2.1.

Impossibility of Expressing Center Artifacts by Fringe Zernike Polynomials

In optical element manufacturing, it is common to describe manufacturing errors using Fringe Zernike polynomials. Reference 16 represents some practical methods for representing irregular surfaces by the use of Zernike polynomials. However, it is impossible to express center artifacts this way. One reason is that the width of a low Zernike polynomial, such as $Z_9$ is too wide to express steep center artifacts. Another reason is that higher Zernike polynomials, such as $Z_{16}$, $Z_{25}$, $Z_{36}$, and $Z_{37}$ have ripples at the outer region of the aperture. To numerically prove this, let us take a precision turned plastic lens as an example as follows.

The surface deviation is measured using UA-3P, a three dimensional measurement system along two perpendicular directions on the lens surface. Since these two profiles do not differ largely from each other, the surface profile can be approximated as rotationally invariant. Figure 1 shows one direction, in which the horizontal axis is the normalized radial coordinate and the vertical axis is the deviation in millimeters. The marginal coordinate corresponds to 26-mm diameter in exact scale. The blue line describes surface profile in one direction and the red line its approximation represented by the rotationally invariant Fringe Zernike polynomials ($Z_1,Z_4,Z_9,Z_{16},Z_{25},Z_{36}$, and $Z_{37}$). Figure 2 represents the difference between the two lines. Although Zernike fitting can provide an approximation for lower frequency errors, it is completely insufficient to express the center artifact. Therefore, we needed to construct a new mathematical method to express center artifacts.

Fig. 1

An example of surface figure error and its Zernike approximation.

2.2.

Approximation of Center Artifacts by Normal Distribution Function

As seen in Sec. 2.1, so as to express the shape of center artifacts mathematically, we should introduce another mathematical representation other than Zernike polynomials. Since center artifacts are sharply convex shaped localized at the center, they can be represented by the normal distribution function. Equation (1) can provide a simple mathematical model for representing a center artifact.

In this mathematical model, the maximum is $A$, the minimum is 0, and full width at half maximum (FWHM) of $f(r)$ is $2.53\sigma$. We do not need the DC component in Eq. (1), because it is automatically considered by Zernike polynomials fitting. When $\sigma$ becomes smaller, the profile of the center profile becomes steeper. In the case of Fig. 2, we decided $A=0.0036$, $\sigma =0.0546$ by least square method. Figure 3 compares the Zernike fitting error and the mathematical model described in Eq. (1).

Fig. 2

Fitting error of center artifact using 12th-order Fringe Zernike polynomials.

Fig. 3

Zernike fitting error and its approximation by Eq. (1).

Figure 4 shows the residual of figure error. This figure represents that the use of normal distribution approximates the center artifact profile with submicron accuracy, which is satisfactory for precision turned plastic lenses. Therefore, we concluded that it is suitable to express the shape of the center artifact by the use of the normal distribution function. Also since the normal distribution function is characterized by its width and height, the tolerance of the drawing is simple and can be easily understood for manufacturing.

Fig. 4

The residual of an approximation using normal distribution function.

2.3.

Impossibility of Expressing the Normal Distribution Function by Even-Order Aspherical Surfaces

Section 2.2 represents the possibility of expressing center artifacts using the normal distribution function. However, since the normal distribution function is generally not prepared in ordinary optical design software, we should rewrite Eq. (1) as the general form for aspherical surfaces.

In Secs. 2.3.1 and 2.3.2, we will discuss the expression of normal distribution by the use of even-order surfaces, the most popular surface type in optical design. Subsection 2.3.1 employs Taylor expansion of normal distribution and Sec. 2.3.2 employs Zernike expansion. We will show that these two methods do not provide sufficient approximation for the normal distribution.

2.3.1.

Approximation by Taylor expansion

First, we considered Taylor expansion, which is the most direct method to obtain polynomials. Equation (2) represents the result of expanding $\exp (-r^2/{\sigma }^2)$ to the power series of $r$.

Eq. (2)

$$\exp (-\frac{r^2}{{\sigma }^2})\sim 1-\frac{r^2}{{\sigma }^2}+\frac{1}{2}\frac{r^4}{{\sigma }^4}-\frac{1}{6}\frac{r^6}{{\sigma }^6}+\cdots +\frac{{(-1)}^n}{n!}{\left(\frac{r^2}{{\sigma }^2}\right)}^n+\cdots .$$

However, the power series of Eq. (2) does not provide suitable approximation using any finite number of terms. Figure 5 compares the power series up to 30th (the upper limit of ordinary even-order aspherical surface in code-V^™) with the original normal distribution function $\exp (-\frac{r^2}{{\sigma }^2})$ for $\sigma =0.2$. Figure 5 represents that the power series is an accurate approximation for $r<0.5$, however, it diverges over the outer area, providing that this power series is not valid for expressing center artifact shape.

