Diffraction Gratings

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
• How tightly can we focus a beam?
• Babinets Principle
• Diffraction Gratings Spectrometers
• Examples of Fraunhofer diffraction
• Randomly placed identical holes X-ray
  crystallography Laser speckle Particle
  counting

Prof. Rick Trebino, Georgia Tech www.physics.gatec
h.edu/frog/lectures
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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 function
where weve dropped the subscripts, 0 and 1,
and
kx kx1/z and ky ky1/z
The ks 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.
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.

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.
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Gaussian Beam Math

• 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.
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Gaussian Beam Spot, Radius, and Phase

• 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
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Gaussian Beam Collimation
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 w0 l 10.6 µm l
  0.633 µm
• _____________________________________________
• .225 cm 0.003 km 0.045 km
• 2.25 cm 0.3 km 5 km
• 22.5 cm 30 km 500 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.
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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.

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

Electric field
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Laser Spatial Modes

• Some Transverse Electro-Magnetic (TEM) modes

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

d
neglecting constant phase delays.
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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-makers formula, z is the lens
focal length! For a lens that's curved on both
faces,
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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
  isnt 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.
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Focusing a laser beam
Lens
A laser beam typically has a Gaussian radial
profile
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
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How tightly can we focus a beam?
Geometrical optics and the previous result
predict a focused spot of width zero. But
geometrical optics neglects the wave nature of
light, and our previous result assumes small beam
angles.
0
Lets reconsider this problem in view of our
knowledge of intererence. Well consider the
k-vectors in pairs of symmetrically propagating
directions and add up all the fields at the focus.
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Beams crossing at an angle
x
q
k2
z
k1
Notice that the fringes are finest when q p/2,
and the beams counter-propagate. In this case,
their field has a fringe spacing of 2p/k l.
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Fringes from the various crossed beams
So lets add up all the sinusoidal electric
fields from every angle q
E
x
where a k sinq
The best we can do is to focus rays from all
angles (q 0 to p/2), so the fringe spacing
will vary from l to 8. Or a 0 to k. q p/2
will require an infinitely big lens, but, hey,
why not?
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The tightest focus possible
So, to find the focused field in this ideal case,
we integrate from a 0 to k
The focused field
E(x)
Width 2p/k l
0 when x p/k l/2
x
The focused irradiance (E2) can have a width as
small as l/2.
21
Babinets Principle
The diffraction pattern of a hole is the same as
that of its opposite!
Holes
Neglecting the center point
Anti-Holes
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The Diffraction Grating

• A diffraction grating is a slab with a periodic
  modulation of any sort on one of its surfaces.

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.
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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 A0 term because A(x,y) cant be
negative.
Ignoring the y0-integration, the x0-integral is
just the Fourier transform
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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.
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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.
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The Grating Equation

• 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
qm
a
qi
Scatterer
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Diffraction-grating spectrometer resolution
How accurate is a diffraction-grating
spectrometer (a grating followed by a lens)?
Two similar colors illuminate the grating.
2w1
d
d cos(qm)
f
Two nearby wavelengths will be resolvable if
theyre 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.
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Diffraction-grating spectrometer resolution
Recall the grating angular dispersion
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
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Diffraction-grating spectrometer resolution
Lets plug in some numbers
l 600 nm m 1 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
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Blazed Diffraction Grating

• By tilting the facets of the grating so the
  desired diffraction ordercoincides with the
  specular reflection from the facets, the
  gratingefficiency 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.
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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
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The Fourier Transform of a randomarray 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
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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.
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Laser speckle is a diffraction pattern.
When a laser illuminates a rough surface, it
yields a speckle pattern. Its the diffraction
pattern from the very complex surface. Dont
try to do this Fourier Transform at home.
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Particle detection and measurement by diffraction
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Moon coronas are due to diffraction.
When the moon looks a bit hazy, youre seeing a
corona. Its a diffraction effect.