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Optical Diffraction and Image Formation
H. Stahlberg
Block Course Biophysics and Structural Biology
Introduction
To extract structural information from an object, it is illuminated by rays, which can be
described either by a stream of particles or by waves. The object scatters the particles or
diffracts the waves: both descriptions are apparently contradictory, but are equally valid. The
single dots on a weakly illuminated photographic film or also the so-called “photo effect
experiment” in physics prove the particle nature of light. On the same time, the phenomenon of
destructive interference proves the wave nature of light. Quantum physics combines these views,
by describing the probability to find a certain particle at a given (approximate) time and
(approximate) location by a wave function. This probability density function then can then show
interference phenomena.
Light
The spectrum of electromagnetic waves ranges from gamma radiation, visible light, to radio
waves (Fig. 1).
Fig. 1
The spectrum of electromagnetic waves
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Electromagnetic waves emerge from a periodically moving charge (Fig. 2). The Maxwell
equations describe the propagation of electromagnetic waves. The parameters that specify a light
wave are its wavelength !, its amplitude A, and its polarization (Fig. 2). The frequency " of this
wave is calculated from eq. 1 (see Fig. 1):
! # " = c;
(1)
Fig. 2
where c = 299’792’458 m/s, being the speed of light in vacuum.
The Hertzian dipole emits a polarized, coherent electromagnetic wave
Optical fields lie at frequencies of 1014 to 1015 Hertz on the electromagnetic spectrum (Fig. 1). At higher
frequencies (x-ray and gamma ray), the particle nature of quantized fields becomes more and more
prevalent. These high-energy photons are emitted by nuclear processes and high-energy particle
accelerators. Particle counters measure such high-energy photons. At lower frequencies in the radio
range, fields are generated and detected by continuous charge fluctuations in antennas. The number of
photons in radio fields is generally so large that particle-like phenomena and counting statistics are not
relevant. In the frequency ranges of microwave, infrared or optical waves, radiating fields are generated
and detected by discrete phenomena such as spontaneous atomic emission or black body radiation rather
than continuous current distributions, but the number of photons in the field is often large enough that
wave analysis of the field is more useful than quantized analysis.
Coherence and Interference
Coherence is a measure of the statistical correlation between fields drawn from different points
in space and time. Fields emerging from different points can only interfere, if they are coherent.
Heated filaments emit short wave trains, which do not have any phase correlation, as the emission of
light is a statistical process. In addition, these wavelets possess wavelengths distributed over the broad
emission spectrum of a hot wire. Many different wave trains with different wavelengths are emitted on
the same time, so that a broad color spectrum is produced. Using diffractive filters, an optical field
produced by such a light source can be filtered to a narrow range of wavelengths that cover only a range
of 0.1 to 1 nm width. On a 1000 nm center wavelength, this quasi-monochromatic field has a carrier
frequency, which is 103 to 104 times greater than its bandwidth (1000 nm wavelength, 1 nm width, means
wavelengths between 999.5 nm and 1000.5 nm). In contrast, a typical gas laser might have a bandwidth
of 1-100 MHz for light at 1014 to 1015 Hertz, giving a bandwidth/frequency ratio of 108. Lasers, therefore,
emit highly coherent light waves.
The bandwidth (!") determines the coherence in the time domain, i.e., the length of the wave
trains that interfere. The size (d) of the source, however, determines the coherence in the space
domain, i.e., the angle within which the field is coherent. The perfect coherent source is a point
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(d=0) emitting a monochromatic (!"=0) spherical wave; if it is located at infinity, the
corresponding wave front is planar.
Two wave trains represented by the oscillation A cos $(x,t) interfere in the region of overlap
(Fig. 3) provided that they have the same polarization (see Fig. 2). Depending on their phase
difference !$, they interfere constructively (!$ = 2% . m; m = integer) or destructively
(!$ = % (2m + 1)). The contrast of interference fringes is a direct measure of the degree of
coherence.
Fig 3
Light waves are mathematically best described by rotating vectors: A cos ($), and in complex notation: A
exp (i $) = A{cos ($) + i sin ($)}. Two light wave trains only interfere, if they overlap.
