1. Laser Beams, Diffraction Gratings and Lenses
Zone plate
Laser-beam diffraction
A lens transforms a Fresnel diffraction problem to a Fraunhofer diffraction problem.
The lens as a Fourier transformer
Diffraction gratings & spectrometers
Examples of Fraunhofer diffraction:
Babinet’s Principle Randomly placed identical holes X-ray crystallography Laser speckle Particle counting
Image from Fio Omenetto
Thanks to Prof. Rick Trebino,

4. Recall the Fraunhofer diffraction formula.
The far-field light field is the Fourier Transform of the apertured field.
E(x,y) = constant if a plane wave
Aperture transmission function
that is:
and:
kx = kx’/z and ky = ky’/z
The k’s are off-axis k-vectors.

5. Fraunhofer Diffraction of a Laser Beam
A laser beam typically has a Gaussian radial profile:
No aperture is involved. w0 ’ w0 z
What will its electric field be far away?
The Fourier transform of a Gaussian is a Gaussian.
In terms of x’ and y’: or
The larger the beam initially, the smaller the beam far away.
where:

6. Angular Divergence of a Laser Beam
The beam diverges.
What will its divergence angle be? w0 q z ’
Recall that:
The half-angle will be:
The divergence half-angle will be:

7. Collimated region (where the spot size remains ~ constant)
Gaussian Beams
The Gaussian beam is the solution to the wave equation, or equivalently, the Fresnel integral, for a wave in free space with a Gaussian profile at z = 0. x
Collimated region (where the spot size remains ~ constant)
R(z) = wave-front radius of curvature
w(z) z
Beam radius w(z)
Beam waist
The beam has a waist at z = 0, where the spot size is w0.
It then expands to w = w(z) with distance z away from the waist.
The beam radius of curvature, R(z), is ∞ at z = 0. It then decreases but eventually increases with distance far away from the waist.

8. Gaussian Beam Expressionz w0
w(z)
R(z)
The expression for a real laser
beam's electric field is given by:
Recall the phase factor in front of the diffraction integrals.
w(z) is the spot size vs. distance from the waist,
R(z) is the beam radius of curvature, and
y(z) is a phase shift.
This is the solution to the wave equation or, equivalently, the Fresnel diffraction integral.

9. Gaussian Beam Spot, Radius, and Phasez w0
w(z)
R(z) zR
The expressions for the spot size, radius of curvature, and phase shift:
where zR is the Rayleigh range, and it's given by:
Twice the Rayleigh range is the distance over which the beam remains about the same size, that is, remains collimated.

10. Gaussian Beam Collimationw0
Tightly focused laser beams expand quickly.
Weakly focused beams expand less quickly, but still expand.
Longer wavelengths and smaller waists expand faster than shorter ones.
Collimation Collimation
Waist spot Distance Distance
size w l = 10.6 µm l = µm
.225 cm km km
2.25 cm km km
22.5 cm km km
As a result, it's very difficult to shoot down a missile with a laser—the beam is most intense at the laser and dims with distance.

11. The Gouy Phase Shift
The phase factor yields a phase shift relative to the phase of a
plane wave when a Gaussian beam goes through a focus.
p/2
-p/2 zR
-zR
y(z) z
Phase relative
to a plane wave:
Recall the i in front of the Fresnel integral, which is a result of the Gouy phase shift.

12. Laser Spatial Modes
Laser beams can have any pattern, not just a Gaussian. And the phase shift will depend on the pattern. The beam shape can even change with distance.
Some beam shapes do not change with distance. These laser beam shapes are referred to as Transverse Electro-Magnetic (TEM) modes. The actual field can be written as an infinite series of them.
The 00 mode is the Gaussian beam. Higher-order modes involve multiplication of a Gaussian by a Hermite polynomial.
Some Transverse Electro-Magnetic (TEM) modes
Irradiance patterns from
Electric field

13. Laser Spatial Modes Some Transverse Electro-Magnetic (TEM) modes
Irradiance
Irradiance patterns from

14. Laser Spatial Modes
Some particularly pretty measured laser modes (with a little artistic license…)
Fio Omenetto

15. Diffraction Involving a Lens
A lens has unity transmission, but it adds an extra phase delay proportion-al to its thickness at a given point (x,y):
where L(x,y) is the thickness at (x,y).
Observation plane
Lens
Object
Illumi-nation f
tlens(x,y)
tobject(x,y)
Compute L(x,y): d
(x,y) R
neglecting constant phase delays (that are independent of x and y).

