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1051-455-20073, Physical Optics
1 Laboratory 6: Diffraction and Imaging
References:Introduction to Optics, Pedrotti & Pedrotti, Chapters 16,18; Optics, E. Hecht, Chapter 10 We are able to determine locations of images and their magnifications by using the concept of light as a “ray” (geometrical optics) via the equation:
1 z 1
z 2
f
where z 1 and z 2 are the distances from object to lens and from lens to image and f is the focal length of the lens, which measures its ability to deviate rays. The model of light as a “wave” adds a fundamental constraint to an imaging system via the inherent process of “diffraction.” The concepts/processes of “interference” and “diffraction” actually are different manifestations of the same phenomenon, differing only in the number of sources involved (interference =⇒ few sources, say 2 - 10 and the light is recombined deliberately; diffraction =⇒ many sources, up to an infinite number, and the light is recombined naturally). Diffraction is the source of the theoretical limit to the performance of an imaging system.
1.1 Theory
This lab will investigate the diffraction patterns generated from apertures of different shapes and observed at different distances. As we have mentioned, the physical process that results in observable diffraction patterns is identical to that responsible for interference. In the latter case, we generally speak of the intensity patterns generated by light after passing through a few apertures whose size(s) generally are smaller than the distance between them. The term diffraction usually is applied to the process either for a single large aperture, or (equivalently) a large number of small (usually infinitesmal) contiguous apertures and the light recombines “naturally” as a part of its propagation. In studies of both interference and diffraction, the patterns are most obvious if the illumination is coherent, which means that the phase of each sinusoidal electric field is strictly deterministic. In other words, knowledge of the phase of the field at some point in space and/or time determines the phase at other points in space and/or time. Coherence has two flavors: spatial and temporal. For spatially coherent light, the phase difference ∆φ ≡ φ 1 − φ 2 of the electric field measured at any pair of locations in space by a vector distance ∆r at the instant of time (∆t = 0) is the same. In equations, we could write this as:
Φ
r 1 , t 1
r 1 + ∆r, t 1
r 2 , t 2
r 2 + ∆r, t 2
=⇒ spatial coherence
If the phase difference measured at the SAME location at two different times separated by ∆t ≡ t 1 − t 2 is the same for all points in space, the light is temporally coherent:
Φ
r 1 , t 1
r 1 , t 1 + ∆t
r 2 , t 2
r 2 , t 2 + ∆t
=⇒ temporal coherence
Light from a laser may be considered to be BOTH spatially and temporally coherent. The properties of coherent light allow phase differences of light that has traveled different paths to be made visible, since the phase difference is constant with time. In intereference, the effect often results in a sinusoidal fringe pattern in space. In diffraction, the phase difference of light from different points in the same large source can be seen as a similar pattern of dark and bright fringes, though not (usually) with sinusoidal spacing. Observed diffraction patterns from the same object usually have very different forms at different distances from the object to the observation plane. If viewed very close to the aperture (in the Rayleigh-Sommerfeld diffraction region), then Huygens’ principle says that the amplitude of the electric field is the summation (integral) of the spherical wavefronts generated by each point in
the aperture. The resulting amplitude pattern may be quite complicated to evaluate. If observed somewhat farther from the aperture, the spherical wavefronts may be accurately approximated by paraboloidal wavefronts. The approximation applies in the near field, or the Fresnel diffraction region. If viewed at a large distance compared to the extent of the object, the light from different locations in the aperture may be accurately modeled as plane waves with different wavefront tilts. This occurs in the Fraunhofer diffraction region.
