Deriving the Axially Symmetric Aberration Polynomial

This page derives Eqs. 5.1 and 5.2 from Modern Optical Engineering. They give the image-plane intersection \((x',y')\) of a ray as a power series in field height, pupil radius, and pupil azimuth. The derivation is useful beyond reproducing the equations: it explains why a centred optical system has only odd-order transverse aberrations and why the trigonometric factors occur in their particular combinations.

The equation numbers and coefficient names below follow Warren J. Smith, Modern Optical Engineering: The Design of Optical Systems, 4th ed., Sec. 5.2. Section 15.3 of the same source supplies the connection to the wave-aberration polynomial.

Geometry and convention

Take the optical axis to be \(z\). Rotational symmetry lets us put the object point in the meridional \((y,z)\) plane without loss of generality:

\[\boldsymbol H=(0,h).\]

Smith measures pupil azimuth \(\theta\) from the positive \(y\)-axis, not from the positive \(x\)-axis. The pupil point is therefore

\[\boldsymbol\rho=(p,q) =\left(s\sin\theta,\;s\cos\theta\right),\]

where \(s\) is its radial pupil coordinate. The ray intersects the chosen image plane at

\[\boldsymbol r'=(x',y').\]

The three scalar products that can be formed from the field and pupil vectors are

(1)\[u=\boldsymbol\rho\mathbin{\cdot}\boldsymbol\rho=s^2,\qquad v=\boldsymbol\rho\mathbin{\cdot}\boldsymbol H =sh\cos\theta,\qquad w=\boldsymbol H\mathbin{\cdot}\boldsymbol H=h^2.\]

Why this is the complete starting point

Let \(Q\) be any rotation or reflection in the transverse plane. A centred system is unchanged by that transformation, so its ray map must obey

\[\boldsymbol r'(Q\boldsymbol\rho,Q\boldsymbol H) =Q\,\boldsymbol r'(\boldsymbol\rho,\boldsymbol H).\]

Reflection symmetry excludes rotated pseudovectors such as \((-q,p)\). Consequently, an ordinary transverse vector can only be assembled from \(\boldsymbol\rho\) and \(\boldsymbol H\), multiplied by scalar functions of the three invariants in Eq. (1):

(2)\[\boldsymbol r' =\boldsymbol\rho\,F(u,v,w)+\boldsymbol H\,G(u,v,w).\]

This compact expression contains the parity result. Each of \(u,v,w\) has total degree two in pupil and field coordinates. Multiplication by either vector adds one more degree, so an analytic centred-system ray map contains orders

\[1,\ 3,\ 5,\ 7,\ldots\]

and no even orders. Tilting or decentring a surface breaks the symmetry used above and permits even-order terms.

First-order terms

The constant parts of \(F\) and \(G\) give

(3)\[\boldsymbol r'_A=A_1\boldsymbol\rho+A_2\boldsymbol H.\]

Taking components immediately gives

\[x'_A=A_1s\sin\theta,\qquad y'_A=A_1s\cos\theta+A_2h.\]

Thus \(A_2\) is the paraxial magnification, while \(A_1\) measures defocus of the selected image plane. At the paraxial image plane, \(A_1=0\).

Third order: the five Seidel terms

Symmetry fixes the available monomials. Fermat’s principle supplies one additional third-order constraint: in canonical pupil coordinates, the transverse ray error is proportional to the pupil gradient of a scalar optical characteristic. The common proportionality factor (\(-l/n\) when the scalar is OPD) can be absorbed into the coefficients.

A fourth-degree characteristic containing the five independent centred-system terms is

(4)\[\Phi_4={B_1\over4}u^2+B_2uv +{B_3+B_4\over2}uw+B_3v^2+B_5vw.\]

Using

\[\nabla_{\boldsymbol\rho}u=2\boldsymbol\rho,\qquad \nabla_{\boldsymbol\rho}v=\boldsymbol H,\qquad \nabla_{\boldsymbol\rho}w=0,\]

its pupil gradient is

(5)\[\begin{split}\begin{aligned} \boldsymbol r'_B &=\nabla_{\boldsymbol\rho}\Phi_4\\ &=\boldsymbol\rho \left[B_1u+2B_2v+(B_3+B_4)w\right]\\ &\quad+\boldsymbol H \left[B_2u+2B_3v+B_5w\right]. \end{aligned}\end{split}\]

For the \(x\) component, \(H_x=0\). Substitution of Eq. (1) and \(2\sin\theta\cos\theta=\sin2\theta\) gives

(6)\[x'_B =B_1s^3\sin\theta +B_2s^2h\sin2\theta +(B_3+B_4)sh^2\sin\theta.\]

For the \(y\) component, both vectors contribute. In particular, the coma factor follows from

\[2qv+hu =s^2h(2\cos^2\theta+1) =s^2h(2+\cos2\theta).\]

The result is

(7)\[y'_B =B_1s^3\cos\theta +B_2s^2h(2+\cos2\theta) +(3B_3+B_4)sh^2\cos\theta +B_5h^3.\]

The five coefficients are spherical aberration \(B_1\), coma \(B_2\), astigmatism \(B_3\), Petzval curvature \(B_4\), and distortion \(B_5\).

