier Transform of the aperture function.
4.4.2 Image plane
In the same way, the optical disturbance at the image plane can be obtained from the Fourier transform of the optical disturbance at the transform plane,
UI (x′, y′) ∝
−∞
dμdν UP (μ, ν)e−ı(μx
′+νy′)
However, UP can also be related to the aperture function using the inverse Fourier transform,
g(x, y) ∝
dμdν UP (μ, ν)eı(μx+νy)
To ensure consistency between these two equations, it must be that x = −x′^ and y = y′. This is an expression of the fact that the image formed by a thin lens is inverted. In this case, the pair of thin lenses surrounding the transform plane act like a single lens. The inversion of the image, which can be easily obtained heuristically using ray tracing, results because of the switching of the sign in the exponent of the inverse Fourier transform. We have gone through a rigorous derivation to show something that we already knew. However, the essential point of this derivation is to show that it is possible to manipulate the image from the diffracting slit by placing devices at the transform plane to manipulate the Fourier transform. For example, consider the case of an optical signal with some kind of pe- riodic narrow band-width noise. We could remove the noise using a diffrac- tion apparatus. First, pass the signal through a diffraction grating. Each frequency in the initial signal is diffracted to a characteristic position on the transform plane. These heights are given by the Fourier transform of the aperture function. To remove the noise, place a physical obstruction in the aperture plane at the points to which the noise frequency radiation is diffracted. When the signal reaches the image plane, the component of the original signal at the frequency of the noise will be absent.
