The Maksutov Approximation
A simple lens has inverse image distance:
1/q = ( 1/p + (n-1) / r1
) / ( 1 - (t/n) * ( 1/p + (n-1) / r1 ) ) - (n-1) / r2
where n is the refractive index, t is the lens thickness, r1
and r2 are the first and second surface radii, respectively,
and p is the object distance.
Differentiating with respect to refractive index:
d(1/q)/dn = [ 1 / r1
- ( t / n2 ) * ( 1/p + (n-1) / r1 )2
] / [ 1 - ( t / n ) * ( 1/p + (n-1) / r1 ) ]2 -
1 / r2
Achromatism occurs when this derivative is zero:
1 / r2 = [
1 / r1
- ( t / n2 ) * ( 1/p + (n-1) / r1 )2
] / [ 1 - ( t / n ) * ( 1/p + (n-1) / r1 ) ]2
Or:
r1 - r2
= (t*r1/n2) *
( 1/p + (n-1) / r1 ) *
( 2 * n - ( r2 + t ) * ( 1/p + (n-1) / r1 ) )
The simple case with infinite object distance ( 1/p is zero ) is
solved here:
r1 - r2
= (n-1)*t/n2 *
( n+1 - (n-1)*( r2 + t - r1 ) / r1 )
The hypothesis here is that the term:
(n-1)*( r2 + t - r1
) / r1
may be ignored. If so, then the approximation is:
r1 - r2
=
(n-1)*t/n2 * ( n + 1 ) =
( (n2-1)/n2 ) * t
However, suppose the approximation is used. Then the ignored
term is:
(n-1)*(t - ( (n2-1)/n2
) * t) / r1
Or:
( (n-1)/n2) * (t / r1)
As long as this term is small relative to n+1, the approximation is
valid.
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