The smallest focal ratio for a camera lens is not 0.5. You CAN make a
lens with a f number less than 0.5.
Just to clarify, the F number of a lens is:
f number = f/D
where f is the focal length and D is the lens diameter.
The reason why someone may have determined a limit of 0.5 is probably
due to the fact that the index of refraction of glass (which most
lenses are made out of) is 1.5.
The focal length of a lens can be approximated by an equation called
the lensmaker's equation. It is as follows:
1/f=(n2-n1)/n1*(1/R1-1/R2)
where f = the focal length, n2 is the index of refraction of glass, n1
is the index of the surrounding medium (for air n=1). R1 and R2 are
the radii of curvature of the front and back surface of the lens. You
should be able to find a diagram of this in an optics book or on the
web somewhere.
Now if we calculate the lensmaker's equation for glass with an index
of 1.5 and surrounded by air, it becomes:
1/f=(1.5-1)/1*(1/R1-1/R2) = 0.5*(1/R1-1/R2)
The largest we can make R1 and R2 is 1/2 of the diameter of the lens
(in which case the lens is a sphere) so R1=D/2 and R2=-D/2. The
lensmaker's equation then becomes:
1/f=0.5*(2/D+2/D)=0.5*(4/D)=2/D
or
1/f=2/D
if we rearrange this equation, we see that:
f number = f/D = 0.5
Now we see why someone may have said that the minimum f number is 0.5.
This is true for a glass lens with an index of 1.5.
However, there are other transparent materials you could use besides
glass that have a higher index of refraction.
For example, quartz has an index of 1.644, sapphire has an index of
1.77, and diamond has an index of 2.417.
All of these materials (and some others too) would be capable of
making a lens with an f number less than 0.5.
As an example, consider a lens made of diamond where the front half of
the lens (facing toward the object) is a half of a sphere (R1=D/2) and
the back half of the lens (facing towards the film) is somewhat less
curved (R2=-D)
evaluating the lensmaker's equation again
1/f=(2.417-1)/1*(2/D+1/D)
1/f=1.417*3/D=4.251/D
f number = f/D = 0.235 |