1. Diffraction Gratings & Lenses
Laser beam diffraction
A lens transforms a Fresnel diffraction problem to a Fraunhofer diffraction problem
The lens as a Fourier Transformer
Babinet’s Principle
Diffraction Gratings & Spectrometers
Examples of Fraunhofer diffraction:
Randomly placed identical holes X-ray crystallography Laser speckle Particle counting
Prof. Rick Trebino, Georgia Tech

2. Recall the Fraunhofer Diffraction formula
The far-field light field is the Fourier Transform of the apertured field.
E(x,y) = const if a plane wave
Aperture transmission function
that is:
and:
kx = kx1/z and ky = ky1/z
The k’s are off-axis k-vectors.

3. Fraunhofer diffraction of a laser beam
A laser beam typically has a Gaussian radial profile:
No aperture is involved. z w0 w1
What will its electric field be far away?
The Fourier transform of a Gaussian is a Gaussian.
In terms of x1 and y1: or
where:

4. Angular divergence of a laser beam
The beam diverges. What will its divergence angle be? w1 q w0 z
Recall that:
The half-angle will be:
The divergence half-angle will be:

5. 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
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), at first decreases but then also increases with distance far away from the waist.

6. Gaussian Beam Math z w(z) w0 The expression for a real laser
R(z)
The expression for a real laser
beam's electric field is given by:
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.
Recall the phase factor in front of the diffraction integrals.

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

8. Gaussian Beam Collimationw0
Twice the Rayleigh range is the distance over which the beam remains about the same size, that is, remains collimated.
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
____________________________________________
Longer wavelengths and smaller waists expand faster than shorter ones.
Tightly focused laser beams expand quickly.
Weakly focused beams expand less quickly, but still expand.
As a result, it's very difficult to shoot down a missile with a laser.

9. The Guoy 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)
Phase relative
to a plane wave:
Recall the i in front of the Fresnel integral, which is a result of the Guoy phase shift.

10. 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

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

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

13. Diffraction involving a lens
A lens has unity transmission, but it introduces a phase delay proportional to its thickness at a given point (x,y):
where L(x,y) is the thickness at (x,y). Compute L(x,y):
t(x,y) d
neglecting constant phase delays.

14. A lens brings the far field in to its focal length.
A lens phase delay due to its thickness at the point (x0,y0):
If we 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!
For a lens that's curved on both faces,

15. A lens brings the far field in to its focal length.
This yields:
If we look in a plane one focal length behind a lens, we are in the Fraunhofer regime, even if it isn’t far away! So we see the Fourier Transform of any object immediately in front of the lens!
E(x,y)
F {t(x,y) E(x,y)}
t(x,y)
A lens in this configuration is said to be a Fourier-transforming lens.

16. Focusing a laser beam f f
A laser beam typically has a Gaussian radial profile:
Lens f
2w0
2w1 f
What will its electric field be one focal length after a lens? or
Look familiar? This is the same result for a beam diffracting!
where:
or:

17. 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 f
2w0
2w1 f
seems to say that, if w0 is huge, we can focus to an arbitrarily small spot w1. What’s going on?
The discrepancy comes from our use of the paraxial approximation 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.

18. Babinet’s Principle Holes Anti- Holes
The diffraction pattern of a hole is the same as that of its opposite!
Holes
Neglecting the center point:
Anti- Holes

19. 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.

20. Diffraction Grating Mathematics
Begin with a sinusoidal modulation of the transmission:
where a is the grating spacing. The Fraunhofer diffracted field is:
We need the t0 term because t(x,y) usually isn’t negative.
Ignoring the y0-integration, the x0-integral is just the Fourier transform:

21. Diffraction orders
Because x1 depends on l, different wavelengths are separated in the +1 (and -1) orders. x1
No wavelength dependence in zero order. z
The longer the wavelength, the larger its diffraction angle in nonzero orders.

22. 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.

23. The Grating Equation a qm qm qi a qi
An order of a diffraction grating occurs if:
where m is an integer.
This equation assumed normal incidence and a small diffraction angle, however. We can derive a more general result, the grating equation, if we use a tilted beam, E(x,y), or if we recall scattering ideas: a
Scatterer A D C B
Potential diffracted wave-front
Incident wave-front qi qm qm a qi
AB = a sin(qm)
CD = a sin(qi)
Scatterer

24. Diffraction-grating spectrometer resolution
How accurate is a diffraction-grating spectrometer (a grating followed by a lens)?
Two similar colors illuminate the grating. f
2w1 dq d
d cos(qm) f
Two nearby wavelengths will be resolvable if they’re separated by at least one spot diameter, 2w1.
The diffraction grating will separate them in angle by dq, which will become f dq at the focal plane of the lens.

25. Diffraction-grating spectrometer resolution
Recall the grating angular dispersion: d f
So two nearby spots will be separated by:
Setting this distance equal to the focused-spot diameter:
where N = # grating lines illuminated = d / a or

26. Diffraction-grating spectrometer resolution
2w0
2w1 f
Let’s plug in some numbers:
l ≈ 600 nm m = N = (50 mm) x (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!
Resolution:

27. Blazed Diffraction Grating
By tilting the facets of the grating so the desired diffraction order coincides with the specular reflection from the facets, the grating efficiency can be increased.
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.

28. 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
Round holes

29. The Fourier Transform of a random array of identical tiny objects
Define a random array of two-dimensional delta-functions:
If Hole(x,y) is the shape of an individual tiny hole, then a random array of identically shaped tiny holes is:
The Fourier Transform of a random array of identically shaped tiny holes is then:
Shift Theorem
Sum of rapidly
varying sinusoids
(looks like noise)
Rapidly Slowly
varying varying

30. 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.

31. 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.

32. Particle detection and measurement by diffraction

33. 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.