Provided that a parallel monochromatic beam is split and combined after traveling along two
different optical paths, we expect to observe interference fringes as long as the path difference is
shorter than the wave trains. We call this maximum path difference the coherence length lc, and
it is measured in the Michelson interferometer (Fig. 4).
Fig. 4 Fringe visibility allows the coherence of optical fields to be determined. In the Michelson interferometer
(left) the optical path between two interfering beams is changed, and the coherence length lc is determined by the
fringe visibility. In Young’s interferometer (right) the distance between two apertures is varied to determine the
spatial coherence $c.
The coherence of a divergent optical field is tested using a screen with two small holes that is
placed between source and observer (Young’s interferometer, Fig. 4). Although the path
difference between light passing through one hole and that propagating through the other hole
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may be shorter than lc, interference fringes fade when the angular separation of the two holes
exceeds $c, the coherence angle, with respect to the source.
Thus, two wave trains are capable to generate an observable interference pattern only if they
emerge from an angular interval <$c and have a path difference < lc. Bandwidth !" and source
size d determine the coherence.
(2)
(3)
lc = c / !"
$c = ! / d
Electrons are described by the Schrödinger equation, whose solution is a wave-function. Hence,
the concepts of coherence discussed here for light waves may be applied to electrons as well. In
the following sections light waves are considered to be coherent and of identical polarization.
Optical Diffraction
We will now explain Huygen’s wavelet theory, and then develop the so-called Fraunhofer
theory for diffraction.
Expansion on the lecture’s material (this is optional reading material)
If you imagine a slide projector’s slide in a parallel light path (and we don’t have any lens here
at the moment), then you can imagine that directly behind the slide in a few millimeters distance
you will only get the shadow image of the slide, which is more or less looking exactly like the
slide. However, and if you don’t have any lens, on a wall far away from the slide you will see the
diffraction pattern from that slide, which corresponds to the intensities (= Amplitudes2) of the
Fourier transformation of the slide image. If the slide, for example, shows some lines, then the
distant wall will show the diffraction peaks from those lines. The Fraunhofer theory explains
how that shadow image close to the slide transforms into the Fourier transformation’s
intensities further away. This involves some mathematics, which we have to develop here. This
calculation goes beyond the lecture’s material and is given here as additional material.
While all electromagnetic phenomena are described by the Maxwell equations, material
interactions, sources, boundaries and receivers differ across the frequency spectrum. The wave
equations describing the propagation of electromagnetic waves in space are an important
corollary of Maxwell’s equations. The spherical wave (SW) is a solution of these equations:
(4)
(A/r) exp[ i {$(r)+ 2%"t}] with $(r) = 2% r /!
(note that exp[ i {$(r)] = cos[$(r)] + i sin [$(r)], see handout S. Grzesiek)
By linear superposition of such spherical waves, boundary conditions imposed by non-isotropy
can be resolved. The intuitive picture proposed by Huygens makes this clear: each wave front
can be viewed as a sum of spherical wavelets that propagate and superimpose or interfere (Fig.
5).
Fig. 5 Huygen’s wavelets emerge from wave fronts that are thought to consist
of many scattering centers. Wavelets superimpose to form propagating wave
fronts.
If a plane wave illuminates a planar screen with transparency t(x,y),
the amplitude of emerging SWs is weighted by t(x,y). The sum of
all these weighted SWs then describes the optical field at any point
behind the screen (Fig.6). As we consider a coherent superposition
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of these SWs, the phase differences count. They are due to path-length differences. Note that
this can be extended to 3D objects (t(x,y,z)) such as 3D crystals used for X-ray diffraction
analyses.
Fig. 6
Geometry to calculate the optical field behind a transparent screen t(x,y)
A 2-dimensional transparency is illuminated by a monochromatic plane wave. The wavelet
emitted by element dxdy at x,y has an amplitude t(x,y), the object transparency. We now want to
calculate the intensity distribution of this wave on the “screen” behind the transparent object. By
doing this, we will see that the diffracted image on the screen turns out to be the Fourier
transformation of the structure in the x,y plane, if the screen is sufficiently far away from the
structure.
