Seidel Aberrations

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General Info:

To model optical systems we will need both light sources and optical components.  

The optical (paraxial) axis is the x-axis. The y-axis is horizontal and the z-axis is vertical.

First we’ll consider light sources. To visualize the various sources we’ll utilize the functions TurboPlot and Screen. 

The units are numerical. That is, you decide the units (meters, centimeters, millimeters, etc.).

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?LineOfRays
Options[LineOfRays]

Zernike Polynomials and Seidel Aberrations

Geometrical (Ray) Optics model:

If the interface between two optical media (e.g., air and glass) is hyperbolic or elliptic, light can be focused or collimated perfectly without regard to possibly steep angles, with respect to (wrt) the optical axis, some of the component light rays might have (see Hecht - Optics Ch. 5).

However, this perfect scenario has two drawbacks. 
1) We generally don’t want the light to stay within the second medium.
and 
2) These surfaces are hard (and expensive) to make.

For these reasons optics with spherical surfaces are predominantly used.

The paraxial approximation: The light rays impinging on an optical system are assumed to make shallow angles wrt the optical axis (so that small angle approximations may be employed in analyses).

Spherical surfaces, in general, will yield diffraction-limited (as good as it gets) imagery only in the paraxial region.

Departures from the idealized conditions of the paraxial approximation (i.e., rays that make steep angles with respect to the optical axis) lead to geometrical aberrations.

Planes of interest:



Seidel aberrations (Ludwig von Seidel, 1821-1896) are the so-called five primary aberrations (spherical, coma, astigmatism, field curvature and distortion).  Spherical aberration, coma and astigmatism deteriorate the image making it unclear while field curvature and distortion deform the image.

Seidel used algebraic power-series analysis of the higher (than second) order terms of an expanded characteristic (aberration) function Φ of an optical system to study off-axis distances (of rays).  Φ describes the deformation of the wavefronts in the region of the system’s exit pupil.
                      

For a systematic treatment see Principles of Optics, Born and Wolf, Ch. 5.

A Somewhat Detailed and Simplified Exposition of Geometric Aberration Theory:

Consider the spherical interface of two optical media with indices of refraction (Subscript[n, 1]) and (Subscript[n, 2]) where S (P) is the source (image) point, V is called the surface vertex and A is where the ray intersects the spherical interface. The length (OverBar[SV]) ((OverBar[VP])) is the object (image) distance.



We will need Fermat’s Principle: “The actual path between two points taken by a beam of light is the one that is traversed in the least time”. 
In modern form Fermat’s principle is, “A light ray, in going from point S to point P must traverse an optical path length (OPL) that is stationary wrt variations of that path.”

Consider the ray originating at S, traveling to A, undergoing refraction and then crossing the optical axis at P having OPL = (Subscript[n, 1])(Subscript[l, o]) +(Subscript[n, 2])(Subscript[l, i]). Employing the law of cosines in triangles SAC and ACP  

we can write OPL = (Subscript[n, 1]) [R^2 + ((Subscript[s, o]) + R)^2  -  2R((Subscript[s, o]) + R) cos(ϕ)]^(1/2) +(Subscript[n, 2])[R^2 + ((Subscript[s, i]) - R)^2  +  2R((Subscript[s, i]) - R) cos(ϕ)]^(1/2).

Since ϕ is the position variable (R = constant) Fermat’s principle, ((d(OPL))/(dϕ)) = 0, yields

((Subscript[n, 1])/(Subscript[l, o])) + ((Subscript[n, 2])/(Subscript[l, i])) = (1/R)( (((Subscript[n, 2]) (Subscript[s, i]))/(Subscript[l, i])) - (((Subscript[n, 1]) (Subscript[s, o]))/(Subscript[l, o]))). This relationship holds between the parameters of a ray going from S to P by way of A on a spherical interface This equation is exact but complicated. We would like to somehow get rid of (Subscript[l, o]) and (Subscript[l, i]). If we assume that A  is very close to V then we can use small angle approximations (either cos(ϕ) ≈ 1 or sin(ϕ) ≈ ϕ) so that (Subscript[l, o]) ((Subscript[l, i])) ≈ (Subscript[s, o]) ((Subscript[s, i])) and 

((Subscript[n, 1])/(Subscript[s, o])) + ((Subscript[n, 2])/(Subscript[s, i])) = (((Subscript[n, 2]) - (Subscript[n, 1]))/R) results.

This approximation results in what is known as (1^st) order theory. It is also known as paraxial or Gaussian Optics.

Now we are in a position to outline the theory of geometrical aberrations.

The theory outlined above could gave been formulated not from Fermat’s principle but from Snell’s law. When this is done the resulting equations have sines in them rather than cosines.

Since we know that the small angle approximation is valid only over a small region where A ≈ V or sin(ϕ) ≈ ϕ, we could improve the theory a bit to include rays that make steeper angles wrt to optical axis by keeping the next term in the power-series expansion of sin(ϕ). That is,

sin(ϕ) ≈ 1 - ((ϕ^3)/(3!)). This is called (3^rd) order theory (Seidel in the 1850s). Departures from the (1^st) order theory are embodied in the five primary Seidel aberrations.

