Jacobi Polynomial and Derivatives (Zernike, Chebyshev, and Legendre) Surface Sag by ELE Optics

Background & Theory
When analyzing optical surfaces, optical systems, fabrication methods, and wavefronts, it is extremely common to describe things using a polynomial set, specifically the Zernike polynomial. There area a variety of reasons for this, but concisely the Zernike polynomial is ortho-normal over a circular aperture, which works well for most optics, and the lower order terms of the Zernike polynomial map well to commonly seen aberrations and features in an optical system. For example, low order terms describe tip, tilt, defocus, and spherical aberration, all common aberrations that may be seen in an optical system. Additionally, a variety of terms map particularly well to describing features that can be commonly seen from traditional fabrication methods or mechanical deformations of a lens surface induced do to thermal and mechanical stress. See Dr. Wyant and Dr. Creath for wonderful descriptions of the Zernike polynomial and it’s uses [1]. See Dr. Mahajan for a deeper discussion of circular Zernike polynomials [2].

However, Zernike polynomials are themselves derived from the more general Jacobi polynomial. Other unique forms of the Jacobi polynomial include the Chebyshev and Legendre polynomials. Both have their applications in optics, but one clear example of the use of the Chebyshev is that, unlike the Zernike polynomial, it is ortho-normal over a rectangular aperture. For systems that utilize a rectangular detector, this can be a useful polynomial set to use for fitting and describing a rectangular wavefront. From a complete description on the derivation of of the Zernike, Chebyshev, and Legendre from the Jacobi see the post on this forum dedicated to the topic.

In terms of applications, surfaces in an optical system can be defined as a Zernike, Chebyshev, Legendre, or Jacobi polynomial surface. These surfaces also describe what are commonly referred to as ‘freeform’ surfaces. Additionally, these polynomials can be used to fit the wavefront in an optical system, model deformations of a surface, and describe a measurement from a metrology system of an optical surface.

Algorithm
For the interested scientist and user, it is essential to know what exactly the software is doing to determine the surface type. The following section lays out the algorithms used:

Step 1: Definition of the Jacobi Polymomial

The classic Jacobi Polynomial [3] is a solution to the Jacobi Differential Equation, which is itself a singular Sturm-Liouville eigenvalue problem [4] of the form:

\frac{d}{dx}[(1-x)^{\alpha+1}(1+x)^{\beta+1}y'] + n(n+\alpha + \beta + 1)(1-x)^\alpha (1+x)^\beta y = 0 \tag{1}

Using a solution of the form:

y = \sum_{v=0}^{\infty}a_v(x-1)^v \tag{2}

produces a recurrence relation [5]:

[\gamma -v(v+\alpha+\beta+1]a_v-2(v+1)(v+\alpha+1)a_{v+1}=0 \tag{3}

for the set v=0,1,.... \gamma is defined as

\gamma \equiv n(n+\alpha+\beta+1)\tag{4}

Solving the recurrence relationship gives us the general form of the Jacobi polynomial:

P_n^{(\alpha, \beta)}(x) = \frac{-1^n}{2^n n!} (1-x)^{-\alpha}(a+x)^{-\beta}\frac{d^n}{dx^n}[{(1-x)^{\alpha + n}(1+x)^{\beta + n}}] \tag{5}

Step 2: Defining Ortho-Normality

Equation 5 only is true for cases where \alpha, \beta > -1. It forms a complete orthogonal set in the interval x \in [-1, 1] with respect to the following weighting function, w(x) :

w(x) = (1-x)^\alpha(1+x)^\beta \tag{6}

Step 3: Normalization

The Jacobi Polynomials are normalized via:

P_n^{(\alpha,\beta)}(1)=\binom{n+\alpha}{n} \tag{7}

They satisfy orthonormality via:

\int_1^1P_i^{(\alpha, \beta)}(x)P_j^{(\alpha,\beta)}(x)w(x)dx=\delta_{i,j} \tag{8}

