Optical Coatings

For symmetric lenses (r 2 = 4r 1 ),
AH = 4A H′′
1 2
AH = 0
1
=
and
t c
A2H ′′ = 4 .
n
rt 1 c
.
2nr 4 t (n 4 1)
1 c
⎧
⎪ f ⎡1
(n 1)
HH ′′ = t c ⎨1
4 ⎢ 4 4
n f n
⎣⎢
⎩⎪
⎛ 1 ⎞
HH ′′ = t c ⎜1
4 ⎟ .
⎝ n ⎠
Q = 2 p(1 4cos v)
⎛ v ⎞
2
= 4 p sin ⎜
⎝ 2
⎟
⎠
⎛ f ⎞
v = arctan ⎜
⎝ 2 s′′
⎟
⎠
2
t ⎤⎫
c ⎪
⎥⎬
rr 1 2⎦⎥
⎭⎪
(1.44)
If either r 1 or r 2 is infinite, l’Hôpital’s rule from calculus must be used.
Thus, referring to page 1.27, for plano-convex lenses in the correct
orientation,
(1.45)
For flat plates, by letting r 1 →∞ in a symmetric lens, we obtain
A 1 H = A 2 H″ = t c /2n. These results are useful in connection with
the following paraxial lens combination formulas.
Hiatus or Interstitium (principal-point separation)
(1.46)
which, in the thin-lens approximation (exact for plano lenses),
becomes
(1.47)
Solid Angle
The solid angle subtended by a lens, for an observer situated at an
on-axis image point, is
where this result is in steradians, and where
(1.48)
is the apparent angular radius of the lens clear aperture. For an
observer at an on-axis object point, use s instead of s″. To convert
from steradians to the more intuitive sphere units, simply divide
Q by 4p. If the Abbé sine condition is known to apply, ß may
be calculated using the arc sine function instead of the arc
tangent.
Back Focal Length
and
f = f" + AH′′
b 2
= f " 4
t c
f b = f " 4
n .
f = f 4 AH
f 1
= f +
rt 2 c
n(r 4 r ) + t (n 4 1)
2 1 c
rt 1 c
n(r 4 r ) + t (n 4 1)
2 1 c
m = s ′′
s
f
=
s 4 f
= s ′′ 4 f .
f
PARAXIAL FORMULAS FOR
LENSES IN ARBITRARY MEDIA
(1.49)
where the sign convention presented above applies to A 2 H″ and to
the radii. If r 2 is infinite, l’Hôpital’s rule from calculus must be used,
whereby
Front Focal Length
(1.50)
(1.50)
(1.51)
where the sign convention presented above applies to A 1 H and to
the radii. If r 1 is infinite, l’Hôpital’s rule from calculus must be used,
whereby
t c
f f = f 4
n .
(1.52)
Edge-to-Focus Distances
For positive lenses,
A = f + s
f 1
B = f + s
b 2
(1.53)
(1.54)
where s 1 and s 2 are the sagittas of the first and second surfaces.
Bevel is neglected.
Magnification or Conjugate Ratio
(1.55)
These formulas allow for the possibility of distinct and completely
arbitrary refractive indices for the object space medium (refractive
index n′), lens (refractive index n″), and image space medium (refractive
index n). In this situation, the effective focal length assumes two
distinct values, namely f in object space and f″ in image space. It is
also necessary to distinguish the principal points from the nodal
points. The lens serves both as a lens and as a window separating
the object space and image space media.
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