For example, if the approximations for (Subscript[l, o]) and (Subscript[l, i]) are improved by keeping the (3^rd) order terms in the expansion a third order expression

((Subscript[n, 1])/(Subscript[s, o])) + ((Subscript[n, 2])/(Subscript[s, i])) = (((Subscript[n, 2]) - (Subscript[n, 1]))/R) + (h^2) [((Subscript[n, 1])/(2 (Subscript[s, o]))) ((((1/(Subscript[s, o]))+ (1/R)))^2) + ((Subscript[n, 2])/(2 (Subscript[s, i])))((((1/R) - (1/(Subscript[s, i]))))^2)] results. The additional term in the right hand side is clearly a measure of the deviation from the first-order theory above. In particular, it corresponds to a dependence of focal length on aperture (of radius h) for non-paraxial rays (spherical aberration).

Spherical aberration: An effect that occurs due to the variation of refraction (or reflection) of light rays depending on the distance from the optical axis they strike a spherical optical component.


Comatic aberration: An effect that results in off-axis point sources such as stars appearing distorted, appearing to have a tail (coma) like a comet. Specifically, coma is defined as a variation in magnification over the entrance pupil. This results from the fact that the principle “planes” are planes only within the paraxial regime. In general, they are principle curved surfaces.


Astigmatic aberration: An effect where rays that propagate in two perpendicular planes have different foci.  


Field Curvature: An effect in which a flat object normal to the optical axis cannot be brought into focus on a flat image plane.


Distortion: An effect described by a deviation from rectilinear projection, a projection in which straight lines in a scene remain straight in an image.

Diffraction Theory of Aberrations:
Since the above results are based on the geometrical (ray) optics model they can be expected to be valid only in the limit very short wavelengths. In fact, the distribution of light in the focal plane has (experimentally) been observed to be of a much more complicated nature than suggested by geometrical optics.

Therefore, the study of aberrations on the basis of diffraction theory is a logical next step.

In this treatment, integrals of the expanded wavefront aberration function Φ are carried out over the unit circle of the exit pupil to calculate the normalized intensity of a point P in the image region. It is found that the intensity at point P results from the (0^th) order “Gaussian” term plus a series of aberration terms.  It makes sense to expand Φ in an appropriate complete set of orthogonal (polynomial) functions. The Zernike polynomial function are employed for this purpose and give a particularly simple form to the aberration terms in the expansion.  For a detailed treatment see Principles of Optics, Born and Wolf, Ch. 9.

Zernike polynomials are sequences of orthogonal polynomials of the form

(Subscript[Z, n]^m)(ρ,ϕ) = (Subscript[R, n]^m)(ρ) cos(m ϕ) (even)
and
(Subscript[Z, n]^(-m))(ρ,ϕ) = (Subscript[R, n]^m)(ρ) sin(m ϕ) (odd)
where 
(Subscript[R, n]^m) = (∑_(k=0)^((n-m)/2))(((((-1))^k)(n-k)!)/(k! ((n+m)/2-k)! ((n-m)/2-k)!))(ρ^(n-2k))

and are used to characterize errors observed in optical imaging systems.

A mapping from (Subscript[Z, n]^m) ⟶ (Subscript[Z, j]) can be assigned (R.J. Noll).

The (Subscript[Z, j]) are associated with optical aberrations.

The first few Zernike polynomials are associated with:
1) Piston, 			 equal to the mean value of the wavefront
2) X-Tilt,	 		 the deviation of the overall beam in the sagittal (horizontal) plane
3) Y-Tilt,			 the deviation of the overall beam in the tangential (vertical) plane
4) Defocus, 			 a parabolic wavefront resulting from being out of focus
5) ° Astigmatism,		 a cylindrical shape along the X or Y axis
6) 45° Astigmatism, 		 a cylindrical shape oriented at ±45° from the X axis
7) X-Coma, 			 comatic image flaring (variation in magnification) in the horizontal direction
8) Y-Coma, 			 comatic image flaring (variation in magnification) in the vertical direction
9) Third order spherical aberration

?SeidelAberrations
?ResidualFitError

Let’s consider a point source with a plano-convex spherical lens and look at the Seidel aberrations when we change the diameter of the source at the lens.

planoconspher = {GaussianBeam[3,0,NumberOfRays->15,FullForm->True],Move[PlanoConvexLens[40,35,15],75,0], Move[Screen[25],160]};
AnalyzeSystem[planoconspher, Axes->True];
FindFocus[planoconspher];
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planoconspher = {GaussianBeam[3,0,NumberOfRays->15,FullForm->True],Move[PlanoConvexLens[40,25,5],75,0], Move[Screen[25],119.936]};
ZernikeFit[ReadTurboRays[TurboTrace[planoconspher, ReportedSurfaces->Last],{SurfaceCoordinates,OpticalLength}]];
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