Useful Formulas for Diffractive Lens Surfaces
Diffracted Wavefront:
OPD(r ) =
Blazed Zone Depth:
d0 =
(
)
pλ 0
nt (λ0 ) − ni (λ0 )
 sin[π (α − m )]
η = sinc (α − m ) = 

 π (α − m ) 
pλ0 nt (λ ) − ni (λ )
λ nt (λ0 ) − ni (λ0 )
where,
α (λ ) =
or,
α (λ , d ) =
or,
α (λ , d ,θ ) =
OPD equation
design depth
2
Diffraction Efficiency:
where:
mλ
c1 r 2 + c 2 r 4 + c3 r 6 + L
pλ0
2
scalar theory
for small angles, at design depth
d nt (λ ) − ni (λ )
for small angles, at any depth
λ
d nt (λ ) cos θ t − ni (λ ) cos θ i
λ
nt (λ )sin θ t = ni (λ )sin θ i +
mλ
Λ
general case
the grating equation
Key to symbols:
OPD(r) = Optical path difference in same units as r [divide by λ for waves]
r = Radial position of interest r = x 2 + y 2
m = Operating diffraction order
λ = Operating wavelength
p = Design order (diffracted order for which zone depth is optimized)
λ0 = Design wavelength for diffractive surface (for which depth is usually optimized)
c1, c2, … = Coefficients of OPD (phase) polynomial for diffracted wavefront
Λ = Local diffraction zone width (grating period) Λmin = Minimum zone width
d = Diffraction zone peak-to-valley depth
d0 = Zone depth for maximum efficiency for the design wavelength at normal incidence
ni(λ), nt(λ) = Refractive index on the incident (i) and diffracted (t) side of the grating
η = Scalar diffraction efficiency
α = Detuning factor for depth errors, wavelength change and material dispersion
θi, θt = Angles of incidence (i) and diffraction (t), relative to local surface normal
Paraxial Quantities:
f (λ ) =
− pλ 0
2mλc1
and
Λ min ≈ 2λ0 ( f /# )
ignoring higher order terms
Hybrid (refractive/diffractive) achromatic singlet lens:
Refractive Component
v ref
Φ dif =
v dif
Φ total
Component power
Φ ref =
Abbe ν-number
ν ref =
n(λ mid ) − 1
n(λ short ) − n(λlong )
ν dif =
λmid
λ short − λlong
Partial dispersion
Pref =
n(λ short ) − n(λ mid )
n(λ short ) − n(λlong )
Pdif =
λ short − λmid
λ short − λlong
ν ref − ν dif
Φ total
Diffractive Component
v dif − ν ref
Key to symbols:
m = Diffracted order of interest
λ = Wavelength of interest
p = Design order (diffracted order for which zone depth is optimized)
λ0 = Design wavelength for diffractive surface
c1, c2, ... = Coefficients of OPD (phase) polynomial for diffracted wavefront
f = Focal length (paraxial)
Λ = Local diffraction zone width (grating period)
f/# = F-number = f / (lens diameter)
Φtotal = Total hybrid lens power (reciprocal of focal length) at design wavelength
Φref , Φdif = Individual component powers at design wavelength
νref , νdif = Abbe ν-numbers for refractive and diffractive components
Pref , Pdif = Partial dispersion values for refractive and diffractive components
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