Step 4: Deriving Zernike, Chebyshev, and Legendre Polynomial Sets

The Jacobi Polynomial reduces to the

• Legendre Polynomial when \alpha=\beta=0
P_n^{(0, 0)}(x) = \frac{-1^n}{2^n n!} \frac{d^n}{dx^n}[{(1-x)^{n}(1+x)^{n}}] \tag{9}

and

w(x) = 1 \tag{10}
• The Chebychev Polynomial of the first kind when \alpha=\beta=-1/2, such that:
T_n(x) = \frac{P_n^{(-1/2,-1/2)}(x)}{P_n^{(-1/2,-1/2)}} \tag{11}
• The Zernike Polynomial when
1. x = 1-2\rho^2
2. \alpha = m
3. \beta = 0
4. n' = \frac{1}{2}(n-m)
resulting in :
P_{n'}^{(\alpha,\beta)}(x) = (-1)^{n'}\frac{R_n^m(\rho)}{\rho^\alpha} \tag{12}

As always, if you find any errors please point them out so we can correct them!

References

1. Wyant, J., and Creath, K., “Basic Wavefront Aberration Theory
for Optical Metrology”

2. Mahajan, V. N. (2013). Optical Imaging and Aberrations: Wavefront analysis. United States: SPIE Optical Engineering Press.

3. “Jacobi Polynomial.” Accessed July 16, 2020. Jacobi Polynomial.

4. Hesthaven, Jan S., and Tim Warburton. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications . Appendix A. Texts in Applied Mathematics. New York: Springer-Verlag, 2008. https://doi.org/10.1007/978-0-387-72067-8.

5. Levin, Oscar. “Discrete MathematicsAn Open Introduction.” Discrete Mathematics: An Open Introduction , OpenMath Books, Solving Recurrence Relations.

6. Aftab, Maham, Logan R. Graves, James H. Burge, Greg A. Smith, Chang Jin Oh, and Dae Wook Kim. “Rectangular Domain Curl Polynomial Set for Optical Vector Data Processing and Analysis.” Optical Engineering 58, no. 9 (September 2019): 095105. Rectangular domain curl polynomial set for optical vector data processing and analysis.

7. Aftab, Maham, James H. Burge, Gregory A. Smith, Logan Graves, Chang Jin Oh, and Dae Wook Kim. “Modal Data Processing for High Resolution Deflectometry.” International Journal of Precision Engineering and Manufacturing - Green Technology Volume 6, no. 2 (April 1, 2019): 255–70. Modal Data Processing for High Resolution Deflectometry | SpringerLink.

8. Schwalm, W.A.: Lectures on Selected Topics in Mathematical Physics: Elliptic Function-sand Elliptic Integrals, vol. 68. Morgan & Claypool publication as part of IOP Concise Physics, San Rafael (2015)

9. Boyd, John P., and Rolfe Petschek. “The Relationships Between Chebyshev, Legendre and Jacobi Polynomials: The Generic Superiority of Chebyshev Polynomials and Three Important Exceptions.” Journal of Scientific Computing 59, no. 1 (April 1, 2014): 1–27. The Relationships Between Chebyshev, Legendre and Jacobi Polynomials: The Generic Superiority of Chebyshev Polynomials and Three Important Exceptions | SpringerLink.

10. Lewanowicz, S. “Properties of the Polynomials Associated with the Jacobi Polynomials.” Mathematics of Computation 47, no. 176 (1986): 669–82. AMS :: Mathematics of Computation.

11. Defez, E., L. Jódar, and A. Law. “Jacobi Matrix Differential Equation, Polynomial Solutions, and Their Properties.” Computers & Mathematics with Applications 48, no. 5 (September 1, 2004): 789–803. Redirecting.

12. Koekoek, J., and R. Koekoek. “Differential Equations for Generalized Jacobi Polynomials.” Journal of Computational and Applied Mathematics 126, no. 1–2 (December 2000): 1–31. Redirecting.

13. Forbes, G. W. “Robust, Efficient Computational Methods for Axially Symmetric Optical Aspheres.” Optics Express 18, no. 19 (September 13, 2010): 19700–712. Robust, efficient computational methods for axially symmetric optical aspheres.