The screen is defined by the plane with the coordinates &,' (Fig. 6). On this screen we measure
the contribution
(5)
{t(x,y) exp(i 2" r/!)}/r dxdy,
where dxdy is the “atomic element” in the sample in the x,y plane. Here, r in the denominator
can be approximated by z, giving a simpler formula
(6)
{t(x,y) exp(i 2" r/!)}/z dxdy,
The trouble arises from the phase term (2" r/!), where r cannot be approximated by z, because r
can vary considerably compared to !, when x and z are varied within the experimental limits.
Instead, r is calculated from Pythagoras’ law: r2 = ((&-x)2 + ('-y)2 + z2)1/2. The square root is
conveniently developed in a power series:
(6)
(1 + ()1/2 = 1 + (/2 - (2/8 + ...
and this yields for r the approximation:
(7)
r = z+ (x+y)2/2z – (x& +y')/z + (&+')2/2z + ...
Omitting the time dependence and terminating the series expansion at second order terms, the
paraxial approximation of a SW is written as
(8)
SW = (A/z) exp[i 2%/! {z+ (x+y)2/(2z) - (x& +y')/z + (&+')2/(2z)}]
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To obtain the amplitude T of the diffracted wave at point &, we simply integrate over all the
weighted SWs emanating from planar object t(x) being illuminated by a plane wave. In eq (9)
only those terms that depend on x remain in the integral:
+#
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(9)
T(&)=exp[i2%{z/! +(&+') /(2! z)}]/z $$ t(x)exp[-i2% {(x& +y')/(! z) - (x+y)2/(2! z)}]dxdy
-#
To obtain a manageable integral, we require that the term (x+y)2/(2! z) to be <1/4 (i.e. gives a
phase contribution < %/2), a condition known as the Fraunhofer approximation:
(10)
z >2(x+y)2/!
With h = 2"&/(! z), k = 2"'/(! z), we then find the amplitude distribution of the diffracted wave
T(h,k) to be essentially the Fourier transform of the transmittance t(x) (see handout S. Grzesiek,
section 8):
+#
T(h,k) = P(h,k) $ t(x,y) exp (-i (xh+yk)) dx
-#
(11)
or
T(h,k) = P(h,k).FT {t(x,y)}
Here, P(k) represents the term in eq. 9 that is in front of the integral. It modulates the phase, but
not the intensity distribution of the diffracted wave. Therefore, the intensity of the diffracted
wave equals |T(k)|2 = |FT{t(x)}|2. For z>>2(&+')2/!, P(h,k) is constant in the observation plane,
and the diffracted wave is identical to the Fourier transform of t(x,y) given in (12):
(12)
+#
T(kx,ky) = $ $ t(x,y) exp {-i (xkx + yky )}dx dy
-#
As a practical example let’s consider an object t(x,y) whose radius is 1 mm and which is
illuminated by a green light beam (! = 500 nm). We may say that the diffraction pattern is
|FT{t(x,y)}|2 provided that it is observed at z > 4m (see eq. 10). If we consider a protein crystal
with a radius of 50 µm, and an X-ray beam with ! = 0.15 nm, the detector should be at z > 33 m.
In practice, however, because the coherence of the X-ray beam is limited and only a small
fraction of the crystal is coherently illuminated, a distance between crystal and camera of 10 cm
is sufficient.
In the X-ray diffraction theory, parallel rays are assumed and the phase difference between a wave scattered
at the center j and a hypothetical wave scattered at the origin O is calculated as the difference of two scalar
products (see handout T. Schirmer, p.2):
(13)
)j = 2" r j (s f - s o)/ !
where r j is the position of the center j, s f is the scattering vector and s o is the vector of the illumination.
Note the similarity of eq. (13) and the argument of the exponential in eq. (11): xk = 2" (x &/z)/ !. Hence,
optical diffraction allows X-ray diffraction to be simulated and understood.
A lens is an amazing optical device. It converts an incoming SW to an outgoing SW of different
curvature. Fig. 7 shows the example of a parallel beam (i.e. a plane, incoming wave) that
converges to a point in the back focal plane.
