Kitabı oku: «Optical Engineering Science», sayfa 11

5.3 Zernike Polynomials

5.3.1 Introduction

In describing wavefront aberrations at any surface in a system, it is convenient to do so by expressing their value in terms of the two components of normalised pupil functions P_x and P_y. Where the magnitude of the pupil function is equal to unity, this describes the position of a ray at the edge of the pupil. With this description in mind, we now proceed to describe the normalised pupil position in terms of the polar co-ordinates, ρ and θ. This is illustrated in Figure 5.4.

Figure 5.4 Polar pupil coordinates.

The wavefront error across the pupil can now be expressed in terms of ρ and θ. What we are seeking is a set of polynomials that is orthonormal across the circular pupil described. Any continuous function may be represented in terms of this set of polynomials as follows:

(5.11)

The individual polynomials are described by the term f_i(ρ,θ), and their magnitude by the coefficient, A_i. The property of orthonormality is significant and may be represented in the following way:

(5.12)

The symbol, δ_ij is the Kronecker delta. That is to say, when i and j are identical, i.e. the two polynomials in the integral are identical, then the integral is exactly one. Otherwise, if the two polynomials in the integral are different, then the integral is zero. The first property is that of normality, i.e. the polynomials have been normalised to one and the second is that of orthogonality, hence their designation as an orthonormal polynomial set.

Equations (5.11) and (5.12) give rise to a number of important properties of these polynomials. Initially we might be presented with a problem as to how to represent a known but arbitrary wavefront error, Φ(ρ,θ) in terms of the orthonormal series presented in Eq. (5.11). For example, this arbitrary wavefront error may have been computed as part of the design and analysis of a complex optical system. The question that remains is how to calculate the individual polynomial coefficients A_i. To calculate an individual term, one simply takes the cross integral of the function, Φ(ρ,θ), with respect to an individual polynomial, f_i(ρ, θ):

By definition we have:

(5.13)

So, any coefficient may be determined from the integral presented in Eq. (5.13). The coefficients, A_i, clearly express, in some way, the magnitude of the contribution of each polynomial term to the general wavefront error. In fact, the magnitude of each component, A_i, represents the root mean square (rms) contribution of that component. More specifically, the total rms wavefront error is given by the square root of the sum of the squares of the individual coefficients. That this is so is clearly evident from the orthonormal property of the series:

(5.14)

5.3.2 Form of Zernike Polynomials

Following this general discussion about the useful properties of orthonormal functions, we can move on to a description of the Zernike circle polynomials themselves. They were initially investigated and described by Fritz Zernike in 1934 and are admirably suited to a solution space defined by a circular pupil. We will suppose initially, that the polynomial may be described by a component, R(ρ), that is dependent exclusively upon the normalised pupil radius and a component G(φ) that is dependent upon the polar angle, φ. That is to say:

(5.15)

We can make the further assumption that R(ρ) may be represented by a polynomial series in ρ. The form of G(φ) is easy to deduce. For physically realistic solutions, G(φ) must repeat identically every 2π radians. Therefore G(φ) must be represented by a periodic function of the form:

(5.16)

where m is an integer

This part of the Zernike polynomial clearly conforms to the desired form, since not only does it have the desired periodicity, but it also possesses the desired orthogonality. The parameter, m, represents the angular frequency of the polar dependence.

Having dealt with the polar part of the Zernike polynomial, we turn to the radial portion, R(ρ). The radial part of the Zernike polynomial, R(ρ), comprises of a series of polynomials in ρ. The form of these polynomials, R(ρ), depends upon the angular parameter, m, and the maximum radial order of the polynomial, n. Furthermore, considerations of symmetry dictate that the Zernike polynomials must either be wholly symmetric or anti-symmetric about the centre. That is to say, the operation r → −r is equivalent to φ → φ + π. For the Zernike polynomial to be equivalent for both (identical) transformations, for even values of m, only even polynomials terms can be accepted for R(ρ). Similarly, exclusively odd polynomial terms are associated with odd values of m.

Overall, the entirety of the set of Zernike polynomials are continuous and may be represented in powers of Px and Py or ρcos(φ) and ρsin(φ). It is not possible to construct trigonometric expressions of order, m, i.e. cos(mφ) and ρsin(mφ) where the order of the corresponding polynomial is less than m. Therefore, the polynomial, R(ρ), cannot contain terms in ρ that are of lower order than the angular parameter, m.

