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Princeton University Physics 312 Spring 2011

Lecture Notes on Fourier Optics

Introduction

We have stressed, wherever applicable, the additional insight one can get by thinking of electrical signals in both the time domain and the frequency domain. We have also noted that this duality applies to everything from quantum mechanics (bras and kets) and condensed matter physics (lattices and inverse lattices) to general relativity (contravariant and covariant vectors). Here, we will indicate how to think about optics in two domains. This will also allow us to use a laser and opaque grid to demonstrate a beautiful theorem, called the convolution theorem, that relates multiplication the two domains. So, for optics, what are the applicable domains? As you might guess, they are space and wavenumber. We can make the following correspondences:

Electrical signals ~ Optics (1)

time t ~ .T distance (2)

angular frequency w ~ k wavevector (3) frequency j ~ Ko wave number. (4)

Just as j = w/21f and has units of l/time, Ko = k/21f and has units of 1/distance. The Fourier transform pairs are {j, t} and {Ko,.T}.

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'f~_ ~(\~"'_ .. f=~ F--" W T(lA'-1Jtw~' Before jumping into optics, lets look at the Fourier transform of a square pulse given in

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the figure above. By the definition of Fourier transform,

1+00

H(f) h(t)exp{ i27r ft}dt (5)-00

21+70 /A cos(27r ft)dt (6)

-70/ 2

A sin(7r fTO)- TO (7)

7rfTo - ATo sinc(7r fTo). (8)

The function sin(x)/x occurs so often that it has its own name. It is called the sinc (pronounced 'sink') function. We say that the Fourier transform of a square pulse is a sinc function. Note where most of the power in the sinc function is, between ±l/TO.

To make the connection with optics, we will need to know exactly what the Fourier trans­form does. Let's spell it out. To find the transform of some signal, it sa.ys to

1. Pick the frequency, or harmonic component, that you are interested in. That is, chose one value of f in exp{ i27rf t};

2. Multiply the function you want to transform by this, that is, form h(t) exp{ i27rf t};

3. Integrate over time.

These three steps are easily verified for the DC (f = 0) component in the above example. The integral of h(t) is just ATo. I sometimes look at the Fourier transform as telling me how much a sine or cosine wave can "fit into" the function in which I'm interested.

Now let's look at optics. I am going to approach it from a completely different perspective than you are probably use to. I will not pretend to be rigorous (for that, see one of my two favorite texts, Optics by Hecht or Principles of Optics by Born and \\Tolf) but instead will argue by analogy. In the following , I will also assume that you a.re familar with optics at the Halliday, Resnick & Walker level.

Let us take a plane wave of light illuminating a single slit of width a. We all know that light is just a traveling wave of the electric field, E(x, t) = Eo exp{i(kx - wt)}. The wavevector is 27r / A. In optics we generally ignore the iwt. We will observe the diffraction pattern of the slit on a screen very far away. The criteria for "far" is that the rays of light from the top and

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bottom of the slit are parallel at the screen. We can calculate the angular location of the first minimum. It is essentially the place where light from the bottom of the slit interferes with light from the top of the slit to give a null.

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Note what the slit is doing. It is taking light that is traveling in the x direction and spreading it out in the y direction. We have an electric field impinging on the slit and the slit is transforming that wave into an angular distribution. Let~ look at a particular incoming k so we can be explicit.

Here is the analogy we want to make. Just as we looked at the square pulse in time and asked how much of each frequency would "fit into" it; we want to look at the aperture as a square "pulse" in space and ask how much of each spatial frequency can "fit into" it. To be explicit,

pulse in time ---7 "pulse" in space (9)

27rf = W ---7 ky (10)

f ---7 K, (11)

TO --+ a. (12)

So now we can write down the distribution for the electric field as a function of ky

j +OO

E(ky) ex - 00 A(y)exp{ikyy}dy (13)

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where A(y) is the aperture distribution for the electric field,

A(y) - 1 Iyl::; a/2 (14)

o Iyl > a/2. (15)

This is just like the previous integral so we can write down the solution,

sin(ky ~'Y2)ex: a-----"'---- (16)

ky a/2 sin(ksin(O) a/2)E(O) (17)

ex: a k sin(0) a/2 .

Of course, we never observe the electric field directly because it is oscillating so fast. Rather we observe the intensity which is proportional to E2. So, we may write,

(18)

where ex = k a sin(0) /2. The first minimum of this function comes at ex = 7r or sin(0) = A/a, just what we found before. In words, we would state this as: the diffraction pattern for a single frequency is the square of the Fourier transform of the aperture distribution. In these domains, spatial distributions transform or map into angles.

This is a very powerful concept. It allows you to "see" the diffraction pattern of almost any aperture. Likewise, given a diffraction pattern, you can work in reverse and figure out what makes it. This is the basis behind electron diffraction, crystallography and many scattering experiments. There are complications arising from the fact that when intensities are measured, as opposed to the electric field, one gives up the phase information. Holography remedies this and instead of recording the magnitude of the Fourier transform squared (the power, if you will), it encodes the Fourier transform of the aperture distribution itself.

Let's move on and demonstrate a fundamental theorem called the convol'ution theor-em. This theorem relates multiplication in one domain, say temporal or spatial frequency, to a convolution in the other domain, time or space. Specifically, it says that the Fourier transform of the product of two functions in one domain equals the convolution of the Fourier transform of the functions . In formulas, for the frequency-time pair,

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1+= 1+=h(t) ® g(t) = _= h(t - T)g(t)dT = _= H(f)G(f)exp{ -i27r ft}df (19)

or similarly,

1+= 1+00H(f) ® G(f) = _= H(f - T)G(f)dT = _= h(t)g(t)exp{i27r ft}dt (20)

We can connect this to optics with the double slit diffraction pattern (Young's interference pattern). The double slit aperture function can be described as a convolution of a single slit of width d (aperture function A(y)) with two "delta functions" separated by the width of the slits a (aperture function B(y)). The resulting diffraction pattern is then magnitude squared of the product of the Fourier transforms of each of these. This is shown pictorially in the attached figure.

REFERENCES

1. Optics, Hecht, Chapter 11, Addison-Wesley, 1987.

2. Principle of Optics, Born & Wolf, Pergamon Press, 6th Ed., 1980.

3. Fundamentals of Physics, Halliday, Resnick, & Walker, Chapter 41, 4th Ed., Wiley, 1993.

4. The Fast Fourier Transform, Brigham, Prentice-Hall, 1974.

Updated L. Page, April 2011

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