We derive a function that can be used to generate random cases of index inhomogeneity, with the desired P-V errors.

**Authored By Erin Elliott, Tolis Deslis**

## Introduction

Modeling index inhomogeneity can be important for lenses that are large in diameter or with large vertex thicknesses. We can approximate the form of the index inhomogeneity with tilt and power terms. We derive an equation that generates sets of Z_{2}, Z_{3}, and Z_{4} Zernike polynomials that satisfy a given P-V error. The Zernikes can be used to model inhomogeneity in an optical system.

## Introduction to inhomogeneity

We want to model index inhomogeneity, or how the index of refraction might vary within a lens. Index inhomogeneity can be a problem for any high-precision system, but is particularly problematic for lenses with large diameters.

We need to add the effect of the index variation to the optical model and determine the effect on the system performance. We’ll do this by adding a Zernike Fringe Phase surface after the lens in question, and assigning appropriate values to the Zernike surface. We will have to make assumptions about the likely form of the index inhomogeneity in order to do this.

Glass grades specify the maximum index variation within a part. For example, homogeneity Grade 3 glass has an index variation of +/- 2*10^{-6}. This corresponds to a highest and lowest possible index value, or a peak-to-valley (P-V) index variation. (We are making the assumption that this variation is constant as a function of wavelength, which is an acceptable assumption.)

Glass is cooled in large boules, and the edges cool faster than the center. This creates a typical index profile for the boules. As blanks are cut from the boules, the index variations are dominated by power and tilt terms.

The figure below is from Ralf Jedamzik, Uwe Petzold, "Optical glass: refractive index homogeneity from small to large parts - an overview," Proc. SPIE 10914, Optical Components and Materials XVI, 109140V (27 February 2019); doi:10.1117/12.2511271.

For a lens, then, it is reasonable to model the index inhomogeneity using Zernike tilt and power terms: Z2, Z3, and Z4.

For simplicity, we will assume that the lens is a plate with equal thickness at all X and Y coordinates, so that the wavefront error due to the index variation is just a constant. (For a convex lens, the actual wavefront deviation will be slightly smaller, since the thickness of the glass falls with radial coordinate. For a negative lens, the thickness of the glass may increase toward the edge of the lens; in those cases, it may be more appropriate to select the edge thickness than the vertex thickness to estimate a worst-case effect.)

The desired P-V phase change needed depends on the total index variation, the wavelength of the system, and the vertex thickness of the lens. We will assume Grade 3 glass, a wavelength of 500 nm, and a lens thickness of 30 mm. The maximum P-V phase change possible due to index variation is then:

So, we want to find the appropriate values of Z2, Z3, and Z4 for the Zernike Fringe Phase surface that give the required P-V error of 0.24 waves. The function described below will generate the appropriate set of Z coefficients.

## The P-V phase

The phase of the Zernike Fringe Phase surface using only the Z2, Z3, and Z4 terms, is:

(1)

We want to know the P-V phase of this function. If we assume that we always select a cross-section of the phase along the direction of the maximum tilt, then we can write the phase in a simplified version as:

(2)

The P-V of the phase will depend on the minimum and maximum of this cross-section. The maximum of this function always occurs at the edge of the aperture. The minimum can occur at the opposite edge of the aperture (if the tilts are larger than the power) or somewhere within the aperture (if the power is larger than the tilts).

The three figures below show plots of the cross-sections for these different conditions.

The 1^{st} one is a cross-section of the phase when Z4 < 0.25 (Z22 + Z32)1/2.

Here, D = 200 mm, Z2 = 0.1, Z3 = 0.05, and Z4 = 0.1*(Z22 + Z32). The minimum of the function occurs somewhere outside the aperture.

The 2^{nd} one is a cross-section of the phase when Z4 = 0.25 (Z22 + Z32)1/2.

Here, D = 200 mm, Z2 = 0.1, Z3 = 0.05, and Z4 = 0.25*(Z22 + Z32). The minimum of the function occurs at -D/2.

The 3^{rd} one is a cross-section of the phase when Z4 > 0.25 (Z22 + Z32)1/2.

