Waves on a spherical refracting surface
Suppose that light is incident parallel to the optic axis
The ray coming in along the optic axis is not deflected. The ray incident at a height y above the optic axis bends as shown.
With air on the left side and glass on the right we have
s in sin n , o r fo r sm all an g les = n
y
R
y
f n
fy y
y
R
y
nR R nR
R
n
nR
n
, ,
11 1
11 1
y
R
y
f n
fy y
y
R
y
nR R nR
R
n
nR
n
, ,
11 1
11 1
This is independent of y, so all incident rays that hit the spherical surface parallel to the optic axis are refracted through F
The refracted ray intersects the optic axis at some point H
Solve for y’ and the distance VG
VH = VC + CG + GH
=
y
y y y
VH = VC + CG + GH
=
y
y y y
1 1 1 1Solve for
y y
y y
VH = VC + CG + GH
=
y
y y y
1 1 1 1Solve for
y y
y y
1 1
1 1
y
y
Now,
1and 1
n n
n n
Now,
1and 1
n n
n n
1 1so that
11 111
n nn
nn
Now,
1and 1
n n
n n
1 1so that
11 111
n nn
nn
1 1independent of y!
1 / 1 1
y Ry
y n y R n n
and we have
Now,
1and 1
n n
n n
1 1so that
11 111
n nn
nn
1 1independent of y!
1 / 1 1
y Ry
y n y R n n
Finally, the image is located at
yVG = +CG = +
1 1Therefore,G F, the focal point
R nRR R R f
n n
and we have
All the light is focused at one point, lying on a plane perpendicular to the optic axis, the so-called focal plane.
Recall that
n
so that
n n
nn 1
From the geometry,
y
s
y
R
y
so i
; ; 1 1
s
n
s
n
Ro i
When the external medium has a refractive index n1, the corresponding formula is
n
s
n
s
n n
Ro i
1 2 2 1
n1
n2