On 10/23/2017 03:15 PM,
whburling@outlook.com wrote:
Text books often define the Raleigh Criterea as Sin(theta) = 1.22* Lambda/D
Should not the criterea be, TAN(theta) = 1.22 * Lambda/D
I am suggesting the criterea should be with tan from looking at its derivation from Bessel function
r = 3.832/pi * L*Lambda/D to first point
r = 1.22 * L*Lambda/D
r/L = 1.22 * lambda/D
tan (theta) = r/L = 1.22 * Lambda/D
if i could draw a picture of r and L, that would be great, but I will have to draw it with words. I am assuming r is the distance between the edge of the aires disc to the center (its radius) and the L is the focal length which is the triangle side
which forms a right angle with the radius. the hypotenuse is not directly defined; hence the application of tan and not sin which requires knowing the hypotenuse value.
I appreciate that if the angle is small, then sin(theta) = tan(theta)
some people say sin(theta) can equal theta but I can not see that.
sin (1E-5) = 1.745E-7 degrees
tan (1E-5) = 1.745E-7 degrees
but 1E-5 does not equal 1.245E-7
Please help me understand. I truly am in the space of trying to understand.
Thank you
bil
Sure, no worries. We all start out there.
First of all, the small angle approximation. sin(theta) ~= theta for
small theta, but it has to be in radians, not degrees. Mathematically,
that's because
sin x = x - x**3/3! + x**5/5! -x**7/7! + ....
Thus the relative error in approximationg sin x ~= x is about
(x**3/3!)/x = x**2/6.
Thus for x < 0.1 radian (~5.7 degrees), the relative error is less than
0.17%, which is pretty good for most things.
Next, the Fourier transform relation.
The conjugate variables in 1-D Fourier optics are x/lambda and u =
sin(theta), so all the angular dependences come out in terms of
sin(theta). That is, Fourier optics connects the spatial and angular
domains. The natural place to measure the spatial pattern is on a
plane, and the natural place to measure the angular pattern is on the
inside of a large sphere. If you cut a ping-pong ball in half, it makes
a useful scatterometer for visual observations. (I have several in my
drawer.)
If the Rayleigh pattern were a function of tan(theta), the fringe
spacing would go to zero as you go to +-90 degrees, whereas in real life
the fringe period can't be less than than lambda. What actually
happens, if you use a very narrow pinhole and project the fringes on the
inside of your ping-pong ball is that you'll see the fringes get a bit
broader as you go towards theta = pi/2 rather than narrower, and they
cut off at some lowish order like 5 or 10 or 100 depending on the
diameter of the pinhole.
If you project the pattern on a plane perpendicular to the beam axis,
the fringes get much wider as you go out in angle.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics
160 North State Road #203
Briarcliff Manor NY 10510
hobbs at electrooptical dot net
http://electrooptical.net
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