Fig. 5

Comparison of the power series up to 30th with the normal distribution function for $\sigma =0.2$.

2.3.2.

Approximation by Zernike expansion

Next, we considered Zernike expansion for the normal distribution function. The example of a center artifact described in Sec. 2.1 cannot be approximated by Fringe Zernike polynomials. However since Zernike polynomials are complete, the normal distribution function can be expressed using a sum of Zernike polynomials.

Equation (3) describes rotationally invariant Zernike polynomials²⁰

Eq. (3)

$$Q_n(t)=\frac{{(-1)}^n}{n!}\frac{d^n}{d{t}^n}{t}^n{(1-t)}^n,$$

where $t$ is the square of the radial coordinate $r$. Table 1 represents the relationship between Eq. (3) and Fringe Zernike polynomials

Table 1

Relationship between Qn(t) of Eq. (3) and Fringe Zernike polynomials.

Definition in Eq. (3)	Fringe Zernike	Description
$Q_0(t)$	$Z_1$	1
$Q_1(t)$	$Z_4$	$2r^2-1$
$Q_2(t)$	$Z_9$	$6r^4-6r^2+1$
$Q_3(t)$	$Z_{16}$	$20r^6-30r^4+12r^2-1$
$Q_4(t)$	$Z_{25}$	$70r^8-140r^6+90r^4-20r^2+1$
$Q_5(t)$	$Z_{36}$	$252r^{10}-630r^8+560r^6-210r^4+30r^2-1$
$Q_6(t)$	$Z_{37}$	$924r^{12}-2772r^{10}+3150r^8-1680r^6+420r^4-42r^2+1$

Fringe Zernike expansion of function $f(t)$ is

Eq. (4)

$$f(t)={\mathrm{c}}_0{Q}_o(t)+c_1{Q}_1(t)+c_2{Q}_2(t)+{\mathrm{c}}_3{Q}_3(t)+{\mathrm{c}}_4{Q}_4(t)+{\mathrm{c}}_5{Q}_5(t)+{\mathrm{c}}_6{Q}_6(t),$$

where $c_n$’s are Fringe Zernike coefficients which are calculated by

Eq. (5)

$${\mathrm{c}}_n=(2n+1)\underset{0}{\overset{1}{\int }}f(t)Q_n(t)\mathrm{d}t.$$

Note that Fringe Zernike polynomials satisfy orthogonality, but the ordinary normalization is not satisfied.

To estimate the accuracy of Fringe Zernike expansion of center artifacts, we calculated Fringe Zernike coefficients $c_n$ for the normal distribution function $\exp (-r^2/{\sigma }^2)$. Equations (4) and (5) lead to Eq. (6) and Table 2 represents the numerical result

Eq. (6)

$${\mathrm{c}}_n={(-1)}^n\frac{(2n+1)}{n!}\underset{0}{\overset{1}{\int }}\exp (-t/{\sigma }^2)\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}{t}^n{(1-t)}^n\mathrm{d}t.$$

Table 2

Fringe Zernike coefficients of exp(−r2/σ2).

n	cn
0	${\sigma }^2(1-e^{-1/{\sigma }^2})$
1	$6{\sigma }^4(1-e^{-1/{\sigma }^2})-3{\sigma }^2(1+e^{-1/{\sigma }^2})$
2	$(60{\sigma }^6+5{\sigma }^2)(1-e^{-1/{\sigma }^2})-30{\sigma }^4(1+e^{-1/{\sigma }^2})$
3	$(840{\sigma }^8+84{\sigma }^4)(1-e^{-1/{\sigma }^2})-(420{\sigma }^6+7{\sigma }^2)(1+e^{-1/{\sigma }^2})$
4	$(15120{\sigma }^{10}+1620{\sigma }^6+9{\sigma }^2)(1-e^{-1/{\sigma }^2})-(7560{\sigma }^8+180{\sigma }^4)(1+e^{-1/{\sigma }^2})$
5	$(332640{\sigma }^{12}+36960{\sigma }^8+330{\sigma }^4)(1-e^{-1/{\sigma }^2})-(166320{\sigma }^{10}+4620{\sigma }^6+11{\sigma }^2)(1+e^{-1/{\sigma }^2})$
6	$(8648640{\sigma }^{14}+982800{\sigma }^{10}+10920{\sigma }^6+13{\sigma }^2)(1-e^{-1/{\sigma }^2})$$-(4324320{\sigma }^{12}+131040{\sigma }^8+546{\sigma }^4)(1+e^{-1/{\sigma }^2})$

