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

FIGURE 2.15 (a) Light passing through a Michelson interferometer for a light beam incident under the angle θ; (b) light passing through a Michelson interferometer for a light beam incident under the angle θ and one arm folded onto the other; (c) calculation of path difference for a light beam incident under the angle θ.

The intensity of the superimposed two beams is obtained as

When a fringe is formed by incident light under the same angle of inclination, one speaks of Heidinger fringes. Heidinger fringes are shown in Figure 2.16. A cone of light was used with a Michelson interferometer for the observation of the ring pattern. In FileFig13. we study graphs of the cross-section of the intensity pattern of Heidinger fringes. They are produced by the Michelson interferometer for the dependence on the angle θ, for fixed wavelength λ and for fixed thickness D. We may also produce with the Michelson interferometer fringes of equal thickness. We fold one beam onto the other beam, as shown in Figure 2.17a, and then consider a similar positioning as discussed for the plane parallel plate. If one of the mirrors of the Michelson interferometer is tilted (Figure 2.17b), we have the same situation as for the wedge-shaped gap (Figure 2.10a). Therefore, the fringe pattern of the Michelson interferometer with one tilted mirror is similar

FileFig 2.13 (I3MICHANS)

Ring pattern of intensity of the Michelson interferometer depending on angle θ for two wavelengths λ and λλ for fixed D.

I13MICHANS

Michelson Interferometer, Dependence on θ

Fringe pattern depending on angle θ for two fixed wavelengths λ and λλ and fixed displacement D. An ideal beamsplitter is assumed. All lengths in mm.

Application 2.13.

1.Observe that the separation of the fringes for wavelengths λ and λλ gets smaller for larger angles and that at the center, when one wavelength λ has a maximum, the other has none.

2.Consider one wavelength only and fixed angle θ. To each maximum cor-

responds an integer m determined by 2D cos θ mπ. Use D/λ x for the ratio, and show that the maxima may be numbered by m(θ) 2x cos(2πθ/360). Make graphs for m(θ) for θ from 0 to 90 and determine the number of rings one has for ratios of x 1, 2, 3, 4. The larger number m belongs to the smallest angle θ. This is different from Young’s experiment and similar ones, where the angle is proportional to the order of interference.

110 2. INTERFERENCE

FIGURE 2.18 (a) Geometry for multiple interference at a plane parallel plate of index n2 and thickness D. The light is incident at an angle θ1 from a medium with index n 1. Using Snell’s law at the first and second surface one may show that the emerging angle is equal to the incident angle θ1. The reflection angle within the plate is θ2; (b) geometry for the calculation of the optical path from point a to point c and c.

2.6 MULTIPLE BEAM INTERFEROMETRY

2.6.1 Plane Parallel Plate

In Section 5.2 we studied the plane parallel plate and considered only two reflected waves (Figure 2.9). We formulated the condition for constructive and destructive interference, but did not investigate the question, “Where does the light travel?” when for destructive interference there is no light in the direction of reflection. Now we want to include in our discussion all internal reflections of the plate and calculate the resulting reflected and transmitted waves as shown in Figure 2.18. We show that the reflected and transmitted intensity is equal to the incident intensity. When we have destructive interference for the reflected light, all light is transmitted and vice versa.

The incident wave is assumed to have magnitude A and makes the angle θ1 with the normal. The plate has thickness D and a refractive index n2. The refractive index on both sides of the plate is assumed to be 1. The magnitudes of the reflected and transmitted waves are Air and Ait , where i is 1, 2, 3, . . . , respectively. The calculation of the optical path difference is done by using

the wave reflected on the first surface and the wave reflected once on the second surface. We assume that both waves propagate in the same direction and calculate the optical path difference, (Figure 2.18b) using the distances [ac] and [bc]. One has [ab] [bc] D/ cos θ2 and [ac] [(2D/ cos θ2)(sin θ2)]] 2D tan θ2. The optical path difference between A1r and A2r is then

The same optical path difference is obtained for transmission. We call r12 the amplitude reflection coefficient for a wave incident from medium 1 and reflected at medium 2, with the corresponding intensity R12. Similarly, we call r21 for a wave incident from medium 2 and reflected at medium 1, with the corresponding intensity R21. The first index indicates the medium in which the wave travels. The second index indicates the medium at which it is reflected. Similarly, we

conservation we have that R12 + T12 1 and R21 + T21 1. The phase difference is

We call r the absolute value of the amplitude reflection coefficients r12 and r21,

The transmitted intensity is obtained by multiplication of At with its complex conjugate At

At At A2(1 − r2)2[1/(1 − r2ei )(1 − r2e−i )].

One has (1−r2ei )(1−r2e−i ) 1−r2ei −r2e−i +r4 1+r4 −r22 cos and gets

Application 2.14. Consider transmitted and reflected intensity depending on the thickness of the plate.

1.Try out different combinations such as n1 < n2 < n3, n1 < n2 > n3, and n1 > n2 < n3, and see the effect of the phase jump of reflection at the denser medium.

2.Choose arbitrary values for r < 1 and observe how the intensity is changing.

Intensity of interference at a plane parallel plate assuming normal incidence. Graph of the reflection and transmission depending on wavelength λ for fixed D and different values of n1, n2, and n3.

I15PLANIDS is only on the CD.

Application 2.15. Consider the transmitted and reflected intensity depending on wavelength.

1.Try out different combinations such as n1 < n2 < n3, n1 < n2 > n3, and n1 > n2 < n3, and see the effect of the phase jump of reflection at the denser medium.

2.Find the wavelength for the last fringe, depending on the thickness D of the plate.

2.6.2 Fabry–Perot Etalon

A plane parallel plate with reflecting surfaces on both sides is called an etalon. The reflecting layers may be made of metal or a structure of dielectric films. We show that for a specific wavelength at a specific spacing of two reflecting surfaces, all incident light will be transmitted while a single reflecting surface will transmit only a small amount.

There the mathematical form of It is called the Airy function.

If [sin /2]2 0 we have the condition of constructive interference for

If [sin /2]2 1 we have a minimum of light transmitted. The condition is

where m is an integer. The graph in FileFig 2.16 shows three transmission patterns for three different absolute values of the reflection coefficient. We have chosen λ .1 and plotted the transmitted intensity as a function of the spacing D, for m 1 and r .7, .9, and .97, respectively. We see that the width of the transmitted intensity depends on the absolute value of the reflectance r of a single plate and becomes narrower when r gets close to 1. For constructive interference, that is, when 2D mλ, it follows that sin2 /2 0. Therefore It is 1, independent of the value of r. We may have r so close to 1 that the transmission of a single plate is almost zero, but the transmission of the pair of plates at the right distance will be one. At this distance the Fabry–Perot etalon has a resonance mode. In experimental Fabry–Perot etalons, the peak transmission will not be exactly one, due to losses such as absorption in the plates. The Fabry– Perot etalon, using high orders, is applied to investigate with high resolution details of a spectral line in a narrow spectral range. The dependence of the width of the spectral line on the reflection coefficient of the etalon is shown in the graph of FileFig 2.17. The transmittance is plotted depending on the wavelength λ for three different reflection coefficients r and fixed distance D.