To describe each polynomial, R(ρ), it is customary to define it in terms of the maximum order of the polynomial, n, and the angular parameter, m. For all values of m (and n), the polynomial, R(ρ), may be expressed as per Eq. (5.17).

(5.17)

Cn,m,i represents the value of a specific coefficient

The parameter, Nn,m, is a normalisation factor. Of course, any arbitrary scaling factor may be applied to the coefficients, Cn,m,i, provided it is compensated by the normalisation factor. By convention, the base polynomial has a value of unity for ρ = 1. Of course, with this in mind, the purpose of the normalisation factor is to ensure that, in all cases, the rms value of the polynomial is normalised to one. It now remains only to calculate the values of the coefficients, Cn,m,i. These are determined from the condition of orthogonality which applies separately for Rn,m(ρ) and may be set out as follows:

(5.18)

The general formula for the coefficients Cn,m,i is set out in Eq. (5.18).

(5.19)

For i = n = 0, the value of the coefficient, Cn,m,i, as prescribed for the piston term, is unity. The value of the normalisation factor, Nn,m, is given in Eq. (5.20).

(5.20)

More completely we can express the entire polynomial:

(5.21a)

(5.21b)

The parameter, m, can take on positive or negative values as can be seen from Eq. (5.16). Of course, Eq. (5.16) gives the complex trigonometric form. However, by convention, negative values for the parameter m are ascribed to terms involving sin(mφ), whilst positive values are ascribed to terms involving cos(mφ).

Zernike polynomials are widely used in the analysis of optical system aberrations. Because of the fundamental nature of these polynomials, all the Gauss-Seidel wavefront aberrations clearly map onto specific Zernike polynomials. For example, spherical aberration has no polar angle dependence, but does have a fourth order dependence upon pupil function. This suggests that this aberration has a radial order, n, of 4 and a polar dependence, m, of zero. Similarly, coma has a radial order of 3 and a polar dependence of one. Table 5.2 provides a list of the first 28 Zernike polynomials.

In Table 5.2, each Zernike polynomial has been assigned a unique number. This is the ‘Standard’ numbering convention adopted by the American National Standards Institute, (ANSI). It has the benefit of following the Born and Wolf notation logically, starting from the piston term which is denominated the zeroth term. If the ANSI number is represented as j, and the Born and Wolf indices as n, m, then the ANSI number may be derived as follows:

(5.22)

Unfortunately, a variety of different numbering conventions prevail, leading to significant confusion. This will be explored a little later in this chapter. As a consequence of this, the reader is advised to be cautious in applying any single digit numbering convention to Zernike polynomials. By contrast, the n, m numbering convention used by Born and Wolf is unambiguous and should be used where there is any possibility of confusion.

5.3.3 Zernike Polynomials and Aberration

As outlined previously, there is a strong connection between Zernike polynomials and primary aberrations when expressed in terms of wavefront error. Table 5.2 clearly shows the correspondence between the polynomials and the Gauss Seidel aberrations, with the 3rd order Gauss-Seidel aberrations, such as spherical aberration and coma clearly visible.

The power of the Zernike polynomials, as an orthonormal set, lies in their ability to represent any arbitrary wavefront aberration. Using the approach set out in Eq. (5.13), it is possible to compute the magnitude of any Zernike term by the cross integral of the relevant polynomial and the wavefront disturbance. Furthermore, the total root mean square (rms) wavefront error, as per Eq. (5.14), may be calculated from the RSS (root sum square) of the individual Zernike magnitudes. That is to say, the Zernike magnitude of each term represents its contribution to the rms wavefront error, as averaged over the whole pupil.

The use of defocus to compensate spherical aberration was explored in Chapters 3 and 4. In this instance, for a given amount of fourth order wavefront error, we sought to minimise the rms wavefront error by applying a small amount of defocus.

Hence, without defocus, adjustment, the raw spherical aberration produced in a system may be expressed as the sum of three Zernike terms, one spherical aberration, one defocus and one piston term. The total aberration for an uncompensated system is simply given by the RSS of the individual terms. However, for a compensated system only the Zernike n = 4, m = 0 term needs be considered. This then gives the following fundamental relationship:

Table 5.2 First 28 Zernike polynomials.