16. A lens brings the far field in to its focal plane.
A lens phase delay due to its thickness at the point (x,y):
Substitute this result into the Fresnel (not the Fraunhofer!) integral:
The quadratic terms inside the exponential will cancel provided that:
Recalling the Lens-maker’s formula, z is the lens focal length!
QED
For a lens that's curved on both faces, cancellation occurs if:

17. A lens brings the far field in to its focal length.
This yields:
Observation plane
Lens
Object
Illumi-nation f
tobject(x,y)
F {tobject(x,y) E(x,y)}
E(x,y)
Fourier transform of object
One focal length behind a lens is the Fraunhofer regime—even if it isn’t far away!
There, we’ll see the Fourier transform of the product tobject(x,y) E(x,y)— the field immediately in front of the lens!
A lens in this configuration is said to be a Fourier-transforming lens.

18. Focusing a Laser Beam f 2w0 ’
Lens ’
A laser beam typically has a Gaussian radial profile:
What will its electric field be one focal length after a lens? or
Look familiar? This is the same result for a beam diffracting! Here, the beam is propagating backwards.
where:
or:

19. How tightly can we focus a laser beam?’
Recall that we showed earlier that a beam cannot focus to a spot smaller than l/2. But this result
seems to say that, if w0 is huge, we can focus to an arbitrarily small spot What’s going on? w0 ’
The discrepancy comes from our use of the paraxial approximation here in diffraction, where we assumed small-angle propagation with respect to the z-axis. So don’t use this result when the focus is extremely tight! A beam cannot be focused to a spot smaller than l/2.
On the other hand, unlike geometrical optics, this formula at least tells us the correct focused spot size—except in the above limit.

20. The Diffraction Grating
A diffraction grating is a slab with a periodic modulation of any sort on one of its surfaces.
Diffraction angle, qm(l)
Zeroth order
First order
Minus first order
The modulation can be in transmission, reflection, or the phase delay of a beam.
The grating is then said to be a transmission grating, reflection grating, or phase grating, respectively.
Diffraction gratings diffract different wavelengths into different directions, thus allowing us to measure spectra.

21. Diffraction Grating Mathematics
Begin with a sinusoidal modulation of the transmission:
where a is the grating spacing. The Fraunhofer diffracted field is:
Ignoring the y-integration, the x-integral is just the Fourier transform:
Substituting for kx
1st order
0th order
-1st order

22. Diffraction Orders x’ z
Because x’ depends on l, different wavelengths are separated in the nonzero orders. x’
No wavelength dependence in zero order. z
The longer the wavelength, the larger its diffraction angle in nonzero orders.

23. Diffraction Grating Math: Higher Orders
What if the periodic modulation of the transmission is not sinusoidal?
Since it's periodic, we can use a Fourier Series for it:
Keeping up to third order, the resulting Fourier Transform is:
A square modulation is common. It has many orders.

24. The Grating Equation qm a qi
Recall that, using scattering ideas, we derived a more general result, the grating equation:
The Grating Equation
Scatterer a qi qm
AB = a sin(qm)
CD = a sin(qi) A D C B
Diffracted wave-front
Incident wave-front
If we now assume normal incidence (qi = 0) and a small diffraction angle, we see that an order of a diffraction grating occurs if:
where m is an integer.
This is the same result we just obtained using diffraction ideas.
But scattering doesn’t provide the relative intensities of each order.

