1. Today • Diffraction from periodic transparencies: gratings
• Grating dispersion
• Wave optics description of a lens: quadratic phase delay
• Lens as Fourier transform engine

2. Diffraction from periodic array of holes
Spatial frequency: 1/Λ
A spherical wave is generated
at each hole;
we need to figure out how the
periodically-spaced spherical waves
interfere
incident
plane
wave

3. Diffraction from periodic array of holes
Spatial frequency: 1/Λ
Interference is constructive in the
direction pointed by the parallel rays
if the optical path difference
between successive rays
equals an integral multiple of λ
(equivalently, the phase delay
equals an integral multiple of 2π)
incident
plane
wave
Optical path differences

4. Diffraction from periodic array of holes
Spatial frequency: 1/Λ
From the geometry
we find
Therefore, interference is
constructive iff

5. Diffraction from periodic array of holes
incident
plane
wave
Grating spatial frequency: 1/Λ
Angular separation between diffracted orders: Δθ ≈λ/Λ
2nd diffracted
order
1st diffracted
“straight-through”
order (aka DC term)
–1st diffracted
several diffracted plane waves
“diffraction orders”

6. Fraunhofer diffraction from periodic array of holes

7. Sinusoidal amplitude grating

8. Sinusoidal amplitude grating
incident
plane
wave
Only the 0th and ±1st
orders are visible

9. Sinusoidal amplitude grating
three
plane
waves
far field
one
plane
wave
three
converging
spherical waves
+1st order
0th order
–1st order
diffraction efficiencies

10. Dispersion

11. Dispersion from a grating

12. Dispersion from a grating

13. Prism dispersion vs grating dispersion
Blue light is refracted at
larger angle than red:
normal dispersion
Blue light is diffracted at
smaller angle than red:
anomalous dispersion

14. The ideal thin lens as a Fourier transform engine

15. Q: how can we “undo” the convolution optically?
Fresnel diffraction
Reminder
coherent
plane-wave
illumination
The diffracted field is the convolution of the transparency with a spherical wave
Q: how can we “undo” the convolution optically?

16. Fraunhofer diffraction
Reminder
The “far-field” (i.e. the diffraction pattern at a large
longitudinal distance l equals the Fourier transform
of the original transparency
calculated at spatial frequencies
Q: is there another
optical element who
can perform a
Fourier
transformation
without having to go
too far (to ∞ ) ?

17. The thin lens (geometrical optics)
f (focal length)
object at ∞
(plane wave)
Ray bending is proportional
to the distance from the axis
point object
at finite distance
(spherical wave)

18. The thin lens (wave optics)
incoming wavefront
a(x,y)
outgoing wavefront
a(x,y) t(x,y)eiφ(x,y)
(thin transparency approximation)

19. The thin lens transmission function

20. The thin lens transmission function
this constant-phase term can be omitted
where
is the focal length

21. Example: plane wave through lens
plane wave: exp{i2πu0x}
angle θ0, sp. freq. u0≈ θ0 /λ

22. Example: plane wave through lens
back focal plane
spherical wave,
converging
off–axis
wavefront after lens :
ignore

23. Example: spherical wave through lens
front focal plane
spherical wave,
diverging
off–axis
spherical wave (has propagated distance ) :
lens transmission function :

24. Example: spherical wave through lens
front focal plane
spherical wave,
diverging
off–axis
plane wave
at angle
wavefront after lens
ignore

25. Diffraction at the back focal plane
diffraction pattern
gf(x”,y”)
thin
transparency
g(x,y)
thin
lens

26. Diffraction at the back focal plane
1D calculation
Field before lens
Field after lens
Field at back f.p.

27. Diffraction at the back focal plane
1D calculation
2D version

28. Diffraction at the back focal plane
spherical
wave-front
Fourier transform
of g(x,y)

29. Fraunhofer diffraction vis-á-vis a lens

30. Spherical – plane wave duality
plane wave oriented
point source at (x,y)
amplitude gin(x,y)
towards
... of plane waves
corresponding to
point sources
in the object
each output coordinate
(x’,y’) receives ...
... a superposition ...

31. Spherical – plane wave duality
produces a spherical wave converging
a plane wave departing
from the transparency
at angle (θx, θy) has amplitude
equal to the Fourier coefficient
at frequency (θx/λ, θy /λ) of gin(x,y)
produces a spherical wave converging
towards
each output coordinate
(x’,y’) receives amplitude equal
to that of the corresponding
Fourier component

32. Conclusions
When a thin transparency is illuminated coherently by a monochromatic plane wave and the light passes through a lens, the field at the focal plane is the Fourier transform of the transparency times a spherical wavefront
The lens produces at its focal plane the Fraunhofer diffraction pattern of the transparency
When the transparency is placed exactly one focal distance behind the lens (i.e., z=f ), the Fourier transform relationship is exact.