FT2224 Applied Optics

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FT222/4 Applied Optics

• Module 7
• Diffractive Optical Elements
• Lenses

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Diffractive Optics.
Optical objects of spatially varying transmission
and/or phase, such that the transmitted beam
generates a desired profile at a given point in
space.
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Nomenclature
A number of different names apply to this general
class, and are used
Commonly called Diffractive Optical Elements
(DOEs)
Circularly symmetric patterns can act as a lens
diffractive micro-lenses
Generally produced by lithography, so often
referred to as Binary Optics
Given sufficient computational power the object
that would give rise to any arbitrary pattern can
be calculated, so know as Holographic Optical
Elements (HOEs) or Computer Generated Holograms
(CGHs).
Objects producing a phase variation only phase
gratings
Regular, grid-like patterns can generate a
regular array of spots - array generators
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Fresnel Diffraction and Zone Plates (For
more detail see Optics Hecht, Sec 10.3 )
Huygens successfully explained wave propagation
as each point on a wave-front itself giving rise
to a circular wave. The field at any point is the
sum of all these propagating elements.
(Fig 10.46 Hecht)
Fresnel and Kirchoff introduced correction terms
to make the wave propagate forward only -
obliquity factor essentially saying that for
any sub-wavefront slightly more light travels
forward rather than backwards.
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Zone construction
Consider a point P0 as the source of
monochromatic spherical waves. A typical wave has
radius r0. A some time later this wave will
reach point P, r0 b from P0. The field at P
can be considered to be made up of contributions
from all points on the previous wave-front.
Zone construction From point P draw spheres
of radius b, b l/2, b2 l /2, b3 l /2, The
resultant zones are each successively a half
wavelength away from P.
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• The field at P can be considered to be made up of
  contributions of wavelets originating from each
  zone.
• As each zone is, on average, l /2 further away,
  the contributions from successive zones will be
  of opposite phase.
• Sum of terms from m zones is
• E E1 - E2 E3 - E4 E 5 - ./- Em.
• Contribution from each zone
• a area
• 1 / (distance from P)
• but area / (distance from P) constant
• contribution from each zone is equal to within
  obliquity factor.

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For m odd E E1/2 (E1/2 - E2 E3/2 ) (
E3/2 - E4 E 5 /2) (E 5 /2 - .)
Em/2 As the obliquity factor varies only slowly
from zone to zone, the contribution in each
bracket is approx. zero. For m zones, m odd E ?
E1/2 Em/2 Similarly, for m even can show E ?
E1/2 - Em/2 For m large obliquity factor -gt
zero, gt Em - gt zero So field at P Ep ? E1/2
How might this describe diffraction at a
circular aperture. Consider an aperture of radius
first zone. What happens as the aperture
expands.
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Zone Plates
As described, contributions from successive zones
cancel each other out. If we remove all odd or
even contributions up to m, then the field at P
would be Ep ? m/2 . E1 as compared to E1/2
with no obstruction. As intensity E2,
intensity is enhanced by x m2 times.
Such a device is a zone plate. As this light is
redirected from elsewhere the zone plate
essentially acts as a lens, focusing light.
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Zone plate radii and focal length.
Each successive path through an opening must
differ by ml/2.
(rm rm) (r0 r0) ml/2 For each
triangle rm Rm 2 r0 21/2 r0 Rm 2 /
2r0 (for r0 gtgt Rm) rm Rm 2 r0 21/2 r0
Rm 2 / 2r0 (for r0 gtgt Rm)
Substituting we get 1/ r0 1/ r0 ml/ Rm
2 This equation is similar in form and content
to the thin lens formula
So setting r0 -gt infnity, r0 become the focal
length. fm Rm 2 / ml or Rm v(f1l m) vm
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Comments
Note that the focal length depends strongly on l
gt large chromatic aberration. (Typical of
diffractive optics)
Also note that there are other weaker focii at
shorter wavelengths, which can be accounted by
considering groups of zones to be blocked.
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Improving grating efficiency
Blocking the light means that most light is
wasted. A more efficient solution is to make a
phase only, transparent object with alternating
layers of a thickness so as to introduce a shift
of /- l /2. Binary Phase grating
This is not exactly what we wanted, but rather
that there is always a shift of l /2 at each
interface. (Very like a blazed grating)
Exact calculations show the inclines should be
curved.
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Alternative Approach Fresnel Lens
Divide lens into l 2p layers Outline square
blocks that contribute only a phase
shift. Remove redundant layers from a
conventional lens gt Fresnel lens gt Zone Plate
approaches ideal Fresnel lens.
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Zone Plate and Diffraction Efficiency

• For a zone plate Rm v(f1l m)
• Width of a zone lt Rm gt v(f1l m) - v(f1l (m-1))
• v(f1l ) v m - vm-1
• R1 v m - vm-1
• For l 500 nm.
• Focal Length f 5mm 1mm 100 mm
• R1 v(f l ) 50mm 22mm 7mm
• lt R10gt .16 R1 8 mm 3.6 mm 1.14 mm
• lt R20gt .11 R1 5.7 mm 2.5 mm .8 mm
• For given resolution of etching (eg 1 mm ) there
  is a maximum diameter
• f/ is limited.

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Multi-level Etching.
Efficiency limited by etching resolution
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Amplitude 10 Binary phase 41 4 Level 81 8
Level 95 16 level 98.7
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• Best Solution gt use multi-level for central
  zones, decreasing to binary grating for outer
  levels.
• Using n masks and etch steps yields 2n etch
  depths
• width of a zone 2n x minimum feature size
• eg for 1mm resolution,
• 3 masks yield 8 steps, ie minimum 8 mm width
  zones
• 2 masks yield 4 steps, ie minimum 4 mm width
  zones

Q For f 1mm, l 500 nm, 1 mm etch
resolution R1 22 mm gt can use 4 mask/etch
steps (16 etch depths) What is the maximum
radius?? Switching to 4 steps, what would be the
maximum radius.? How far could the radius be
extended with binary etching.?