1.1.1 Rayleigh-Sommerfeld Diffraction: Spherical Waves
A spherical wave of light with frequency ν 0 emitted by a source at the origin and observed at r = [x, y, z] has the form:
f [x, y, z, t] =
A 0
r
exp [+i (kr − ω 0 t)]
A 0
p x^2 + y^2 + z^2
exp
+2πi
à p x^2 + y^2 + z^2 λ 0
− ν 0 t
If emitted at a different location (say, r 0 = [x 0 , y 0 , z 0 ], then the distances in the denominator and the exponent must be adjusted:
f [x, y, z, t; x 0 , y 0 , z 0 ] =
A 0 [x 0 , y 0 , z 0 ] q (x − x 0 )^2 + (y − y 0 )^2 + (z − z 0 )^2
exp
⎣+2πi
q (x − x 0 )^2 + (y − y 0 )^2 + (z − z 0 )^2 λ 0
− ν 0 t
The amplitude at an observation point generated by light from a number of such sources is the integral over the source function:
g [x, y, z, t] =
Z Z Z
source
f [x, y, z, t; x 0 , y 0 , z 0 ] dx 0 dy 0 dz 0
Z Z Z
source
A 0 [x 0 , y 0 , z 0 ] q (x − x 0 )^2 + (y − y 0 )^2 + (z − z 0 )^2
e
+2πi
(^) √ (x−x 0 )^2 +(y−y 0 )^2 +(z−z 0 )^2 λ 0 −ν^0 t
dx 0 dy 0 dz 0
Often, the source is constrained to a 2-D plane [x 0 , y 0 ] and observed at a 2-D plane [x, y; z 1 ] so the integral is 2-D as well:
g [x, y, t; z 1 , z 0 ] = e−^2 πiν^0 t
Z Z
source
A 0 [x 0 , y 0 ; z 0 ] q (x − x 0 )^2 + (y − y 0 )^2 + (z 1 − z 0 )^2
e+2πi
√(x−x 0 )^2 +(y− λy 0 )^2 +(z 1 −z 0 )^2 (^0) dx 0 dy 0
We often discard the time dependence and specify the source as a 2-D function A 0 [x 0 , y 0 ; z 0 ] ≡ f [x 0 , y 0 ] in the plane z = z 0 :
g [x, y; z 1 , z 0 ] =
Z +Z∞
−∞
f [x 0 , y 0 ] q (x − x 0 )^2 + (y − y 0 )^2 + (z 1 − z 0 )^2
e+2πi
√(x−x 0 )^2 +(y− λy 0 )^2 +(z 1 −z 0 )^2 (^0) dx 0 dy 0
The mathematical solution of this integral is difficult for all but the simplest source functions, so we will not consider this case further.
1.1.2 Fresnel Diffraction
If the distance between the object and the observation plane is assumed to be large compared to the 2-D extent of a planar object, then the diffraction integral is significantly simpler (though perhaps still challenging). Under these conditions in the Fresnel diffraction region, the wavefront emitted by each source point in the 2-D object plane is assumed to have a paraboloidal shape (rather than
1.1.3 Fraunhofer Diffraction
At large distances from the object plane, the diffraction is in the far field or Fraunhofer diffraction region. Here, the pattern of diffracted light usually does not resemble the object at all. The size of the observed pattern varies in proportion to the reciprocal of the object dimension, i.e., the larger the object, the smaller the diffraction pattern. Note that increasing the size of the object also produces a brighter diffraction pattern, because more light reaches the observation plane. The mathematical relation between the shape and size of the output relative to that of the input is a Fourier transform, which is a mathematical coordinate transformation that was “discovered” by Baron Jean-Baptiste Joseph de Fourier in the early 1800s. For the same input pattern f [x, y], the diffraction pattern in the Fraunhofer region has the form:
F 2 {f [x, y]} ≡ F [ξ, η] =
Z Z +∞
−∞
f [x, y] (exp [+2πi (ξx + ηy)])∗^ dx dy
Z Z +∞
−∞
f [x, y] exp [− 2 πi (ξx + ηy)] dx dy
In words, the input function f [x, y] is transformed into the equivalent function F [ξ, η], where the coordinates ξ, η are spatial frequencies measured in cycles per unit length, e.g., cycles per mm. In optical propagation, the end result is a function of the original 2-D coordinates [x, y], which means that the coordinates [ξ, η] are “mapped” back to the space domain via a scaling factor. Since the coordinates of the transform have dimensions of (length)−^1 and the coordinates of the diffracted light have dimensions of length, the scale factor applied to ξ and η must have dimensions of (length)^2. It is easy to show that the scaling factor is the product of the two length parameters available in the problem: the wavelength λ 0 and the propagation distance z 1. The pattern of diffracted light in the Fraunhofer diffraction region is:
g [x, y] ∝ F 2 {f [x, y]}|λ 0 z 1 ξ→x,λ 0 z 1 η→y ≡
Z Z +∞
−∞
f [α, β] exp
− 2 πi
μ α
x λ 0 z 1
y λ 0 z 1
dα dβ
In mathematical terms, this is a “linear, shift-variant” operation; it is linear because if the input amplitude is scaled by a constant factor, the output amplitude is scaled by the same factor. It is shift variant because a translation of the input does not produce a corresponding transformation of the output. Because Fraunhofer diffraction is shift variant, it may NOT be represented as a single convolution. However, once the Fourier transform is understood, it is very easy to visualize Fraunhofer diffraction patterns of many kinds of objects. The study of Fourier transforms allows us to infer some important (and possibly counterintuitive) properties of Fraunhofer diffraction:
- Scaling Theorem: If the scale factor of the aperture function f [x, y] increases, then the resulting diffraction pattern becomes brighter and “smaller,” i.e., the scale factor is proportional to the reciprocal of the scale factor of the input function.
- Shift Theorem: Translation of f [x, y] adds a linear term to the phase of the diffraction pattern, which is not visible in the irradiance. Thus translation of the input has no visible effect on the diffraction pattern.
