DIFFRACTIVE OPTICAL ELEMENTS
4.3 ZONE PLATE DESIGN AND SIMULATION
4.3.2 Zone Plate Computation for Design and Simulation
The radii for the rings in the zone plate are computed to give a specified focal length.
-8-4-2 10.5
-0.5
fp
-1 2468-6
FIGURE 4.8 Focusing of zone plate.
ZONE PLATE DESIGN AND SIMULATION 69
q q
q A B
D
C
fp d′
si
z
x E
FIGURE 4.9 Geometry for determining zone plate equations.
4.3.2.1 Derivation of Equations for Zone Plate Design For the blocking zone plate (Figure 4.6a), in-phase rays at the focal point are passed through clear rings of the DOE. Those rays that would be out of phase at the focal point are blocked by opaque rings. The phase for each ray can be determined by its geometric length.
The equations can be extended to surface relief zone plate (Figure 4.6b) by allowing aπphase change in place of the blocked or opaque regions. This is equivalent to a change in sign, so that the previously out-of-phase parts are now in phase. This is normally used in diffractive elements to increase efficiency. The resulting intensities obtained here are doubled in this case. Aπphase change is achieved in a surface relief zone plate by exact control of the etch depth so that propagation through this depth of material will cause aπphase change relative to propagation through the same depth in air.
We now compute the radiisi for the rings of a zone plate that focuses at a focal length offp (Figure 4.8) [134]. Figure 4.9 shows at B the edge of theith ring at distancesi from the axis for the zone plate focusing to a point C at the equivalent focal length on axisfp. For the ray from theith ring at radiussi to be at a transition between in phase and out of phase relative to that from the on-axis center of the zone plate, the additional distance traveled (XD) to the focal point is an odd number of half-wavelengths:
d= λ
2(2i−1) (4.8)
From equation (4.8), similar trianglesABCandDBA,λ=(2π)/ k, andAD≈AB, si
fp = (π/ k)(2i−1)
si = π(2i−1) ksi
(4.9)
70 DIFFRACTIVE OPTICAL ELEMENTS
We select an equivalent focal length for the zone plate and define a parameter
σ2=fp/ k (4.10)
Substitutingfpfrom equation (4.10) into equation (4.9), si2
σ2 =π(2i−1) (4.11)
Therefore, opaque rings can be placed at intervalss2i≤ρ≤s2i+1, where the radius isρ,s0=0, and integersi >0:
si=σ
π(2i−1) (4.12)
The opaque rings are marked black on the cross section at the left-hand side of Fig- ure 4.8. Because the zeros ofU[cos{ρ2/(2σ2)}] occur whenρ=siin equation (4.12), the transmission function for the zone plate may be written as
T(ρ)=U
cos ρ2
2σ2 , U(x)=
1 forx≥0
0 forx <0 (4.13) Equation (4.13) is used to compute the cross section for the zone plate.
4.3.2.2 Derivation of Equations for Transverse Field at Focal Plane To find the intensity of the field at the output plane, we use scalar diffraction theory (Sec- tion 3.3). Fresnel diffraction can be written from equation (3.45) as
g(x0, y0, z0)= − j λz0
ejkz0
y
x
f(x, y) exp
j k 2z0
(x−x0)2+(y−y0)2
dxdy (4.14) For circular symmetry ρ=
x2+y2, equation (4.14) can be written using a Fourier–Bessel transform [49, 134] as
g(ρ0, z0)= −j2π λz0
ejkr0 b
0
ρf(ρ) exp
jkρ2 2z0
J0
kρρ0 z0
dρ (4.15)
wherer0=z0+(x20+y02)/(2z0) andJ0is a Bessel function of first kind and zeroth order.
Inserting the zone plate design equations (4.13) and (4.10) into equation (4.15) in place off(ρ),Bis the amplitude of the incoming wave, and by changing variables
ZONE PLATE DESIGN AND SIMULATION 71
usingw=kρ2/(2z0) (dω=(kρ)/(z0)dρ) orρ2=(2z0ω)/ k, we obtain [134]
g(ρ0, z0)=B
kb2/(2z0)
0
ejwU
cos wz0
f J0
ρ0 2kw
z0 1/2
dw (4.16)
where the phase term−jejkr0 is neglected. From equation (4.16), for the zone plate withnrings and transparent ring locations (left-hand side of Figure 4.8),
g(ρ0, z0) (4.17)
= B π/2
0
ejwJ0
ρ0 2kw
z0 1/2
dw +B
5π/2 3π/2
ejwJ0
ρ0
2kw z0
1/2
dw+ · · · +B
π(n−1/2)
π(n−3/2)
ejwJ0
ρ0
2kw z0
1/2
dw (4.18)
4.3.2.3 Performance of Zone Plate On thezaxis,ρ0=0, and asJ0=1, the field at the focal point of the zone plate fornrings is from equation (4.18)
g(ρ0, z0)=B
[1+j]+[1+1]+ · · · +[1+1]
=B[n+j]
Equation (4.18) shows that incoming light is focused to a higher intensity at the focal point.
The solid line in Figure 4.10 shows the normalized output intensity at the back focal plane for the zone plate illuminated with collimated coherent light. Note that
0.8
0.6
0.4
0.2
0
10 20 30 40
Lens Zone plate
Radius
Absolute amplitude
50 60
FIGURE 4.10 Comparison of profile at focal plane with a lens.
72 DIFFRACTIVE OPTICAL ELEMENTS
the zone plate will divert light into many higher orders because it is binary. Therefore, there are other higher order focus points at greater distances than the first order. Light in negative orders is spread around a circle and not focused at a point, so its peak intensity is significantly less than that at the focal point of a lens.
The field across the focal plane,z0=f, for a lens can be obtained from equa- tion (4.15) by allowing the lens function to cancel the first exponent in the in- tegral and setting f(ρ)=1. By changing variables to ρ=kρρ0/z0 and using x
0 ρJ0(ρ)dρ=xJ1(x), this can be written as g(ρ0, z0)= −j2π
λf ejkr0bJ1(bkρ0/f)
kρ0/f (4.19)
AsJ1(bkρ0/fp)=0, whenbkρ0/fp=1.22π, the distance 2ρsacross the main lobe, or the spot size, is
2ρs =2.44λf
2b (4.20)
We consider the surface relief zone plate. If the opaque blocking regions are re- placed by the material of the zone plate with additional height or depth such that 180◦(π) phase shift occurs in these regions, then we add the same intensity again for these regions. The intensity at the peak for the surface relief zone plate is therefore twice that for the blocking zone plate. The intensity of the spot at the focal length will decrease if the etch depth deviates from that computed for 180◦ phase shift.
Figure 4.11 shows the computed profile, rescaled, at the focal plane for a surface relief zone plate: This represents a significant improvement over the blocking zone plate shown previously because the side lobes fall off faster.
16 14 12 10 8 6 4
10
0 20 30 40
Radius
Absolute amplitude
50 60
FIGURE 4.11 Computed profile at the focal plane for a surface relief zone plate.
GERCHBERG–SAXTON ALGORITHM FOR DESIGN OF DOEs 73
4.4 GERCHBERG–SAXTON ALGORITHM FOR DESIGN OF DOEs