Figure 6 compares normal distribution functions for $\sigma =0.2,0.1,0.05$ with their Fringe Zernike expansion and Table 3 lists the coefficients explicitly. Figure 6 explains that the Fringe expansion of the normal distribution function for small $\sigma$ is insufficient to approximate the original functions even if all terms up to the 37th (upper limit of Fringe Zernike polynomials) are used. Thus, the set of Fringe Zernike polynomials is inappropriate to express the shape of the normal distribution function.

Fig. 6

Comparison of normal distribution functions for $\sigma =0.2,0.1,0.05$ and their Fringe Zernike expansion.

Table 3

Fringe Zernike coefficients for normal distribution function.

Coefficients	σ
	0.2	0.1	0.05
$c_0$	0.040000	0.010000	0.002500
$c_1$	$-0.110400$	$-0.029400$	$-0.007463$
$c_2$	0.155840	0.047060	0.012313
$c_3$	$-0.170330$	$-0.062012$	$-0.016982$
$c_4$	0.157875	0.073546	0.021400
$c_5$	$-0.128731$	$-0.081267$	$-0.025508$
$c_6$	0.094161	0.085104	0.029253

The discussion in Secs. 2.2, 2.3.1, and 2.3.2 leads to the conclusion that the shape of a steep center artifact can be expressed by neither finite number even-order terms nor by Fringe Zernike expansion.

2.4.

Expressing the Normal Distribution Function by Power Terms Including Odd-Order

Since we proved that the normal distribution function is not represented by even-order aspherical surfaces in Sec. 2.3, we introduced odd-order aspherical surface in order to express center artifact. According to our investigations of the characteristics of odd-order surfaces, we have analogically predicted that center artifact shape can be sufficiently represented by a small finite number of power terms including odd-order terms which can be used in most optical design software.

Equation (7) represents a polynomial of order $N$ including odd-order

When the curve $z=g(r)$ passes through $N+1$ points $(r_k,z_k)k=0,1,\cdots ,N$, the coefficients $a_k$ are the solution of the following linear equation:

We considered the function $\exp (-r^2/{\sigma }^2)$ and choose $N+1$ points as $(r_k,z_k)=\{\frac{k}{N},\exp [-{(\frac{k}{N})}^2/{\sigma }^2]\}$ at even intervals in the region $r\in [\mathrm{0,1}]$. By resolving the linear equation Eq. (8), the coefficients $a_k$’s were obtained. Figure 7 shows the maximum approximation error curves of Eq. (7) with respect to $\sigma$.

Fig. 7

Maximum approximation error versus $\sigma$ for Eq. (7).

Figure 7 explains that the approximation error is nearly saturated when the degree $N$ exceeds 6. In practice, Fig. 8 shows that the normal distribution function can be fully approximated by polynomials of order $N=6$ for $\sigma =0.2,0.1$, and 0.05.

Fig. 8

Approximation curves for $\sigma =0.2,0.1$, and 0.05.

The discussion of this section gives an explicit method for approximating center artifacts by polynomials including odd-order.

3.1.

Optical Design

Using the new mathematical model we proposed in Sec. 2, we discuss how image quality is affected by center artifacts. For this purpose, we considered a projection lens which is a modified design from Ex. 2 of Ref. 22. The specification is listed in Table 4. Table 5 represents the lens data and Fig. 9 shows the optical configuration of this design. This design consists of eight lenses in which both sides of the second lens are aspherical. Equation (9) is the aspherical description of this design.

Table 4

Design specifications for projection optics.

Table 5

Lens design data.

Fig. 9

Optical configuration of the lens represented in Table 4.

Since the size of a pixel for the display device is $10\mu {\mathrm{m}}^2$, the Nyquist frequency is 50 lines per millimeters. Thus, Fig. 10 represents the designed MTF curve of this lens in the region, with a maximum frequency twice its Nyquist frequency. The horizontal axis represents spatial frequency and the vertical axis MTF values. As shown in Fig. 10, using aspherical lenses, we obtained excellent optical performance. However, since precision turned plastic lenses sometimes have center artifacts at the lens axis, the image quality of the field center degenerates remarkably in this case. Therefore, the tolerance of PV value and width of center artifacts must be properly obtained.

