What Causes the Sagnac Effect and How to Calculate Reduction in Light Intensity?

[MathematicalPhysicist]

in this webpage: http://www.mathpages.com/rr/s2-07/2-07.htm we have an explanation to the saganc effect and in page 2 and 3 i got stumbled:
i can't get to the quadratic equation in page 3 i.e:
[c^2-R^2*w^2*cos(theta)]*T^2+-[2R^2*w*sin(theta)]*T-2R^2[1-cos(thata)]=0
from this:
c^2*T^2=2R^2*[1-cos(w*T)*cos(theta)+-sin(wT)sin(theta)]
in page 2.

i tried to use limits because in the paper the said that w*T is extremely small, but what i got is
c^2*T^2=2R^2*[1-cos(theta)]
(i took limits to both the sin and cos functions with w*T as the angle), but as you see it doesn't match, then what have idone wrong?

 

[MathematicalPhysicist]

no one knows?

 

[MathematicalPhysicist]

for those too much lazy or don't understnad the equations i wrote here they are in the attachment.

 

[MathematicalPhysicist]

sorry but the other picture is too much big (anyway you can see it in the link i first gave i know ).

 

[DW]

in this webpage: http://www.mathpages.com/rr/s2-07/2-07.htm we have an explanation to the saganc effect and in page 2 and 3 i got stumbled:
i can't get to the quadratic equation in page 3 i.e:
[c^2-R^2*w^2*cos(theta)]*T^2+-[2R^2*w*sin(theta)]*T-2R^2[1-cos(thata)]=0
from this:
c^2*T^2=2R^2*[1-cos(w*T)*cos(theta)+-sin(wT)sin(theta)]
in page 2.

i tried to use limits because in the paper the said that w*T is extremely small, but what i got is
c^2*T^2=2R^2*[1-cos(theta)]
(i took limits to both the sin and cos functions with w*T as the angle), but as you see it doesn't match, then what have idone wrong?

Its just math. Perhaps no one cares, but why assume no one knows? That is "your" speculation. Try again, but this time as it said, keep powers of omega*T up to second order, NOT JUST FIRST ORDER.

 

[Chronos]

w is the angular velocity of the device and T is the time required for the light pulse to travel from one mirror to the other in the foward and reverse direction. w*T therefore is a measure of distance, not angle. You need to plug in the radius of the ring to derive the angle of rotation, which is microscopic.

 

[MathematicalPhysicist]

Its just math. Perhaps no one cares, but why assume no one knows? That is "your" speculation. Try again, but this time as it said, keep powers of omega*T up to second order, NOT JUST FIRST ORDER.

i hoped that with asking: "no one knows?" some users will be incentived to answer, anyway i don't understand neither your answer nor chronos answer.
when you mean omega*T up to second order do you mean that the deravartive of the cos and sin of w*t will be of 2degree?

and to chronos when you multiply omega with T you get the unit of rad which can be translated to degrees with which you measure the length of the arc.
anyway chronos, when you mean the radius of the ring are you referring to capital R in this paper?

 

[DW]

when you mean omega*T up to second order do you mean that the deravartive of the cos and sin of w*t will be of 2degree?

No, I mean that you have to carry out the taylor series expansion of the cos function to second order in omega*t, not just to first. The sin function's second order term is zero so even though you are really to carry them both out to second order be sure to pick up the second order term from the cos expansion.
In other words use:
$$sin(\omega T) \approx \omega T$$
and
$$cos(\omega T) \approx 1 - \frac{\omega ^{2}T^{2}}{2}$$

 

[MathematicalPhysicist]

No, I mean that you have to carry out the taylor series expansion of the cos function to second order in omega*t, not just to first. The sign function's second order term is zero so even though you are really to carry them both out to second order be sure to pick up the second order term from the cos expansion.
In other words use:
$$sin(\omega T) \approx \omega T$$
and
$$cos(\omega T) \approx 1 - \frac{\omega ^{2}T^{2}}{2}$$

now that's more clear, thanks.

 

[bshweta]

can anyone answer this question
In Sagnac interferometer replace the beam splitter by a fourth mirror and calculate how many loops around the interferometer can the light make before its intensity is reduced by 99% ?