Association of Technical Opticians

y = semi diameter (diameter/2)
r = radius at the apex
p = aspheric variable*
e = eccentricity
k = conic constant

y^2 = 2*r*x^2 - p*x^2

For a flattening surface
p = 1 - e^2

For a steepening surface
p = 1 / (1 - e^2)

-or-

Using a conic constant
p = 1 + k

R1=front radius of curvature
R2=back radius of curvature
i=image distance
o=object distance
n=index of refraction

Reduction of Sperical Aberration

(R1+R2)/(R1-R2)=[2((n^2)-1)/(n+2)]*[(i+o)/(i-o)]

RA=radial astigmatism
Pt=tangential power
Ps=sagittal power
ang=angle off optical axis

Radial Astigmatism
RA=Pt-Ps

-or-

RA=P*tan^2(ang)

small order aproximation*
RA=P(ang^2)

*small order aproximation measured in radians and below 20 degrees

Pt=power in the tangential meridian
Ps=power in the sagittal meridian
P=power at optical axis
n=index of material
ang=degrees off axis

Tangential Meridian
Pt=P[(2n+sin^2(ang))/(2n*cos^2(ang)]
-or-
Pt=Ps/cos^2(ang)

small angle approxiamation*
Pt=P[1+(4*(ang^2)/3]

Sagittal Meridian
Ps=P[1+(sin^2(ang)/2n)]

small angle aproximation*
Ps=P[1+(ang^2)/3)]

*small angle aproximations use off axis measure in radians less than 20 degrees

Tavg=Average light transmission
n=index of material

Average Transmission
Tavg = [(4*n)/(1+n)^2]^2/[1-(((1-n)/(1+n))^2)^2]