(5.23)

The rms wavefront error has thus been reduced by a factor of six by the focus compensation process. Furthermore, this analysis feeds in to the discussion in Chapter 3 on the use of balancing aberrations to minimise wavefront error. For example, if we have succeeded in eliminating third order spherical aberration and are presented with residual fifth order spherical aberration, we can minimise the rms wavefront error by balancing this aberration with a small amount of third order aberration in addition to defocus. Analysis using Zernike polynomials is extremely useful in resolving this problem:

As previously outlined, the uncompensated rms wavefront error may be calculated from the RSS sum of all the four Zernike terms. Naturally, for the compensated system, we need only consider the first term.

(5.24)

For the fifth order spherical aberration, the rms wavefront error has been reduced by a factor of 20 through the process of aberration balancing. In terms of the practical application of this process, one might wish to optimise an optical design by minimising the rms wavefront error. Although, in practice, the process of optimisation will be carried out using software tools, nonetheless, it is useful to recognise some key features of an optimised design. By virtue of the previous example, optimisation of spherical aberration should lead to an OPD profile that is close to the 5th order Zernike term. This is shown in Figure 5.5 which illustrates the profile of an optimised OPD based entirely on the relevant fifth order Zernike term. The graph plots the nominal OPD again the normalised pupil function with the form given by the Zernike polynomial, n = 6, m = 0.

In the optimisation of an optical design it is important to understand the form of the OPD fan displayed in Figure 5.5 in order recognise the desired endpoint of the optimisation process. It displays three minima and two maxima (or vice versa), whereas the unoptimised OPD fan has one fewer maximum and minimum. Thus, although the design optimisation process itself might be computer based, nevertheless, understanding and recognising the how the process works and its end goal will be of great practical use. That is to say, as the computer-based optimisation proceeds, on might expect the OPD fan to acquire a greater number of maxima and minima.

Figure 5.5 Fifth order Zernike polynomial and aberration balancing.

One can apply the same analysis to all the Gauss-Seidel aberrations and calculate its associated rms wavefront error.

(5.25a)

(5.25b)

(5.25c)

(5.25d)

θ represents the field angle

Equations (5.25a)–(5.25d) are of great significance in the analysis of image quality, as the rms wavefront error is a key parameter in the description of the optical quality of a system. This will be discussed in more detail in the next chapter.

Worked Example 5.2 A plano-convex lens, with a focal length of 100 mm is used to focus a collimated beam; the refractive index of the lens material is 1.52. It is assumed that the curved surface faces the infinite conjugate. The pupil diameter is 12.5 mm and the aperture is situated at the lens. What is the rms spherical aberration produced by this lens – (i) at the paraxial focus; (ii) at the compensated focus? What is the rms coma for a similar collimated beam with a field angle of one degree?

Firstly, we calculate the spherical aberration of the single lens. With the object at infinity and the image at the first focal point, the conjugate parameter, t, is equal to −1. The shape parameter, s, for the plano convex lens is equal to 1 since the curved surface is facing the object. From Eq. (4.30a) the spherical aberration of the lens is given by:

r_max = 6.25 mm (12.5/2); f = 100 mm; n = 1.52; s = 1; t = −1

By substituting these values into the above equation, the spherical aberration may be directly calculated:

where A = 4.13 × 10⁻⁴ mm ρ = r/r_max

From Eq. (5.23), the uncompensated rms wavefront error is A/√5 and the compensated error is A/√180. Therefore the rms values are given by:

Φrms(paraxial) = 185 nm; Φrms(compensated) = 30.8 nm

Secondly, we calculate the coma. From (4.30b), the coma of the lens is given by:

Again, substituting the relevant values for f, n, rmax, s, and t, we get:

where A = 3.24 × 10⁻³ mm ρ = r/r_max  ry = r sin ϕ

From (5.25b)

We are told that θ = 1° or 0.0174 rad. Therefore, Φrms = 6.66 × 10−6or 6.66 nm

5.3.4 General Representation of Wavefront Error

We have emphasised the synergy between Zernike polynomials and the classical treatment of aberrations in an axially symmetric optical system, i.e. the Gauss-Seidel aberrations. However, in practice, in real optical systems, these axial symmetries are often compromised, either by accident or by design. Some systems are deliberately designed whereby not all optical surfaces are aligned to a common axis. These will inevitably introduce non-standard wavefront aberrations into the system. Most significantly, even with a symmetrical design, component manufacturing errors and system alignment may introduce more complex wavefront errors into the system. Naturally, alignment errors create an off-axis optical system ‘by accident’. Manufacturing or polishing errors might produce an optical surface whose shape departs from that of an ideal sphere or conic in a somewhat complex fashion. For example, the effects of these errors may be to introduce a trefoil term (n = 3, m = 3) into the wavefront error; this is not a standard Gauss-Seidel term.

