OPTICAL DIFFRACTION
3.2 UNCERTAINTY PRINCIPLE FOR FOURIER TRANSFORMS
42 OPTICAL DIFFRACTION
frequency domain. On taking the inverse transform, the tiger can be seen without the bars. Targets with spatial periodicity can be recognized, and early submarines could be identified by the periodic ribs in their structure.
Correlation and convolution are widely performed computations. The Fourier transform is normally used to perform convolution or correlation on a computer or in optics because convolution and correlation become multiplication in the frequency domain and the low cost of the fast Fourier transform in computers or the optical Fourier transform in optics makes the computation faster than a straight convolution or correlation computation [83]. Correlation is the basis for comparing images or searching for patterns or words in huge databases. Convolution implements linear systems such as smoothing, prediction, and many other signal processing functions.
Chapter 14 of my previous book [83] describes an optical data flow computer that I designed as a PI to DARPA and that was later partially constructed for NSA. It performs ultrafast correlation.
UNCERTAINTY PRINCIPLE FOR FOURIER TRANSFORMS 43
transform. Consequently, we expect a limit to accurately predict both the location of the pulse and its frequency simultaneously. In the limit, a delta function in time, representing exact knowledge of time (t→0), has all frequencies equally, which prevents accurate determination of its frequency. Similarly, a single frequency, a sinusoid, represented by a delta function in frequency, giving exact knowledge of frequency (ω→0), continues for all time, which prevents accurate measurement of its time of occurrence.
3.2.1.1 Proof of Uncertainty Principle for Fourier Transforms in Time We note that the constant on the right-hand side of the equation (3.5) may vary depending on the definitions of width used fortandω: possibilities are width at half maximum, root mean square, standard deviation, and Rayleigh width between nulls.
We follow the approach for complex functionsf used in Ref. [134]. A proof for real functionsf is given in Ref. [169].
E= ∞
−∞|f(t)|2dt, E= 1 2π
∞
−∞|F(ω)|2dω (3.7) Using Schwartz’s inequality [134], p(t)q(t)dt2≤
|p(t)|2dt
|q(t)|2dt, with p(t)=tf andq(t)=df∗/dt, we can write, reversing the direction of the equation,
∞
−∞t2|f|2dt ∞
−∞
df dt
2dt≥ ∞
−∞(tf) df∗
dt
dt
2 (3.8)
We show that the square root of the right-hand side of equation (3.8) leads to E/2, which isEtimes the right-hand side of the uncertainty principle, equation (3.5);
that is∞
−∞(tf)(df∗/dt) dt=E/2. First, the square root of the right-hand side of equation (3.8) may be written as
∞
−∞tfdf∗ dt dt
= 1 2
∞
−∞t
fdf∗
dt +f∗df dt
dt
(3.9)
= 1 2
∞
−∞td(ff∗) dt dt
=1 2
∞
−∞t d|f|2
dt
dt = 1
2 ∞
−∞td|f|2
= 1 2
t|f|2∞−∞− ∞
−∞|f|2dt
= −E
2 (3.10)
where we used integration by parts and equation (3.7) for obtaining the last line.
Also, ast→ ∞,√
|t||f| →0⇒t|f|2=0, so the first term of the last line in equa- tion (3.10) is zero.
Now we show that the left-hand side of equation (3.8) leads to the product of width squared in time and width squared in frequency for the square of the uncertainty principle on the left-hand side of equation (3.5). Taking a derivative of a time function is equivalent to multiplying by jω in the frequency domain and taking a second
44 OPTICAL DIFFRACTION
derivative is equivalent to multiplying byω2in the frequency domain. Therefore, if F(ω) is the Fourier transform off, then
∞
∞
df dt
2dt= 1 2π
∞
∞ ω2|F(ω)|2 dω (3.11) Substituting equation (3.11) into the left-hand side of equation (3.8) and equa- tion (3.10) into the right-hand side of equation (3.8) gives
∞
−∞t2|f|2dt 1 2π
∞
−∞ω2|F(ω)|2 dω≥ E2
4 (3.12)
If we define (t)2= 1
E ∞
−∞t2|f(t)|2dt and (ω)2= 1 E
1 2π
∞
−∞ω2|F(ω)|2dω (3.13) then substituting equation (3.13) into equation (3.12) gives
(t)2(ω)2≥ 1
4 (3.14)
The inequality holds after taking square roots of both sides, so we get tω≥ 1
2 (3.15)
which proves the uncertainty principle for Fourier transforms, equation (3.5).
For equality to hold in equation (3.15), equality must also hold in equation (3.8) or
|df/dt| = |kft|. This occurs only for functionsfwith a Gaussian magnitude because the Fourier transform of a Gaussian function is also Gaussian, exp{−π(bt)2} ←→
exp{−π(f/b)2}.
