Astronomy 03/403, all 999 6 Lecture, December 999 6 A single vacuum-dielectric interface Practical optical systems always involve lots of vacuum-dielectric interfaces, so we should use the formalism above to obtain the relevant amplitude transmission reflection coefficients his can be done easily with Equations 58 59 by taking medium zero to be vacuum medium to be a dielectric with index n, so that ε Y0, E Y, E µ ε Y0, M Y, M µ (6) (where are related by Snell s law), by taking medium to have zero thickness, so that the characteristic matrix is simply the unit matrix: L 0 M N M O 0 Q P (6) hen for E waves the amplitude transmission reflection coefficients are Y t 0 E Y0 + Y ε + µ (63) Y0 Y re Y0 + Y + ε µ ε µ, (64) while for M waves we get Y0 tm Y0 + Y ε + µ (65) Y Y rm 0 Y0 + Y + ε µ ε µ (66) or normal incidence on nonmagnetic bµ g materials these pairs of equations both reduce to 999 University of Rochester All rights reserved
Astronomy 03/403, all 999 t + n n r + n (67) (68) Among other things these expressions imply a limitation to the quantum efficiency of semiconductor detectors when their bare surfaces are illuminated normally, since in this case the quantum efficiency can be no larger than the (power) transmission of the front surface: some of the incident power is reflected n the argot of optics this is called a reflection loss Most detector semiconductors have rather high index for instance, n 34 40 respectively for silicon germanium, for which the maximum quantum efficiency for bare surfaces is Y η Y t 0 4n + n a f 070 064 (69) Reflection losses can be reduced or eliminated altogether by using thin dielectric layers as antireflection coatings, as you will find in this week s homework (problem ) 6 he abry-perot interferometer Let us return to the plane-parallel dielectric slab of igure 5, for we are now in a position to calculate the spectrum of its transmission easily or simplicity we assume that the medium is nonmagnetic surrounded by vacuum, that light is incident normally; then Y Y ε cos θ n (60) E M πnd πnd κ cos θ, (6) so the characteristic matrix of the slab is M L NM cosκ isin κ / Y iy sinκ cosκ O L QP N M πnd i πnd cos sin n πnd πnd insin cos O Q P (6) Of course Y for the media on either side of the slab, so the amplitude transmission coefficient, given by Equation 58, is t m + m + m + m πnd πnd cos i n+ nk J sin, (63) the (power) transmission is the square modulus of t times the ratio of Ys ( in this case), or 999 University of Rochester All rights reserved
Astronomy 03/403, all 999 τ tt* πnd πnd πnd π + K J nd cos i n sin cos + i n+ K J sin n n 4 πnd πnd π + + K J L + M + K J O nd 4cos n sin n 4 n n Psin 4 + L N M n + n KJ O Q P πnd sin + NM n πnd sin n KJ QP (64) his function has the expected series of transmission maxima: τ whenever the sine term vanishes, which takes place whenever the argument of the sine is an integer multiple of π: πdnt a f (65) πm m 0,,,, which is identical to Equation 54 he dielectric slab is more useful as a filter or a spectrometer if the surfaces are more reflective than the bare dielectric we just considered Note that the bare surface has a power reflectance given from Equation 68 by r n n + K J, (66) so that r a f a f a f a f n+ + n 4n n + n +, (67) r r n n + a f (68) n + n 4n n We can put this in Equation 68 to obtain τ + r r πnd sin K J (69) Note that the interference is determined by the phase lags from path lengths through the slab, which do not depend upon the values of reflectivity at the surfaces hus Equation 69 is more general than Equation 64, because the surface reflectivity can be independent of the refractive index: for example, by coating it with a reflective metal or dielectric film t is often written as 999 University of Rochester 3 All rights reserved
Astronomy 03/403, all 999 τ + sin φ, (60) where φ 4πnd / κnd, where r r K J (6) is called the coefficient of finesse he width of the transmission maxima varies with, larger values leading to narrower peaks he function in Equation 60 is called the Airy function, again in honor of the former Astronomer Royal of Engl ( not to be confused with the Airy disk, or the other Airy function, the latter describing the supernumerary arcs of a rainbow) t is plotted in igure 6 ransmission, τ 08 06 04 r 005 r 050 0 0 0 4 6 8 0 4 φ / r 095 igure 6: the Airy function, the transmission of the abry-perot interferometer t is of course possible to perform this derivation for obliquely-incident waves, but not as quickly as this, so we will just give the answer: it turns out that the only change necessary is in the argument of the sine in the denominator of Equation 60 (cf Equation 54) 4πnd φ κnd cos θ (6) 63 abry-perot spectrometers or large values of the coefficient of finesse, the transmission between peaks is very small, which suggests that isolating a single peak is a good way to make a narrow-b (high-resolution) filter urthermore, if the reflecting surfaces are placed on independently movable substrates, with a non-solid dielectric 999 University of Rochester 4 All rights reserved
Astronomy 03/403, all 999 medium in between, the wavelength at which the peak occurs is tunable his is the basis of the abry- Perot spectrometer, a class of astronomical instruments that is good for observation of images of astronomical objects in spectral-line light Let s compute the resolution of such a filter by calculating the wavelength interval corresponding to the half-peak-transmission points n terms of φ, these are found by setting φ/ + sin, (63) φ / or sin ± (64) f the peaks are very narrow (that is, if is large), φ is very close to πm, the small-angle approximation may be used: φ / ± (65) b g he relation between a small interval of φ a small interval of wavelength, δ φ δ, is thus governed by b g δ φ b g d d 4πnd π θ φδ δ H G 4 nd cos K J d d φ δ πm δ, δ (66) so from which we get δ φ π δ / d / i± m, (67) WHM δ/ + δ/ (68) πm mq Here we have defined the finesse of the abry-perot interferometer, Q π / n these terms the (reflectance-limited) spectral resolution of the interferometer is K J WHM, r 999 University of Rochester 5 All rights reserved mq (69) Note that the spectral resolution gets smaller (improves, in the spectroscopic sense) if the finesse Q or the order number m increases he isolation of a high-order peak of the Airy function is illustrated in igure 6 f one is interested in observations at a certain wavelength 0, say because there is a spectral line there, one can use one abry- Perot set in a low order there, in series with a higher-order one A wide-b filter of the type discussed in
Astronomy 03/403, all 999 5 can be used to reject the higher orders of the low-order abry-perot; this interferometer, if narrow enough, rejects all but one order of the high-order abry-perot he abry-perot orders can be tuned by adjustment of d (or n) to scan the detector s response across the spectral line An example of this concept is explored in extra-credit Problem #7 in this week s homework High-order abry-perot interferometer ransmission Low-order abry-perot interferometer Long-wavelength-pass broadb filter Scan System (product of three above) 0 Wavelength igure 6: ransmission of a tem abry-perot spectrometer Because φ depends upon, reflectance isn t the only thing that can limit the resolution n Problem 4 of this week s homework, you will show that is a abry-perot is used in a diverging or converging beam of angular radius θ, the resolution can be no better than K J WHM, θ o keep beam divergence or convergence from spreading the beam unnecessarily, we want or b θ K J << θ H G K J WHM, WHM, r θ (630), (63) << mq (63) or high resolution / >000g, this usually means that the high-order abry-perot needs to be illuminated with a collimated beam, in order to keep θ as small as possible 999 University of Rochester 6 All rights reserved