Paraxial Analysis - ThirdOrder

n - Refractive index in the object space of the surface
n' - Refractive index in the image of the surface
cv - Radius of curvature
th - Thickness in the image space of the surface

Ray quantities (real ray heights and angles)

y - Ray height at the surface
u - Ray angle in the object space of the surface
u' - Ray angle in the image space of the surface
i - Angle of incidence
i' - Angle of refraction
d - Object distance
d' - Image distance

Marginal and Chief rays
The marginal ray is the ray that passes through the edge of the stop and is referred to as the "a" ray. All ray quantities referring to the marginal ray will be suffixed with an "a", e.g. y > ya. The chief ray is the ray that passes through the center of the stop and is referred to as the "b" ray. All ray quantities referring to the chief ray will be suffixed with a "b", e.g. y > yb.

Equations

Basic raytrace equations

ynu trace

y' = y + th*u (Transfer between surfaces)
n'*u' = n*u - p*y (Refraction at surface)

yui trace:

y' = y + th*u
i = u + cv*y
u' = u + ( n/n'-1)*i

Auxiliary quantities

i = u + y*cv
i' = u' + y*cv
d = -y/u
d' = -y/u'

Nomenclature

SPH3 - Spherical
CMA3 - Coma
AST3 - Astigmatism
PTZ3 - Petzval Field Curvature
DIS3 - Distortion

Ray specification

(x,y) - Fractional pupil coordinates
(0,h) - Fractional field height

x = r*sin(j) (NOTE: j measured from y axis)

y = r*cos(j)

x²+ y² = r²

Ray aberration (third order)

ey(r,j) = SPH3 * r²* y + CMA3 * h * (r² + 2y²) + (3 * AST3 + PTZ3) * h²* y + DIS3 * h³
= SPH3 * r³* cos(j) + CMA3 * h * (2 + cos(2j)) + (3 * AST3 + PTZ3) * h²* r * cos(j) + DIS3 * h³
ex(r,j) = SPH3 * r²* x + CMA3 * h * 2 * x * y + ( AST3 + PTZ3) * h²* x
= SPH3 * r³* sin(j) + CMA3 * h * r²* sin(2j) + ( AST3 + PTZ3 ) * h²* r * sin(j)

FRED typically evaluates:

ey(r,0) = SPH3 * y³ + CMA3 * h * 3 * y² + (3 * AST3 + PTZ3) * h²* y + DIS3 * h³

ex(r,p/2) = SPH3 * x³ + (AST3 + PTZ3) * h²* x

Total aberration coefficients

g = -1/(2*w') Finite image (w' at final image plane)
= 1/(2*n*y) Infinite image (n,y at final surface)

SPH3 = g * S( SPH3 )_k
CMA3 = g * S( CMA3 )_k
AST3 = g * S( AST3 )_k
PTZ3 = g * S( PTZ3 )_k
DIS3 = g * S( DIS3 )_k

Aberration coefficients for surface i

ha = marginal ray height (ya)
hb = chief ray height (yb)

Sa = (n - n') * (n/n') * ha * (ua' + ia)
Sb = (n - n') * (n / n') * hb * (ub' + ib)
D = (n - n') * (k * cv³+ 8 * d) Aspheric term: k=conic, d=4th order aspheric coeff
LI = Lagrange invariant
SPH3_k = Sa * ia * ia + D * ha⁴
CMA3_k = Sa * ia * ib + D * ha³* hb
AST3_k = Sa * ib * ib + D * ha²* hb²
PTZ3_k = (n - n') / (n * n') * LI²* cv
DIS3_k = Sb * ia * ib + LI * (ub' - ub) * (ub' + ub) + D * ha * hb³