Our analysis is mostly concerned with the amplitudes for the interfering beams. We begin, however, with understanding the path difference between the beams, and beams 1 and 2 in particular. For a normal incidence ( a = 0) the path difference equals twice the optical thickness that depends on the physical thickness of the plate d and the index of refraction of the material h (ray 2 crosses the plate twice): > Gamma:=2*eta*d; For oblique incidence one needs a little trigonometry. The length that ray2 has traversed inside the plate is given as two times d /cos( b ), to turn it into an optical path length we multiply by h . Thus the delay for ray2 is given as > ray2:=2*eta*d/cos(beta); On the other hand the drawing of the wavefront for rays 1/2 (green dashed line) shows that ray1 is delayed by a geometric distance (we use for air h _a=1): > ray1:=2*sin(alpha)*d*tan(beta); Thus we have a path delay for ray2 of > Gamma:=ray2-ray1; Snell's law relates a and b through h . > Snell:=sin(alpha)=eta*sin(beta); > Gamma:=subs(sin(alpha)=rhs(Snell),Gamma); > Gamma:=simplify(Gamma); Note that there is an analogy to Bragg scattering in this expression. We are now interested in the case of monochromatic light of a given wavelength l . If the optical path difference G equals an integral multiple of the wavelength, we can have constructive or destructive interference. Note that the inclusion of a phase shift by p in one ray will switch constructive and destructive interference between the two beams. Also note that for a F-P cavity the phase changes occur for the internal reflections, and that a wave passing though directly is in phase with a wave that undergoes internal reflections, since the phase changes 'cancel', i.e., for each internal reflection pair the two changes by p add up to 2 p . That means that for observing transmission through a F-P cavity, one can ignore any phase changes, while one has to worry about them for observing reflected interfering beams, since the first ray reflected at the interface 'backsilvered mirror - cavity' experiences no phase change, and is therefore different from the ones that have undergone internal reflections. This is in analogy to the plate discussed above. See diagram below. > CI:=Gamma=m*lambda; > DI:=Gamma=(m-1/2)*lambda; The conditions are satisfied for m =1,2,3,... For the Fabry-Perot cavity shown in the diagram above one has to take into account that the internal angle of incidence b that appears in the interference conditions is replaced by the angle a, i.e., the roles of a and b have been reversed. This is to account for the fact that the light rays are incident with a on the glass plate, refract to the angle b , at which point the parallel plate interference analysis is applied. When the transmitted waves exit the second glass plate (not shown in the diagram above), they refract from b back to a . The medium that is responsible for the optical path length difference is air, i.e., one has to set h =1. However, one wonders whether the index of refraction for the glass shouldn't come into play, since the non-reflecting beam is traversing an optical path length in the glass before meeting the wavefront that exits the cavity after multiple reflections. The answer to this worry is that the two interfering beams pass through exactly the same amount of glass, and that one should compare their optical paths outside the second glass plate as shown in the diagram below: Note that the interference conditions for the rays 1 and 2 passing through the F-P cavity are also met for rays 2 and 3, 3 and 4, etc. How they add up to form a pattern based on the actual amplitudes of the waves, which depend on the reflectivity and transmissivity of the coatings will be discussed below using the parallel plate example. The resolvance of a Fabry-Perot interferometer is a measure of how a change in wavelength translates into a change in the angle for the interference condition. Thus, we just need to differentiate the CI condition. We carry this out for the parallel plate, and define a mapping for lambda. Note that we should really have expressed the CI in terms of a , since that is the parameter that one can control, but we note that a and b are simply related by Snell's law. > lambda:=unapply(solve(CI,lambda),beta); > dlda:=diff(lambda(beta),beta); If we consider incident rays near the normal, a (and thus b ) is small, and one can replace sin( b ) by b . Using Snell's law > Snell; > dlda:=subs(sin(beta)=alpha/eta,dlda); we find that d l /d a =-2 d / m a . (valid for small a only) This is an interesting relationship. It states when expressed for d a as a function of d l that the resolution d a increases with the order m , and that the spacing d is playing an important role as well. We can increase d a by reducing d (check this out?!) For small incident angles the CI condition can be Taylor expanded > CI1:=convert(taylor(lhs(CI),beta=0,3),polynom)=rhs(CI); > CI2:=subs(beta=alpha/eta,CI1); We are now interested in the spacing between constructive interference fringes. Suppose a bright fringe appears at a =0 with order m . The order m -1 forms a CI at a . > CI3:=subs(alpha=0,lhs(CI2))-lhs(CI2)=simplify(rhs(CI2)-subs(m=m-1,rhs(CI2))); > eq_dlda:=dl=solve(dl/da=dlda,dl); > resolv:=lhs(eq_dlda)/rhs(CI3)=rhs(eq_dlda)/lhs(CI3); The phase difference is given as > delta:=2*Pi*Gamma/lambda; We assume an idealization, namely that there are no losses to the transmitted/reflected light. How much amplitude is present in the 1,2,3,... and 1',2',3',... rays? We assume that there are no losses, and that the two surfaces are controlled by the same reflection and transmission coefficients that add to unity, r^2 + t^2 = 1. We take an incident amplitude E . After a transmission/reflection split we have for the conserved energy of the wave: E ^2 = ( E r )^2 + ( E t )^2. This implies that the amplitudes are E r and E t respectively. It is important to realize that for each process (reflection or transmission) the amplitude is multiplied by a corresponding factor. In particular, it should become clear that the multiply reflected wave gets weaker in amplitude, since at each contact with the interface it is split

 
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