Fig. 10

MTF curve for the design described in Table 4 (blue line: on-axis, green-line: marginal tangential and radial).

3.2.

Mathematical Representation of Actual Center Artifacts

Aspherical lenses are directly generated from blanks of poly(methyl methacrylate) (PMMA). Figure 11 represents the photograph of an actual aspherical lens. Figures 12(a) and 12(b) represent the measured deviation of the manufactured plastic lens from the designed value. Figure 12(a) is the front surface and Fig. 12(b) the rear surface. Table 6 represents the Zernike approximation obtained by direct calculation of inner products of actual surface figure errors and Zernike polynomials.

Fig. 11

Photograph of the actual turned aspherical lens.

Fig. 12

Measured deviation of the turned lens from the designed shape.

Table 6

Zernike expansion coefficients of the front and rear surfaces.

By removing the component that was expressed by Fringe Zernike polynomials, Fig. 13 represents the approximations of center artifacts using the normal distribution function. The parameters in Eq. (1) for each surface are shown in Table 7. Parameter A represents the height of the center artifact. The parameter $\sigma$ shows the width of the center artifact and has no dimension since the radial coordinate is normalized. The solution of linear Eq. (8) for each $\sigma$ provides its aspherical coefficients.

Fig. 13

Approximation of center artifacts using normal distribution function.

Table 7

The parameters of actual fabricated lens for Eq. (1).

In solving Eq. (8), we normalized $A=1$ for convenience and Table 7 lists the solutions. Since the coefficients listed in Table 8 are normalized to unify the peak, multiplying actual A (artifact height) to the coefficients provides the actual aspherical coefficients. In addition, when we converted these parameters to match those of an optical simulation program, we had to take care that the polynomial equals the function of the normalized radial coordinate. Figure 14 compares the measured figure errors and the final approximation shapes of center artifacts which contain Fringe Zernike coefficients and odd-order terms.

Fig. 14

Measured figure errors and the final approximation shapes.

Table 8

Aspherical coefficients of approximation polynomials.

At the end of this subsection, we discuss the approximation accuracy of the mathematical model for center artifacts. Table 9 shows the PV error of the measured figure and PV residual errors of three approximation methods discussed above.

Table 9

P-V errors of measured figure and P-V residuals of three approximation methods (units are in millimeters)

Because of the narrow width of the center artifact in the front surface, Fringe Zernike fitting does not approximate the figure error. Specifically, the Zernike fitting residual of the first surface remains $4.6\mu \mathrm{m}$ in PV value, which cannot be negligible. However, our new model that employs both Fringe Zernike and normal distribution reduces the residual by 75%. The polynomial model provides almost the same residual as the normal distribution model. Consequently, our mathematical model described in this paper can provide a better description of figure errors that contain center artifacts.

3.3.

Optical Simulation

By adding aspherical coefficients for the actual center artifact to the designed lens data, we could evaluate how center artifacts affect the optical performances such as MTF and PSF.

The aspherical surface is represented by Eq. (10):

In this formula, the first two terms are the original designed form. The additional term, $\{\sum _{k=1}^6{c}_k{Q}_k(r^2)+\sum _{k=1}^6{a}_k{r}^k\}$, represents the figure error. The former $\sum _{k=1}^6{a}_k{r}^k$ $\sum _{k=1}^6{c}_k{Q}_k(r^2)$ is Fringe Zernike components error and the latter $\sum _{k=1}^6{a}_k{r}^k$ the polynomial representation of center artifacts.

Figure 15 compares the designed MTF, the MTF with Fringe Zernike components error, and the MTF with both Zernike components and center artifacts for the axial image point. Furthermore, Fig. 16 compares the designed PSF, the PSF with Zernike components, and the PSF with both Zernike components and center artifacts. These figures represent that the Zernike components are so small that the values of MTF and PSF remain as high as designed value. On the contrary, the center artifacts obviously affect the PSF. Figure 16(c) shows characteristic diffraction rings around the center, and they affect the imaging quality, such as MTF as shown in Fig. 15(c).

Fig. 15

Comparison of MTF of axial image.

Fig. 16

Comparison of PSF of axial image.

These characteristic diffraction rings and deterioration in MTF are very common in practical lens manufacturing. Hence, the mathematical model of center artifacts can represent the image degeneration of image quality as practical optical systems.