In an alternative form, we scale equation (3.12) by changing variables to frequency νin place of angular frequencyω=2πν(and dω=2πdν).
∞
−∞t2|f|2dt ∞
−∞ν2|F(ν)|2 dν≥ E2
4 (3.16)
The uncertainty principle for Fourier transforms in time can then be written in terms of time and frequency widths as
tν≥ 1
2 (3.17)
UNCERTAINTY PRINCIPLE FOR FOURIER TRANSFORMS 45
FIGURE 3.3 Relation between beam angleθand spatial frequencyλs: (a) wave striking vertical screen and (b) triangle in (a).
3.2.2 Uncertainty Principle for Fourier Transforms in Space
Because of the interchangeability of space and time, we can derive an uncertainty prin- ciple for Fourier transforms in space in a similar manner. More simply, we replace timet with spacesand temporal angular frequencyω=2πνwith spatial angular frequencykx=2πfxto obtain the equivalent uncertainty principle for Fourier trans- forms in space, the one that arises in diffraction. Note that space has up to three dimensions, unlike time. For simplicity, we consider only 1D.
skx≥ 1
2 or sfx≥ 1
2 (3.18)
The uncertainty principle may also be formulated in terms of emitted beam angle instead of spatial frequency. We consider a wave of wavelengthλstriking a plane at an angleθwith the normal (Figure 3.3a). The propagation vectorkcan be decomposed intoxandzcomponents according tok2=k2x+kz2. A wave component propagates down the screen inxdirection with propagation constantkxcorresponding to a spatial frequencyfxwith wavelengthλx=1/fx. Figure 3.3b shows a right triangle, extracted from Figure 3.3a, from which
sin θ= λ
λx =λfx for small angles fx=θ
λ (3.19)
where for small angles,θ→θand sin(θ)→θ. Substituting equation (3.19) into equation (3.18) gives an uncertainty principle for angle and distance:
sθ≥ λ
2 (3.20)
46 OPTICAL DIFFRACTION
(a) 2x 3x 4x
s s 0
s x= 1 z
(b) d1
d2
zL
1s x = zL
/2
Slit /2
Slit
s/2sin /2 Intensity
FIGURE 3.4 Fraunhofer diffraction as an interference phenomenon: (a) finite width source and (b) two slits.
In this case, it is not possible to make both the source sizesand the beam angleθof light emission arbitrarily small. In fact, arrays in radar and sonar and beam expanders in optics increasesto reduce beam angle in order to channel electromagnetic energy into a narrower beam for more accurate detection, imaging, or direction finding. A phased array radar or sonar scans this more piercing narrow beam electronically or it can be rotated. In the limit, if the source becomes infinitely small, it radiates in all directions and an angle cannot be distinguished. If, on the other hand, the beam angle is infinitely narrow, the source should be of infinite dimension.
3.2.2.1 Relation Between Diffraction and Interference It is instructive to view diffraction from an interference perspective. We consider a plane wave front at a source of widths(Figure 3.4a). A similar problem is presented by the two slits shown in Figure 3.4b. In either case, we consider two rays starting sapart. At a screen at the Fraunhofer distance, the two beams will interfere to produce a spatial Fourier transform inx. A null occurs at pointP on the screen when the two beams arrive out of phase (πphase difference) with each other for destructive interference.
This corresponds to the difference in distance traveled by the beams,d2−d1=λ/2.
For interference, we add the two beams,E1+E2, and then compute their combined intensity,(E1+E2)(E1+E2)∗. As discussed in Chapter 6, if we assume that the two beams arrive at the target with equal powerA, one of the form exp{i(kd1−ωt)} and the other exp{i(kd2−ωt)}, the intensity of the field on the screen can be written as
I=2A2(1+cos(kxd)) (3.21)
where d =d2−d1. From Figure 3.4b,d =ssin(θ/2)≈sx/zL, so that equation (3.21) becomes, usingkx=2π/λ,
I=2A2
(1+cos
2πsx λzL
(3.22)
SCALAR DIFFRACTION 47
whereλzL is characteristic of shrinkage when optical diffraction is used to com- pute a Fourier transform. For example, when λ=600 nm and distance to screen zL=10 cm, the spatial Fourier transform at the screen is smaller than would be calculated by multiplying byλzL=600×10−9×10−1=6×10−8. The shrinkage is the basis of holographic optical memory [83] and espionage in which a page of data is shrunk to the size of a period in a letter for covert communication.
From equation (3.22), peaks or fringes occur on thexaxis at distancesxf when (2πsxf)/(λzL)=2π orxf =(λzL)/(s). As shown in time in Figure 3.4b, the distance to the first null from the axis is proportional to 1/s. However, for space there is reduction by multiplication byλzL.