Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007

Transmission through a Fabry–Perot depending on separation of plates D for three different reflection coefficients (three different g), m 1, and wavelength

λ 0.1.

I16FABRYS

Fabry–Perot Transmission Depending on D

Normal incidence. Parameters: reflection coefficient, wavelength λ, refractive index. See for global definition. The finesse πg/2 is λ/ λ. All lengths in mm.

Application 2.16.

1.How do the positions of maxima depend on D and λ?

2.Give a formula for the separation of fringes and verify with the data from the graph.

118 2. INTERFERENCE

3.Condsider dependence on θ for θ equal 0.001 to .7 for constant λ .001 and D .02. Make a graph and derive for the fringe number f (θ) (2/y)(Dn2 cos θ). Plot f (θ) and observe that the highest number corresponds to the smallest angle. This is contrary to Young’s experiment; see also FileFig 2.13.

FileFig 2.17 (I17FABRYLS)

Transmission through a Fabry–Perot depending on wavelength λ for three different reflection coefficients r (three different g), m 1, and thickness

D 0.0025.

I17FABRYLS is only on the CD.

Application 2.17. The bandwidth of a peak is the width at half-height, given by bw 2λ/πg. Calculate two bandwidths, bw1 and bw2, with ratio bw1/bw2 5 and make a graph. Verify the data by reading the bw values from the graph.

2.6.3 Fabry–Perot Spectrometer and Resolution

A Fabry–Perot etalon may be used as a spectroscopic device when varying the spacing between the two reflecting surfaces over a small interval. The result is that the first-order resonance wavelength λ0 2D0 is varied around a wavelength interval and therefore we have for different Di the resonance maximum, corresponding to λi . Scanning D gives us a maximum depending on λ and we get the spectral distribution of the incident signal, as one may measure with a grating spectrometer. Two spectral lines of wavelength difference λ may be seen separated or not, depending on the resolution of the Fabry–Perot spectrometer. To calculate the resolution, that is, the wavelength difference λ λ2 − λ1 of the spectral lines of wavelength λ1 and λ2, we assume λ2 > λ1 (Figure 2.20a). The two resonance lines are considered resolved when the crossing of the “right” side of one line with the “left side” of the other line has the value 21 . This condition may be expressed, using Eqs. (2.91) and (2.92), as

FIGURE 2.20 (a) Spectral lines of wavelength λ1 at resonance distance D − and λ2 at resonance distance D + ; (b) one spectral line at resonance distance λ 2D with half-height at D + .

To obtain the value of (we look at a single line at resonance (Figure 2.20b), and get D λ/2. The value of is obtained from one line at half-height

We use the sum formula for sin(a − b) sin(a) cos(b)− cos(a) sin(b) and obtain with sin(2πD/λ) 0 and cos(2πD/λ) −1,

In first order we have for the resonance distance D λ/2, and obtain for λ/ λ πg/2. As mentioned above, πg/2 is called the finesse F . It characterizes the resolving power of the Fabry–Perot spectrometer. Considering the order m (integer) for the resonance distance of λ, that is, D mλ/2, we have more generally for the resolving power λ/ λ

It depends only on the reflection coefficient of the single plate. The inverse, λλ is called the resolution.

120 2. INTERFERENCE

FIGURE 2.21 A Fabry–Perot ring pattern obtained using the green line of the mercury spectrum (from Cagnet, Francon, and Thrierr, Atlas of Optical Phenomena, Springer-Verlag, Heidelberg, 1962).

In FileFig 2.18 we show a graph of the transmitted intensity of two different wavelengths λ1 and λ2 for several orders, using the same value of the finesse. One observes that the resolution may be changed when changing the reflectivity r. In FileFig 2.19 we have calculated for two wavelengths the transmission through a Fabry–Perot, depending on very small angles with respect to the normal. In Figure 2.21 a photo is shown of a ring pattern produced with a Fabry–Perot for the green Hg-line in the visible spectral region.

FileFig 2.18 (I18FABRYRDS)

Graph of Fabry–Perot resonances of two wavelengths λ1 and λ2 depending on separation D of the plates.

I18FABRYRDS is only on the CD.

Application 2.18.

1.Make three choices of λ2 and determine the reflection coefficient r to have them resolved in first-order.

2.Choose λ1 and λ2 and determine the reflection coefficient r2 to have the lines resolved in second-order, and reflection coefficient r3 for the third-order.

3.Introduce λ2 λ1 + λ1 λ1(1 + 2/mπg) and make changes to r so that lines are separated for the first-, second-, and third-order.

122 2. INTERFERENCE

FIGURE 2.22 (a) Waves u1 to uN have their origin at source points 1 to N. The source points are spaced by a. The waves travel in the x direction and have the angle θ with respect to the axis of the setup; (b) the optical path difference between adjacent waves is δ a sin θ, or in the small angle approximation, aY/X.

Similar to the discussion in the section on intensities, we introduce complex notation and write

We note that unlike the case of the plane parallel plate, we can not ignore the Nth power in the summation formula. The expression in brackets of Eq. (2.109) can be rewritten for the numerator as

exp i[2π(−Nδ/2λ)]{exp i[2π(Nδ/2λ)] − exp i[2π(−Nδ/2λ)]}

Substituting δ a sin θ and taking A2 1/N2 for normalization, we can write I as

Equations (2.116) and (2.117) have their main maxima when both numerator and denominator are zero. This is shown in the first graph of FileFig 2.20, where numerator y(θ) and denominator y1(θ) are plotted separately. One observes between the main maxima N − 2 side maxima and N − 1 side minima. From the trace of the numerator one sees that two of the side maxima do not appear. They are at the flank of the main maxima, and one side minima is located at the main maxima. The main maxima and side maxima and minima are shown in the second graph of FileFig 2.20. The interference pattern generated by an array of sources has a wide application. It is used in the discussion of Xray diffraction and is used in the discussion of the diffraction grating.

124 2. INTERFERENCE

Intensity IA of the array, both numerator and denominator are plotted on one graph depending on angle θ Y/X, where Y is the coordinate on the observation screen, and X the distance from the experiment to the screen. N 5, wavelength λ 0.0005, periodicity constant a 0.1. The main maxima of IA are obtained for 0/0 of IA. There are N − 2 side maxima, N − 1 side minima.

I20ARRAYS

Interference Pattern of N Sources

Parameters: Opening a, wavelength λ, number or lines N. Graph as function of θ, because of small angle θ Y/X. Normalization to 1. For comparison of maxima, the numerator is plotted separately.

1.Observe that the main maxima are at angles when numerator and denominator are both zero.

2.The N − 1 side minima are at angles when the nominator has minima.

3.There are N side maxima when the nominator has maxima, but only N − 2 appear.

4.What happens when changing the wavelength?

5.What happens when changing the periodicity constant?

6.What happens when changing N?

2.7RANDOM ARRANGEMENT OF SOURCE POINTS

In Section 2.6 we assumed for the calculation of the interference of N source points that the optical path differences for all waves are equal. The source points were arranged periodically, having the periodicity distance a. We saw that the incident intensity was redistributed into main maxima and side maxima and minima.

We now study the opposite case where the periodicity constant is no longer a constant but has a random distribution in an interval to be specified. In Figure 2.23 we have the waves uα propagating in direction θ and the optical path difference between the (α − 1)th and the αth wave is δα (instead of qδ). All δα are not the same. From the discussion of the periodic arrangement of N source points we have for the superposition of these waves

In this expression δα /λ may be larger than 1. One subtracts a wavelength from δα until δα /λ is smaller than 1. The value of the trigonometric functions is the same, before and after the reduction process. We call γα the reduced value of