- Modulation Theorem: If an aperture can be expressed as the product of two functions, the amplitude of the diffraction pattern is their convolution.
- Filter Theorem: if an aperture pattern is the convolution of two patterns, the amplitude of the resulting diffraction pattern is the product of the amplitudes of the individual patterns.
Example: Consider Fraunhofer diffraction of a simple 2-D rectangular object:
f [x, y] = RECT
h (^) x a
y b
i ≡
1 if |x| < a 2 and |y| < b 2 1 2 if^ |x|^ =^
a 2 and^ |y|^ <^
b 2 or^ |x|^ <^
¯ b 2
¯ (^) and |y| = b 1 2 4 if^ |x|^ =^
a 2 and^ y^ =^
b 2 0 if |x| > a 2 and > b 2
The integral evaluates rather easily:
g [x, y] ∝
Z Z +∞
−∞
RECT
α a
β b
exp
2 πi(xα + yβ) λ 0 z
dα dβ
2
Z (^) y=+ b 2
y=− b 2
Z (^) x=+ a 2
x=− a 2
exp
2 πixα λ 0 z
exp
2 πiyβ λ 0 z
dα dβ
Z (^) x=+ a 2
x=− a 2
exp
∙μ −
2 πix λ 0 z
α
dα ·
Z (^) y=+ b 2
y=− 2 b
exp
∙μ −
2 πiy λ 0 z
β
dβ
exp
h³ − (^2) λπix 0 z
α
i
³ − (^2) λπix 0 z
α=+ α 2
α=− α 2
exp
h³ − (^2) λπiy 0 z
β
i
³ − (^2) λπiy 0 z
β=+ β 2
β=− β 2
exp
h −i πaxλ 0 z
i − exp
h +i πaxλ 0 z
i
³ − (^2) λπix 0 z
exp
h −i πybλ 0 z
i − exp
h +i πybλ 0 z
i
³ − (^2) λπib 0 z
= |a|
sin
h πax λ 0 z
i
³ πax λ 0 z
⎠ (^) · |b|
sin
h πby λ 0 z
i
³ πby λ 0 z
⎠ (^) ≡ |ab| SINC
x ¡ (^) λ 0 z a
y ¡ (^) λ 0 z b
g [x, y] ∝ (ab)^2
Ã
SIN C
x ¡ (^) λ 0 z a
y ¡ (^) λ 0 z b
Examples are shown in the figure:
Fraunhofer diffraction patterns from two slits of different widths; a wider slit has a narrower and taller Fraunhofer diffraction pattern.
Where the 2-D “SINC” function is defined as the orthogonal product of two 1-D SINC functions:
SIN C [x, y] ≡ SIN C [x] · SINC [y] ≡
sin [πx] πx
sin [πy] πy
which has the pattern shown in the figure.
function f [x, y; z = 0] is illuminated by a unit amplitude monochromatic plane wave with wavelength λ 0. The light propagates into the Fraunhofer diffraction region at a distance z 1 , where the resulting amplitude pattern is:
E [x, y; z 1 ] =
E 0
iλ 0 z 1
exp
+2πi
z 1 λ 0
exp
+iπ
x^2 + y^2
λ 0 z 1
F
x λ 0 z 1
y λ 0 z 1
This pattern illuminates the 2-D aperture function p [x, y] and then propagates the distance z 2 into the Fraunhofer diffraction region (determined by the support of p). A second application produces the amplitude at the observation plane:
E [x, y; z 1 + z 2 ] = E 0
Ã
iλ 0 z 1
e+2πi^
z 1 λ (^0) e+iπ^
(x^2 +y^2 ) λ 0 z 1
! Ã
iλ 0 z 2
e+2πi^
z 2 λ (^0) e+iπ^
(x^2 +y^2 ) λ 0 z 2
· F 2
F
x λ 0 z 1
y λ 0 z 1
· p [x, y]
ξ= (^) λ 0 xz 2 , (^) λ 0 yz 2
= E 0
μ −
λ^20 z 1 z 2
e+2πi^
z 1 +z 2 λ (^0) e+iπ^
(x^2 +y^2 ) λ 0
(^1) z 1 +^ z^12
· (λ 0 z 1 )^2 (f [−λ 0 z 1 ξ, −λ 0 z 1 η] ∗ P [ξ, η])|ξ= x λ 0 z 2 ,^ y λ 0 z 2
= E 0
μ −
z 1 z 2
e+2πi^
z 1 +z 2 λ (^0) e+iπ^
(x^2 +y^2 ) λ 0
(^1) z 1 +^ z^12
μ f
∙μ −
z 1 z 2
x,
μ −
z 1 z 2
y
∗ P
x λ 0 z 2
y λ 0 z 2
E 0
MT
e+2πi^
z 1 +z 2 λ (^0) e+iπ^
(x^2 +y^2 ) λ 0
(^1) z 1 +^ z^12
μ f
x MT
y MT
∗ P
x λ 0 z 2
y λ 0 z 2
where the theorems of the Fourier transform and the definition of the transverse magnification from geometrical optics, MT = − z z^21 , have been used. Note that if the propagation distances z 1 and z 2 must both be positive in Fraunhofer diffraction, which requires that MT < 0 and the image is “reversed.” The irradiance of the image is proportional to the squared magnitude of the amplitude:
|E [x, y; z 1 + z 2 ]|^2 =
E 0
MT
¯f
x MT
y MT
∗ P
x λ 0 z 2
y λ 0 z 2
2
In words, the output amplitude created by this imaging “system” is the product of some con- stants, a quadratic-phase function of [x, y], and the convolution of the input amplitude scaled by the transverse magnification and the scaled replica of the spectrum of the aperture function,
P
h x λ 0 z 2 ,^
y λ 0 z 2
i
. Since the output is the result of a convolution, we identify the spectrum as the
impulse response of a shift-invariant convolution that is composed of two shift-variant Fourier trans- forms and multiplication by a quadratic-phase factor of [x, y]. This system does not satisfy the strict conditions for shift invariance because of the leading quadratic-phase factor and the fact that the input to the convolution is a scaled and reversed replica of the input to the system. That said, these details are often ignored to allow the process to considered to be shift invariant. We will revisit this conceptual imaging system after considering the mathematical models for optical elements.