As argued, Zernike polynomials are widely used in the analysis of wavefront error both in the design and testing of optical systems. From a strictly analytical and theoretical point of view the description of wavefront error in terms of its rms value is the most meaningful. However, for largely historical reasons, wavefront error is often presented as a ‘peak to valley’ error. That is to say, the value presented is the difference between the maximum and minimum OPD across the pupil. Historically, the wavefront error for a system might have been derived from a visual inspection of a fringe pattern in an interferogram. The maximum deviation of fringes is relatively straightforward to estimate visually from a fringe pattern which might have been produced photographically. However, the rms wavefront error is more directly related to system performance. Calculation of the rms wavefront error across a pupil is a mathematical process that requires computational data acquisition and analysis and has only been universally available in more recent times. Therefore, the use of the peak to valley description still persists.

One particular disadvantage of the peak to valley description is that it is unusually responsive to large, but highly localised excursions in the wavefront error. More generally, as a rule of thumb, the peak to valley is considered to be 3.5 times the rms value. Of course, this does depend upon the form of the wavefront error. Table 5.3 sets out this relationship for the first 11 Zernike terms (apart from piston). For comparison, a standard statistical measure is also presented – namely for a normally distributed wavefront error profile, the limits containing 95% of the wavefront error distribution (±1.96 standard deviations).

The values presented in Table 5.3 are simply the ratio of the peak to valley (p-to-v) error for that particular distribution. To overcome the principal objection to the p-to-v measure, namely its heightened sensitivity to local variation a new peak to valley measure has been proposed by the Zygo Corporation. This measure is known as P to Vr or peak to valley robust. In this measure, the wavefront error is fitted to a set of 36 Zernike polynomials. Although this process is carried out by computational analysis, the procedure is very simple. Essentially the calculation process exploits the orthonormal properties of the polynomial set and calculates the contribution of each Zernike term using the relation set out in Eq. (5.12). Following this process, the maximum and minimum of the fitted surface is calculated and the revised peak to valley figure calculated. Of course, the reduced set of 36 polynomials cannot possibly replicate localised asperities with a high spatial frequency content. Therefore, the fitted surface is effectively a smoothed version of the original and the peak to valley value derived is more representative of the underlying physics.

Table 5.3 Peak to valley: Root mean square (rms) ratios for different wavefront error forms.

Table 5.4 Comparison of Zernike numbering systems.

It must be stated, at this point, that the 36 polynomials used, in this instance, are not those that would be ordered as in Table 5.1. That is to say, they are not the first 36 ANSI standard polynomials. As mentioned earlier, there are, unfortunately, a number of competing conventions for the numbering of Zernike polynomials. The convention used in determining the P to Vr figure is the so called Zernike Fringe polynomial convention. The logic of ordering the polynomials in a different way is that this better reflects, in the case of the fringe polynomial set, the spatial frequency content of the polynomial and its practical significance in real optical systems.

5.3.5 Other Zernike Numbering Conventions

The ordering convention adopted by the Fringe polynomials expresses, to a significant degree, the spatial frequency content of the polynomial. As a consequence, the polynomials are ordered by the sum of their radial and polar orders, rather than primarily by the radial order. That is to say, the polynomials are ordered by the sum n + m, as opposed to n alone. For polynomials of equal ‘fringe order’ they are then ordered by descending values of the modulus of m, i.e. |m|, with the positive or cosine term presented first.

Another convention that is very widely used is the Noll convention. The Noll convention proceeds in a broadly similar way to the ANSI convention, in that it uses the radial order, n, as the primary parameter for sorting. However, there are a number of key differences. Firstly, the sequence starts with the number one, as opposed to zero, as is the case for the other conventions. Secondly, the ordering convention for the polar order, m, as in the case of the fringe polynomials, follows the modulus of m rather its absolute value. However, the ordering is in ascending sequence of |m|, unlike the fringe polynomials. The ordering of the sine and cosine terms is presented in such a way that all positive m (cosine terms) are allocated an even number. In consequence, sometimes the sine term occurs before the cosine term in the sequence and sometimes after. Table 5.4 shows a comparison of the different numbering systems up to ANSI number 65.