Sinusoidal amplitude grating
…
+1st order
Λ
θ
0th order (or DC term)
–θ
incident
plane
wave
diffraction angle
-1st order
…
MIT 2.71/2.710
04/06/09 wk9-a- 5
spatial frequency
diffraction efficiencies
Example: binary phase grating
s
|gt| [a.u.]
1
q=+5
0.75
0.5
0.25
q=+4
0
−30
q=+3
q=+2
q=+1
incident
plane
wave
q= –1
q= –2
q= –3
q= –4
glass
refractive index n
MIT 2.71/2.710
04/06/09 wk9-a- 9
q= –5
0
x [!]
10
20
30
pi/2
0
−pi/2
−pi
−30
q=0
−10
pi
phase(gt) [rad]
Λ
−20
Duty cycle = 0.5
−20
−10
0
x [!]
10
20
30
Grating dispersion
…
Λ
air
white
…
MIT 2.71/2.710
04/06/09 wk9-a-10
glass
Grating:
blue light is diffracted at
smaller angle than red:
Prism:
blue light is refracted at
larger angle than red:
anomalous dispersion
normal dispersion
Today
•
•
•
•
Fraunhofer diffraction
Fourier transforms: maths
Fraunhofer patterns of typical apertures
Fresnel propagation: Fourier systems description
– impulse response and transfer function
– example: Talbot effect
Next week
•
•
•
Fourier transforming properties of lenses
Spatial frequencies and their interpretation
Spatial filtering
MIT 2.71/2.710
04/08/09 wk9-b- 1
Fraunhofer diffraction
Fresnel (free space) propagation may be expressed as a
convolution integral
MIT 2.71/2.710
04/08/09 wk9-b- 2
Example: rectangular aperture
x
y
z
sinc pattern
free space
propagation by
x0
MIT 2.71/2.710
04/08/09 wk9-b- 3
input field
l→∞
far field
Example: circular aperture
x
y
z
Airy pattern
2r0
free space
propagation by
l→∞
MIT 2.71/2.710
04/08/09 wk9-b- 4
input field
far field
How far along z does the Fraunhofer pattern appear?
Fresnel (free space) propagation may be expressed as a
convolution integral
cos(πα2)
α
For example, if (x2+y2)max=(4λ)2, then z>>16λ to enter the Fraunhofer regime;
if (x2+y2)max=(1000λ)2, then z>>106λ; in practice, the Fraunhofer intensity pattern is recognizable at smaller z than
these predictions (but the correct Fraunhofer phase takes longer to form)
MIT 2.71/2.710
04/08/09 wk9-b- 5
long short
propagation distance z
Fourier transforms
•
One dimensional
– Fourier transform
– Fourier integral
•
Two dimensional
– Fourier transform
– Fourier integral
(1D so we can draw it easily ... )
Re[G(u)]=
MIT 2.71/2.710
04/08/09 wk9-b- 6
g(x) [real]
Re[e-i2πux]
x
dx
Frequency representation g(x)=cos[2πu0x]
x
Re[G(u)]=
x
Re[G(u)]=
δ(u+u0)
½
−u0
MIT 2.71/2.710
04/08/09 wk9-b- 7
Re[e-i2πux]
G(u)
δ(u−u0)
½
+u0
dx =0, if u0≠u
dx =∞, if u0=u
G(u)=½ δ(u+u0)+½ δ(u−u0)
u
The negative frequency is physically meaningless,
but necessary for mathematical rigor; it is the price to pay for the convenience of using
complex exponentials in the phasor representation
Commonly used functions in wave Optics
Text removed due to copyright restrictions. Please see p. 12 in
Goodman, Joseph W. Introduction to Fourier Optics.
Englewood, CO: Roberts & Co., 2004. ISBN: 9780974707723.
Images from Wikimedia Commons, http://commons.wikimedia.org
MIT 2.71/2.710
04/08/09 wk9-b- 8
Goodman, Introduction to Fourier Optics (3rd ed.) pp. 12-14
Fourier transform pairs
Functions with radial symmetry
Table removed due to copyright restrictions. Please see Table 2.1 in
Goodman, Joseph W. Introduction to Fourier Optics. Englewood, CO: Roberts & Co., 2004.
ISBN: 9780974707723.
jinc(ρ)≡
Images from Wikimedia Commons, http://commons.wikimedia.org
MIT 2.71/2.710
04/08/09 wk9-b- 9
Goodman, Introduction to Fourier Optics (3rd ed.) p. 14
Fourier transform properties
Text removed due to copyright restrictions.
Please see pp. 8-9 in Goodman, Joseph W. Introduction to Fourier Optics.
Englewood, CO: Roberts & Co., 2004. ISBN: 9780974707723.
A general discussion of the properties of Fourier transforms may also be found here
http://en.wikipedia.org/wiki/Fourier_transform#Properties_of_the_Fourier_transform.
IMPORTANT! A note on notation: Goodman uses (fX, fY) to denote spatial frequencies along the (x,y) dimensions, respectively. In these notes, we will sometimes use (u,v) instead. MIT 2.71/2.710
04/08/09 wk9-b-10
Goodman, Introduction to Fourier Optics (3rd ed.) pp. 8-9
The spatial frequency domain: vertical grating
y
v
x
y
v
x
Space
domain
MIT 2.71/2.710
04/08/09 wk9-b- 11
u
u
Frequency
(Fourier)
domain
The spatial frequency domain: tilted grating
y
v
x
y
v
x
Space
domain
MIT 2.71/2.710
04/08/09 wk9-b-12
u
u
Frequency
(Fourier)
domain
Superposition: two gratings
+
Space
domain
MIT 2.71/2.710
04/08/09 wk9-b-13
+
Frequency
(Fourier)
domain
Superposition: multiple gratings
discrete
(Fourier
series)
continuous
(Fourier
integral)
Space
domain
MIT 2.71/2.710
04/08/09 wk9-b-14
Frequency
(Fourier)
domain
Spatial frequency representation of arbitrary scenes
0
MIT 2.71/2.710
04/08/09 wk9-b-15
The scaling (or similarity) theorem
Space
domain
MIT 2.71/2.710
04/08/09 wk9-b-16
Frequency
(Fourier)
domain
The shift theorem
Space
domain
MIT 2.71/2.710
04/08/09 wk9-b-17
Frequency
(Fourier)
domain
The convolution theorem
multiplication
MIT 2.71/2.710
04/08/09 wk9-b-18
convolution
MIT OpenCourseWare
http://ocw.mit.edu
2.71 / 2.710 Optics
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.