1.2 Equipment:
- He:Ne laser
- microscope objective to expand the beam; larger power gives larger beam in shorter distance;
- pinhole aperture to “clean up” the beam;
- positive lens with diameter d ∼= 50 mm and focal length f / 600 mm, to collimate the beam;
- positve lens with diameter d ∼= 50 mm and focal length f > 200 mm, to compute the Fourier transform;
- aluminum foil, needles, and razor blades to make your own objects for diffraction;
- set of Metrologic transparencies;
- digital camera to record diffraction patterns.
1.3 Procedure
- Set up the experimental bench as in the figure with the observing screen close to the aperture (within a foot or so) to examine the results in the Fresnel diffraction region. Measure and record the relevant distances. A number of apertures are available for use, including single and multiple slits of different spacings, single and multiple circular apertures, needles (both tips and eyes), razor blades, etc.. In addition, aluminum foil and needles are availble to make your own apertures.
Apparatus for viewing Fresnel diffraction patterns (and Fraunhofer patterns if z 1 is sufficiently large).
(a) Begin with a single slit, a square aperture, or a circular aperture. Note the form of the diffraction pattern. For example, sketch how its “brightness” changes with position and note the sizes and locations of any features. For a slit or circular aperture, you should note light and dark regions in the pattern; measure the positions of some maxima and minima (at least 5). Use the data to derive a scale of the pattern. Sketch the pattern noting the scale. (b) Repeat the previous step with a “wider” slit or aperture. Note the difference in the results. (c) Vary the distance between the screen and the diffracting object. Repeat measurements. What is the relation between the shange in distance and the change in scale of the pattern? Repeat for 5 different distances where the character of the pattern remains the same.
wide spacing), #8 (concentric circles with medium spacing), #9 (concentric circles with narrow spacing), and “crossed” gratings #10 (wide spacing), #11 (medium spacing), and #12 (narrow spacing).
(f) Now overlay a periodic structure (grid) with a circular aperture and observe the pattern. The overlaying of the two slides produces the product of the two patterns (also called the modulation of one pattern by the other). (g) Examine the image and diffraction pattern of the transparency Albert (Metrologic slide #18). Note the features of the diffraction pattern and relate them to the features of the transparency. (h) Examine the pattern generated by a Fresnel Zone Plate (Metrologic slide #13) at different distances. The FZP is a circular grating whose spacing decreases with increasing distance from the center. Sketch a side view of the FZP and indicate the diffraction angle for light incident at different distances from the center of symmetry. You might also overlap another transparency (such as a circular aperture) and the FZP and record the result. I guarantee that this result will not resemble that of part d.
(i) If time permits, you can also find the diffraction patterns of other objects, such as the tip and/or the eye of the needle.
2 Questions
- This experiment demonstrates that interaction of light with an obstruction will spread the light. For example, consider Fresnel diffraction of two identical small circular apertures that are separated by a distance d. How will diffraction affect the ability to distinguish the two sources? Comment on the result as lens diameter d is made smaller.
- The Fresnel Zone Plate (Metrologic slide #13) may be viewed as a circularly symmetric grating with variable period that decreases in proportion to the radial distance from the center. It is possible to use the FZP as an imaging element (i.e., as a lens). Use the model of diffraction from a constant-period grating to describe how the FZP may be used to “focus” light in an optical imaging system. This may be useful for wavelengths (such as x rays) where imaging lenses do not exist.