Fifth order: constructing every allowed angular term

At fifth order, \(F\) and \(G\) in Eq. (2) must be quadratic in \(u,v,w\). The following coefficient grouping is chosen so that the final components have Smith’s \(C_1,\ldots,C_{12}\) notation:

(8)\[\begin{split}\begin{aligned} F_4={}&C_1u^2+2C_3uv+C_5uw+C_6v^2 +2C_9vw+C_{11}w^2,\\ G_4={}&(C_2-C_3)u^2+(C_4-C_5)uv+(C_7-C_8)uw\\ &+2(C_8-C_9)v^2+(C_{10}-C_{11})vw+C_{12}w^2. \end{aligned}\end{split}\]

There is no missing physics in the coefficient differences: they are just a change of basis from the monomials in \(u,v,w\) to the traditional aberration coefficients. Because \(H_x=0\),

\[x'_C=pF_4,\qquad y'_C=qF_4+hG_4.\]

For example, the \(C_3\) terms combine as

\[\begin{split}\begin{aligned} x'_{C_3} &=2C_3p\,uv=C_3s^4h\sin2\theta,\\ y'_{C_2,C_3} &=2C_3q\,uv+(C_2-C_3)hu^2\\ &=(C_2+C_3\cos2\theta)s^4h. \end{aligned}\end{split}\]

Applying the same substitution to every monomial gives

(9)\[\begin{split}\begin{aligned} x'_C={}&C_1s^5\sin\theta+C_3s^4h\sin2\theta\\ &+(C_5+C_6\cos^2\theta)s^3h^2\sin\theta\\ &+C_9s^2h^3\sin2\theta+C_{11}sh^4\sin\theta, \end{aligned}\end{split}\]

and

(10)\[\begin{split}\begin{aligned} y'_C={}&C_1s^5\cos\theta +(C_2+C_3\cos2\theta)s^4h\\ &+(C_4+C_6\cos^2\theta)s^3h^2\cos\theta\\ &+(C_7+C_8\cos2\theta)s^2h^3\\ &+C_{10}sh^4\cos\theta+C_{12}h^5. \end{aligned}\end{split}\]

The coefficient groups are fifth-order spherical \(C_1\), linear coma \(C_2,C_3\), oblique spherical \(C_4,C_5,C_6\), elliptical coma \(C_7,C_8,C_9\), fifth-order Petzval/astigmatism \(C_{10},C_{11}\), and fifth-order distortion \(C_{12}\).

The displayed seventh-order term

Equations 5.1 and 5.2 show only the leading seventh-order spherical term. In invariant notation it is simply

\[\boldsymbol r'_{D_1}=D_1u^3\boldsymbol\rho,\]

so

\[x'_{D_1}=D_1s^7\sin\theta,\qquad y'_{D_1}=D_1s^7\cos\theta.\]

The omitted seventh- and higher-order structures are represented by the ellipses.

The assembled equations

Adding the first-, third-, fifth-, and displayed seventh-order blocks produces the requested equations:

Equation 5.1

\[\begin{split}\begin{aligned} y'={}&A_1s\cos\theta+A_2h\\ &+B_1s^3\cos\theta+B_2s^2h(2+\cos2\theta) +(3B_3+B_4)sh^2\cos\theta+B_5h^3\\ &+C_1s^5\cos\theta+(C_2+C_3\cos2\theta)s^4h\\ &+(C_4+C_6\cos^2\theta)s^3h^2\cos\theta +(C_7+C_8\cos2\theta)s^2h^3\\ &+C_{10}sh^4\cos\theta+C_{12}h^5 +D_1s^7\cos\theta+\cdots . \end{aligned}\end{split}\]

Equation 5.2

\[\begin{split}\begin{aligned} x'={}&A_1s\sin\theta\\ &+B_1s^3\sin\theta+B_2s^2h\sin2\theta +(B_3+B_4)sh^2\sin\theta\\ &+C_1s^5\sin\theta+C_3s^4h\sin2\theta\\ &+(C_5+C_6\cos^2\theta)s^3h^2\sin\theta +C_9s^2h^3\sin2\theta\\ &+C_{11}sh^4\sin\theta+D_1s^7\sin\theta+\cdots . \end{aligned}\end{split}\]

Sanity checks

The finished equations satisfy several useful checks:

Meridional ray: at \(\theta=0\), every term in \(x'\) vanishes, so a ray in the meridional plane stays in that plane.
Mirror ray: replacing \(\theta\) by \(-\theta\) leaves \(y'\) unchanged and reverses \(x'\), as reflection symmetry requires.
Axial object: with \(h=0\), the intercept is radial:

\[\boldsymbol r' =\left(A_1s+B_1s^3+C_1s^5+D_1s^7+\cdots\right) (\sin\theta,\cos\theta).\]
Order count: every displayed monomial has total degree \(1\), \(3\), \(5\), or \(7\) in \(s\) and \(h\).

Using the polynomial with KrakenOS

KrakenOS traces the exact surface geometry; it does not need this truncated polynomial to propagate a ray. The polynomial is useful as an interpretable fit to a set of traced image-plane intercepts:

Choose signed field samples \(h\) and pupil samples \((s,\theta)\).
In this convention calculate \(\theta=\operatorname{atan2}(p,q)\). The more common \(\operatorname{atan2}(q,p)\) measures from \(x\) and would interchange the sine and cosine factors.
Trace each ray to a common reference image plane and record \((x',y')\).
Build the basis columns displayed in Eq. 5.1 and Eq. 5.2, then solve for the coefficients by linear least squares.
Inspect the residual. A structured residual indicates omitted seventh- or higher-order aberration, pupil distortion, or a broken centred-system assumption.

If \(h\) and \(s\) are normalized to unit field and unit pupil, the fitted coefficients are the full-field/full-pupil transverse contributions in the chosen image-length unit. Changing either normalization rescales the coefficients, so the normalization must always accompany reported values.