Here, D = 200 mm, Z2 = 0.1, Z3 = 0.05, and Z4 = 0.5*(Z22 + Z32). The minimum of the function occurs inside the aperture.

We can calculate the cutoff between these regions by setting the derivative of the cross-section equal to zero (Equation 3), and solving for the radial coordinate (r) at the minimum (Equation 4). Setting r_{min} = D/2 and solving for Z4 gives the cutoff where the minimum of the function is at the edge of the aperture (Equation 5). If Z4 is greater than the cutoff in Equation 5, the minimum occurs at r_{min}, and the P-V is calculated at rmin and the opposite edge of the aperture. If Z4 is less than the cutoff, the minimum occurs outside the aperture, and the P-V is calculated at +/- (D/2).

(3)

(4)

(5)

We can use Equation 2 to write down the equation for the P-V in each region.

For |Z4| < Z4 cutoff, the P-V = (D/2) - (-D/2). This gives:

(6)

For Z4 > the cutoff, P-V = (rmin) - (D/2). This gives:

(7)

For Z4 < -Z4 cutoff, P-V = (rmin) - (-D/2). This gives:

(8)

## Solving for Z2, Z3, and Z4

The P-V equations 6 through 8 can be solved for a set of Zernike coefficients (Z2, Z3, Z4) with the correct target P-V value. The solutions can be used in macros and in other programs to generate random sets that correctly capture the expected P-V phase error due to uncertainty in the index of refraction. Care must be taken to avoid unrealistic solutions, though, as described below.

|Z4| < Z4 cutoff

In the case where Z4 is less than the cutoff value, the minimum of the function occurs outside the aperture. In this case, calculation of the peak-to-valley phase change doesn’t depend on the value of Z4. We can generate a random value of Z2, and then solve for Z3 to get the correct P-V phase.

To create a random set of Z2, Z3, and Z4 with the correct P-V, we’ll use the following steps:

- Generate a random value for Z2 that doesn’t exceed the required P-V:

(9a) - Solve for Z3 using the PV1 (Equation 6) to get:

(9b) - Select a random value for Z4 from within the cutoff range:

(9c)

Z4 > Z4 cutoff

In the case where Z4 is greater than the cutoff value, the minimum of the function occurs inside the aperture. Now, the value of Z4 does affect the final P-V phase. We can solve for Z4 using Equation 7, but we need an additional restriction on Z3 to ensure that the solution is real.

To create a random set of Z2, Z3, and Z4 with the correct P-V, we’ll use the following steps:

- Generate a random value for Z2 that doesn’t exceed the required P-V:

(10a) - Generate a random value for Z3. Limit the range of possible tilt values so that the target P-V value isn’t exceeded.

(10b) - Solve for Z4. Use the PV2 (Equation 7) set equal to the target P-V phase (PV), and solve for Z4 to find the solution below.

(Note that the above condition on Z3 insures that the quantity under the first square root will be positive.)

(10c)

Z4 < - Z4 cutoff

In the case where Z4 is negative and less than the negative of the cutoff value, the solution for Z4 is the negative of the solution above. The steps to generate the set of coefficients are the same, with Equation 10c replaced by Equation 11.

(11)

## A few random cases

A set of random cases, generated using the steps described above, is shown below. For each case shown, a random choice is made to use one of the three solutions: |Z4| < Z4 cutoff, Z4 > Z4 cutoff, or Z4 < -Z4 cutoff. The target P-V phase in all cases is 0.1 waves, as shown in Table 1. In Figure 6, the full two-dimensional phase, from Equation 1, is plotted for each case.

The table below shows a random sets of Zernike coefficients, all with a P-V phase of 0.1 waves, generated using the method described in Section 3.

The figure below shows plots of the two-dimensional phase, for the nine random cases.

## References

- Ralf Jedamzik, Uwe Petzold, "Optical glass: refractive index homogeneity from small to large parts - an overview," Proc. SPIE 10914, Optical Components and Materials XVI, 109140V (27 February 2019); doi:10.1117/12.2511271

KA-01976

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