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

yR

The refracted ray intersects the optic axis at some point H

n

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