25. Blazed Diffraction Gratings
By tilting the facets of the grating so the desired diffraction order coincides with the specular reflection from the facets, the grating efficiency of a particular order can be increased.
Facet
Efficient diffraction
Specular means angle of incidence equals angle of reflection.
Input beam
Inefficient diffraction
Even though both diffracted beams satisfy the grating equation, one is vastly more intense than the other.
The analysis using the diffraction integral allows us to model this type of grating and determine the relative intensities of each order.

26. Diffraction-Grating Spectrometer Resolution
How accurate is a diffraction-grating spectrometer (a grating followed by a lens)?
Two similar colors illuminate the grating. f
2w0 ’ dq
For simplicity, assume normal incidence onto the grating. d
d cos(qm) f
Two nearby wavelengths will be resolvable if they’re separated by at least one spot diameter,
The diffraction grating will separate them in angle by dq, which will become f dq at the focal plane of the lens.
2w0 ’

27. Diffraction-Grating Spectrometer Resolution
Recall the grating angular dispersion: d f dq f
So two nearby spots will be separated by:
Setting this distance equal to the focused-spot diameter, :
2w0 ’
where N = # grating lines illuminated = d /a or

28. Diffraction-Grating Spectrometer Resolution
Let’s plug in some numbers: f
2w0 ’
2w0 f
l ≈ 600nm m = N = (50mm) × (2400 lines/mm) = 120,000 lines
For simple order-of-magnitude estimates, 4 / p ≈ 1:
And the resolution, dl/l, depends only on the order and how many lines are illuminated!
Resolving Power:

29. Babinet’s Principle Holes Anti- Holes
The diffraction pattern of a transmission function is the same as that of its opposite!
Holes
Neglecting the center point:
From page 509, Hecht, Optics, 4th ed. The Ealing Corporation and Richard B. Hoover
Anti- Holes

30. Fraunhofer Diffraction: Interesting Example
Hole Diffraction
pattern pattern
Randomly placed identical holes yield a diffraction pattern whose gross features reveal the shape of the holes.
Square holes
From page 483, Hecht, Optics, 4th ed.
Round holes

31. The Fourier Transform of a Random Array of Identical Tiny Objects
Define a random array of two-dimensional delta-functions:
where (xi,yi) are random points
Shift Theorem
Sum of rapidly
varying sinusoids
(looks like noise)
If thole(x,y) is the shape of an individual tiny hole, then a random array of identically shaped tiny holes is:
The Fourier transform of this random array is then:
Rapidly varying Slowly varying

32. X-Ray Crystallography
The tendency of diffraction to expand the smallest structure into the largest pattern is the key to the technique of x-ray crystallography, in which x-rays diffract off the nuclei of crystals, and the diffraction pattern reveals the crystal molecular structure.
This works best with a single crystal, but, according to the theorem we just proved, it also works with powder.
From page 486, Hecht, Optics, 4th ed.

33. Laser speckle is a diffraction pattern.
When a laser illuminates a rough surface, it yields a speckle pattern. It’s the diffraction pattern from the very complex surface.
Don’t try to do this Fourier Transform at home.
Laser illumination
Incoherent illumination
Speckle
Image from
Technically, both images are diffraction patterns, but the incoherent illumination is so complex in time and space you can’t see the speckle.

34. Particle Detection and Measurement by Diffraction

35. Moon coronas are due to diffraction.
When the moon looks a bit hazy, you’re seeing a corona. It’s a diffraction effect.
Image/Text/Data from the University of Illinois WW2010 Project.

36. Frontiers of Optics Ultrafast optics Nonlinear optics
Non-diffracting beams
Ultrahigh intensity
Arbitrary-waveform generation
Ultrahigh-resolution spectroscopy
Ultracold atoms
Laser accelerators
Meta-materials
Super-resolution imaging

37. X-ray crystallography solved DNA.
The vast majority of humankind’s greatest discoveries have resulted directly from light and its measurement.
Microscopes led to biology.
Telescopes led to astronomy.
X-ray crystallography solved DNA.
Spectrometers led to quantum mechanics. [ [
The Michelson interferometer led to relativity.
And technologies, from medical imaging to GPS, result from light measurement!

38. With your newfound optics knowledge, perhaps you’ll make the next one.