Surface retrieval by gradient based transmission test

The surface retrieval with ERT as presented by Ceyhan [69] is focused on finding the deviation of the real surface from the design. The design values are necessary input parameters and the deviation is expected to be relatively small. The retrieval is based on a reverse ray tracing through the lens under test (Fig.23) from the angle of the outgoing rays (β') to the surface of interest Z1. The following derivations will concentrate on a ray propagating only in two dimensions for simplicity reasons where y is the transversal coordinate and z is pointing along the initial propagation direction of the rays.

Fig.23: Tracing a ray trough a plano-convex lens for the derivation of the relation between transmitted ray slopes and the surface under test Z1.

A lens as device under test is made of two surfaces (Z1, Z2) that influence the propagation direction of the ray. For the measurement the first will be the one to be retrieved. A ray parallel to the optical axis with distance y will incident on this surface with an angle α to the surface normal at that position which is related to the local surface slope

dZ₁

dy =tanα . (5.18)

The surface will lead to a refraction of the ray which can be described by Snell's Law as

sinα n₁=sin α ' n₂ , (5.19)

Aspherical surfaces

where n2 is the refractive index of the lens, n1 is the refractive index of the surrounding medium and α' is the angle between the surface normal and the ray inside of the lens. This angle can be replaced by

α'=α +β , (5.20)

due to the negative relationship between α and β, leading to

sinα n₁=sin(α +β )n₂ (5.21)

or with the sum formula

sin( x+ y)=sin x cos y+sin y cos x , (5.22)

to

sinα n₁=(sinα cos β +sin β cosα)n₂ . (5.23)

Dividing both sides by cos α leads to

tan α n₁=(tanα cos β +sin β )n₂ , (5.24)

which can be rearranged to find a definition for the surface slope according to Eq. (5.18) depending on β to be

dZ₁

dy =tanα = n₂sin β

n₁−n₂cos β . (5.25)

Adding the refraction at the second surface (Z2) described by

sin β n₂=sin β ' n₁ , (5.26)

where

sin β =sin β ' n₁

n₂ , (5.27)

to the definition of the surface slope gives dZ₁

dy =tanα = n₁sin β '

n₁−n₂cos β . (5.28)

Rearranging the Pythagorean identity

cos β²+sin β²=1 (5.29)

to

cos β =

√

and inserting Eq. (5.26) yields

Aspherical surfaces

cos β =

√

⁽

⁾

Inserting this into Eq. (5.25) gives dZ₁

dy =tanα = n₁sin β '

n₁−n₂

√

⁽

⁾

and replaces the unknown parameter β. Integrating over the transversal coordinate results in Z₁(y )=

∫

−y

y n₁sin β ' ( y ) n₁−n₂

√

⁽

n₂

)

which shows the relationship between the sag representation of the surface to be retrieved and the angles of the outgoing rays which are the measurement outcome of the ERT. However, this is strictly limited to lenses where the second surface is perfectly flat. In other cases, the incidence angle of the ray at the second surface depend on the local slope at the intersection point as illustrated in Fig.24.

Fig.24: Tracing a ray trough a double-convex lens for the derivation of the relation between transmitted ray slopes and the surface under test Z1.

The local slope of the second surface can be defined as dZ₂

dy2

=tan γ . (5.34)

With the refraction at the second surface described by

Aspherical surfaces

sin(γ −β )n₂=sin (γ −β ' )n₁ , (5.35)

Eq. (5.24) can be developed in a similar fashion as described in Eqs. (5.22) to (5.22) (sin γ cos β −sin β cosγ )n₂

and inserting it into Eq. (5.23) to combine the first refraction with this second

dZ₁

one obtains a relation between the angle of the outgoing ray and the local slope of the first surface at the intersection point of the ray.

Eq.(5.20) offers the possibility to determine the first surface of aspherical shape from a lens with a spherical second side by evaluating the angles of the outgoing rays β' which are the usual outcome of an ERT measurement. However, it contains several more dependencies:

• the refractive index of the surrounding medium n1,

• the refractive index of the lens medium n2,

• the angle β,

• the second surface Z2 and

• the lateral coordinate y2 where the ray intersects with Z2 which is depending on the lens thickness t.

With air as the surrounding medium, n1 can be assumed to be unity. For the lens parameters n2 and t the usual deviation from the design can be assumed to be so small that its effect on the result is

Aspherical surfaces second surface should be characterized beforehand by an interferometer, a surface profiler or an auto-collimator. An accurate knowledge about the second surface is mandatory for the exact evaluation of the first surface. The unknowns left to determine are the angle β and the position y2. This can be achieved by using an optimization process. For this purpose the complete lens has to be modeled numerically. The model must contain a certain mathematical description of the first surface Z1,M . Though other descriptions can be applied, it is common sense to use the customary ISO-standard description (Eq. (5.2)) of the surface sag of a rotational-symmetric aspherical surface shape for this purpose. Its parameters, the radius of curvature R, the conic constant κ and the aspherical coefficients A2n of 2n-order are variables of the optimization process performed by the least squares minimization over i rays

with the deviation of the actual shape (Z1,M) from the modeled (Z1) shape of the first surface

^M=Z₁−Z₁^M . (5.41)

The modeled surface can be expressed using Eq. (5.20) as

dZ₁^M

where n1 from Eq. (5.20) was assumed to be 1 and Z2 must be known from prior measurement and therefore, does not need to be modeled. The ideal minimization shall lead to the condition

lim

Δ^M→0

(cosβ−cosβ^M, y₂−y₂^M, sinβ'−sinβ'^M)=0 . (5.43) For very small values of Δ^M, the last unknown in the denominator of Eq. (5.20) and Eq. (5.21) may be substituted by cosβ^M with the assumption that cosβ ≈ cosβ^M. Henceforth, the deviation from Eq. (5.19) can be expressed by subtracting Eq. (5.21) from Eq. (5.20) which leads to

d Δ^M

The retrieval method described above was tested using artificial surface data

Z₁^Sy=Z₁^by ; R , , A_{2 n}^Sy  , (5.45) as the sum of a base aspherical surface according to design (Z1b) and a defined periodic surface deformation

Aspherical surfaces

^Sy =a⋅cos2  v y . (5.46)

where a is the amplitude and v is the spatial frequency of the oscillation.

The resulting differences between the simulated and the retrieved surface were in good agreement with standard deviations σ <1 nm for fairly low spatial frequencies. However, for higher frequencies the deviations were found to be of significance (σ > 25 nm). The reason for the increasing deviation was found in the limited capabilities of the model function Z1M to represent an aspherical surface with higher frequency deformations which are strongly bound to the used number of aspherical coefficients A2n. The higher the spatial frequency of the deformations the more coefficients are needed. A spatial frequency of v = 1/3 lines/mm would require an order of n = 20. The usual residual surface deformations on real lenses are expected to be in an even much higher spatial frequency domain. The sufficient modeling of such a surface for an highly accurate retrieval, where Eq. (5.19) could be minimized and the assumption cosβ ≈ cosβ^M holds, would demand a tremendously large number of coefficients. As mentioned above, the modeled surface is described using the standard description in Eq. (5.2). As discussed in section 5.1, this polynomial set is inefficient since its terms are not orthogonal to each other leading to a strong cancellation between the terms. Furthermore, they become numerically unstable for orders beyond m = 20. Therefore, one is in a strict need for an improved polynomial representation for rotationally symmetric aspherical surfaces.