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Fig. 7 a) A lens converts a parallel beam into a beam that converges in the focal plane f, because the optical path
*0 in the center of the lens is longer than that at the periphery (*1+ *2). An image of a point source in the infinity is
thus formed in the focal plane. b) If a diffraction grating t(x,y) is placed before the lens, the diffracted beams are
converging into the respective diffraction spots in the back focal plane, i.e., the lens brings the Fraunhofer
diffraction pattern into the back focal plane. If the grating is placed in the front focal plane, the amplitude in the
back focal plane is exactly the Fourier transform of the grating t(x,y).
As documented in Fig. 7b, a lens produces the Fourier transform of a transparency t(x,y) placed
in the front focal plane. From Fig. 7b it follows that the transform may be scaled using lenses
with different focal lengths. In addition, transparencies may also be placed behind the lens to
generate the diffraction pattern. In this case, a quadratic phase term occurs that is not relevant as
long as the intensity |T(kx,ky)|2 is considered.
A lens is therefore a perfect device to study diffraction. It is used to make the diffraction process
visible, and to check micrographs of periodic objects.
Image Formation
As shown in Fig. 8 a lens forms the image of a point source by converting the divergent SW to
an SW that converges into the image point. From simple geometric considerations, the lens law
and magnification M of a lens (eq. 14) are derived (see also Fig. 8).
(14)
1/a+1/b=1/f and M=x’/x=-b/a
t(x,y)
T(&,') L(&,') T’(&,')
i(x’,y’)
Fig. 8 Image formation. Rays emerging from an object point propagate parallel to the optical axis if they cross the
front focal point. Rays that propagate parallel to the optical axis before the lens will cross the back focal point. The
two distinct rays cross at the image point.
The formation of an image may also be described as a sequence of Fourier transforms. Three
steps may be considered:
1. The wave diffracted by the object t(x,y) travels first over a distance a; this propagation is
described by eq. 9 and yields basically the Fourier transform of t(x,y).
2. The lens converts the curvature of the diverging wave to one that converges.
3. The wave now propagates over distance b, inducing a second Fourier transform step.
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Resolution of a Lens System
To obtain the highest possible resolution, the lens must transmit the highest diffraction orders
without distortions. This is achieved by using a lens with a high numerical aperture, which
collects the diffracted light at high angles, and transmits this light without distortion. The
resolution is given by the width of the point image as determined by the first minimum (Fig. 11).
Two (incoherent) spots can only be resolved if the position of one spot image is outside the first
diffraction minimum of the other one.
Fig. 11 A perfect lens produces a diffraction limited image of a point source. This image is the impulse response
h(x’,y’) in eq. 19. Two points cannot be resolved, when the maximum of one point image is closer to the other point
image than r1/M, the minimum of this other point image. Two extreme cases can be considered: a telescope that
produces the image of two stars in its focal plane, and a microscope that produces the image of two fluorescent
molecules at distance b (that can be often treated an virtually infinitely positioned screen). In both cases the
resolution is 1.22 !a/d (where a=f for b>>a).
Because the resolution is limited by diffraction, the focus is not the only place where an image is
sharp. If the image plane is moved along the optical axis, the resolution changes only a little.
This region is called depth of field, and it is easily understood from Fig. 12.
Fig. 12 The depth of field is the result of diffraction-limited resolution.
To calculate the depth of field *f we argue that the resolution disk should have the same
diameter as the cone formed from the rays converging to the image plane. Hence
(22)
*f = 2.44 ! / (NA)2
Aberrations
Unfortunately lenses are not perfect. Spherical glass surfaces can only be made perfect for a
single magnification and a monochromatic illumination. Two types of glasses may be combined
to eliminate to some extent the dispersion (wavelength dependence of the refractive index). Such
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a lens combination is called achromatic. Nevertheless, it has only perfect properties for points
close to the axis. As lenses are produced from pieces of glass, fabrication errors may introduce
deviations from spherical surfaces. Aberrations are corrected to a great degree in modern lenses
and microscope objectives. Figure 12 summarizes the major lens aberrations.
Fig. 12 Lens aberrations
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