Prisms and Diffractive Optics • Prisms •Tunnel diagrams • uses of different types • GRIN lenses • Diffractive optics
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Prisms What are they good for?
• Fold - “Erect” or “rotate” images - Change direction of propagation - Displace the beam - Fold system for compactness • Retro-reflection • Disperse (vs. λ) • Control beam parameters – Anamorphic telescopes – Vary angle, position, path length • Amplitude or pol. division – Beam splitters ECE 5616 Curtis
Folding prisms Bouncing pencils to analyze image orientation
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What the rotations are called
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Folding prisms Bouncing pencils to analyze image orientation
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Tunnel diagrams Tool to simplify ray-tracing Unfold the path and squash the dimensions by n: useful to determine the angular field of the prism as well as the size of the beam which will pass, and see what aberration might it cause. Dove Prism is equivalent to a tilted plane of glass. Will have lots of astigmatism if used in converging or diverging beam
Right angle prism equivalent to block of glass
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Tunnel diagrams Tool to simplify ray-tracing
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More Complex Example
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3D Tunnel diagrams Tool to simplify ray-tracing
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Anamorphic prisms Often better than cylindrical telescope Problems: • Compression largest near TIR – tolerances and polarization dependence • Angular bandwidth quite low (works best for collimated beams)
Advantages • Lower aberrations than cylinders • Cheap
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Prism Deviation of Ray
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Thin Prism Use paraxial approximation When A is small angle and light incident near normal to prism face “thin prism”
Which gives:
Differentiate with respect to n results in
D ≈ V ECE 5616 Curtis
Thin prism tricks Beam is deviated by angle (n-1)a. If prism is rotated about its axis, the beam is deflected in a circle. Two cascaded prisms give arbitrary x,y deviation. For small a, control of deviation can be quite fine.
As above, if α is small, control of displacement can be quite fine.
Variable path length.
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GRIN lenses Common in fiber/telecom applications
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GRIN lenses Good example and important lens technolgy Standard Lens increases the OPL in the middle and decreases towards edges by change in thickness of the glass. 2 2
t ( x, y ) ≈ e
jk o
x +y 2f
What if index of plane of glass d0 thick varies radially as no(1-1/2α2)(x2+y2) ? The transmittance can be written as (ignore constant phase terms)
t ( x, y ) ≈ e ECE 5616 Curtis
k o d o noα 2 ( x 2 + y 2 ) j 2
This gives a focus length
f =
1 noα 2 d o
Diffractive optics Classes
Introduction/terminology
• Diffractive optical element: Modification of the optical wavefront via subdivision and individual modification of the phase and/or amplitude of the segments. • Grating: linear segments = uniform diffraction angle • Computer generated hologram: A DOE in which the structure has been calculated numerically •Holographic optical element: DOE in which the structure is generated by the interference of optical wavefronts.
Discretization •Binary optic: phase or amplitude structure with two levels. Typically created via a single etch step. • Dammann grating: Binary optics with repetitive pattern, generates N beams (fan out) • Multilevel optic: Same as binary but with M etch steps to achieve N=2M levels. • Kinoform: Phase DOE with smoothly varying profile (limit of N) • Blazed: Grating with linear (sawtooth) segments
Fabrication • Direct machining: aka “ruling” or diamond turning, fab via mechanical machining. Often used for masters. • Lithography • Direct write: scan laser or e-beam over photoresist • Interference (holography) inc near field • Masks: grey-scale, multiple exposure • Replication ECE 5616 Curtis
Diffraction gratings Basics
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Diffraction gratings Basics
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Resolving power aka number of spots
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Resolving power aka number of spots
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Estimation of grating R Why gratings are interesting Holographic gratings of 1800 lp/mm are typical in the visible. A 10 mm beam and first-order diffraction would yield R = 10 ×1800 = 18,000 or a minimum resolvable wavelength shift of 0.03 nm in the visible. For a prism at the minimum deviation condition (symmetrical incident and exit angles) the resolving power can be shown to be
In the visible a b = 25 mm prism would give resolving power
or δλ ~ 0.5 to 0.17 nm, roughly an order of magnitude lower resolution than a grating. ECE 5616 Curtis
Bandwidth aka Free spectral range
When will diffractions be confused with the neighboring order?
Thus first-order grating spectrometer could operate from 400 to 800 nm. ECE 5616 Curtis
Efficiency Overview by type
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Efficiency Overview by type
Thick phase grating: Bragg selectivity can give single order and theoretically 100% DE. BUT, very sensitive to incident wavelength and/or angle (unlike thin).
Note for minimum feature size, N reduces F/# linearly (ouch). ECE 5616 Curtis
Diffractive lenses λ dependence of angles Reading a diffractive optic at λ’ and order m’ that was designed for λ and order m.
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Diffractive lenses λ dependence of angles Reading a diffractive optic at λ’ and order m’ that was designed for λ and order m. Change in angle is perfectly analogous to refracting into a slab of index neff. Note that this index can be < 1.
Definition of focal length. 1. Diffracts to set of focii. 2. For neff≠1, each suffers spherical aberration. ECE 5616 Curtis
Diffractive lenses λ dependence of efficiency (1/4) For a kinoform (N= ∞)
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Diffractive lenses λ dependence of efficiency (2/4)
Which gives us the diffracted electric field vs. angle for a uniform Einc
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Diffractive lenses λ dependence of efficiency (3/4) Efficiency of a blazed grating designed for wavelength λ and order m with index n read at wavelength l’ and order m’ with index n’
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Diffractive lenses λ dependence of efficiency (4/4)
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Hybrid refractive/DOEs For DOE
From earlier If used at same order (m=m’) Find change in power over l
From earlier
Solve for V.
This is a) the same for all DOEs, b) negative and c) very strong. Let’s design an achromatic f=25.4 mm with a BK7 singlet:
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Hybrid Continued
In Zemax, thin DOEs are usually modeled with infinitely thin surface with extremely large index perturbations ie 10,000 at design wavelength and index scaling with wavelength
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Reading W. Smith “Modern Optical Engineering” Chapter 7
