Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007
.pdf2.3. SUPERPOSITION OF HARMONIC WAVES |
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2.3.2.1 Use of the Time Average
The time average of the square of a cosine function depending on t, taken over the interval of one period, that is, from 0 to T , is
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cos t]2 dt 1/2. |
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(1/T ) |
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(2.14) |
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Using for the cos-function the “second” factor of Eq. (2.8), we get |
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[cos{2π(x/λ − t/T ) − 2π(δ/2)/λ}]2dt 21 |
(2.15) |
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As result, we obtain for the intensity of the wave described in Eq. (2.8), |
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I [2A cos{2π(δ/2)/λ}]2(1/2). |
(2.16) |
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This is the square of the first factor (Eq. (2.13)) multiplied by 1/2. |
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2.3.2.2 Complex Notation
Complex notation has its advantages, for example, when adding several harmonic waves. Then, the intensity is calculated by the product z z and the time dependence is automatically eliminated.
We rewrite the sum of the two real expressions of Eq. (2.5) in complex notation:
z z1 + z2 A exp i(2π(x)/λ − 2πt/T ) |
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+ A exp i(2π(x − δ)/λ − 2πt/T ). |
(2.17) |
Here the real part of z, that is, Re(z), is the amplitude of u of Eq. (2.8). The superposition of the two waves may be written in complex notation as
z z1 + z2 A exp{−i2πt/T + i(2π(x)/λ)} |
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× {1 + exp(−i2πδ/λ)}. |
(2.18) |
In the second factor of Eq. (2.18) we may factor out exp −i2πδ/2λ and then have
z A exp{−i2πt/T + i(2π(x)/λ)} exp(−i2πδ/2λ) |
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× {exp i2πδ/2λ + exp −i2πδ/2λ}. |
(2.19) |
Using |
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{exp i2πδ/2λ + exp −i2πδ/2λ} 2 cos 2πδ/2λ |
(2.20) |
one gets |
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z A exp{−i2πt/T + i(2π(x)/λ)} exp(−i2πδ/2){2 cos 2πδ/2λ}. |
(2.21) |
For simplicity we collect all phase factors into one term, call |
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exp{−i2πt/T + i(2π(x)/λ) − i2πδ/2λ} exp i , |
(2.22) |
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2. INTERFERENCE |
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and have |
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z [2A cos(2πδ/2λ)] exp i . |
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(2.23) |
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For the intensity we use the square of the absolute value of z, that is, the product |
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of z times its complex conjugate z , |
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z z [2A cos(2πδ/2λ)] exp i [2A cos(2πδ/2λ)] exp −i |
(2.24) |
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or |
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z z [2A cos(2πδ/2λ)]2. |
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(2.25) |
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Comparing the intensity in Eq. (2.16) with Eq. (2.25), one sees that they |
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are different by the factor 1 . This factor |
1 is not significant, since one usually |
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normalizes the results for the calculation of interference and diffraction pattern. |
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FileFig 2.4 (I4COSINTS)
Graphs of the real part of the superposition of two waves depending on space and time, its square, and the time average of the “second factor.” The time-dependent superposition of two waves is given in complex notation. Its real part and the square of the real part are plotted. Then the product zz is plotted, depending on x, and appearing at a constant. The “fringe pattern” is plotted (i.e., zz ), depending on the optical path difference.
I4COSINTS is only on the CD.
2.3.3 Normalization
To compare an interference pattern with observation, one may normalize both. Let us assume we have an interference pattern with a maximum in the center. The scale of the observation data may be changed in such a way that at the center one has the intensity equal to 1. The intensity calculated from the formula will also be changed: one divides by the value calculated at the center, and obtains also for the intensity 1. For example, let us assume that an interference pattern is described by f (x) and that for x 0 we have f (0) b. Then writing
I (x) f (x)/f (0) I0f (x) |
(2.26) |
with I0 1/b results in I (x) equals 1 for x 0.
In FileFig 2.4 we have plotted the real part of u and the time average and compared it with zz , which is independent of time. We also plot what one may call the “fringe pattern,” that is, zz , depending on the optical path difference.
2.4. TWO-BEAM WAVEFRONT DIVIDING INTERFEROMETRY |
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2.4TWO-BEAM WAVEFRONT DIVIDING INTERFEROMETRY
2.4.1 Model Description for Wavefront Division
The classical experiments by Young, Fresnel, and Lloyd were performed to demonstrate the wave theory of light. All three experiments use one incident wave and divide the light from the source into two waves. The two waves are superimposed after an optical path difference is introduced and a pattern showing the maxima and minima is observed. One calls such a pattern, consisting of maxima and minima, a fringe pattern. The process of splitting up the incident wave at two small openings into two new waves involves diffraction, considered in detail in Chapter 3. The incident wave we consider is monochromatic, propagates in the x direction, vibrates in the y direction, and has a large lateral extension in the z direction. The experimental setup produces two waves by wavefront division, propagating under an angle with respect to each other (see Figure 2.4a). When we observe the interference pattern at a faraway screen (see Figure 2.4b) the
FIGURE 2.4 Observation of interference pattern generated by two sources. The generated waves have a constant phase relation: (a) geometry of the experiment with water waves (see also Figure 2.3); (b) Observation at a screen far away from the source; (c) using a lens to reduce the distance X in (b) to distance X in (c). In this case only parallel rays meet at one point in the focal plane of the lens.
90 2. INTERFERENCE
FIGURE 2.5 Schematic ofYoung’s experiment. Light is emitted by the source S0: (a) according to Huygen’s principle, the first aperture A1 produces a spherical wave which generates two spherical waves with fixed phase relation at P1 and P2 and produces a fringe pattern; (b) coordinates for the calculation of the fringe pattern generated by screen 2 and observed on screen 3. The two openings on screen 2 are separated by a. The distance from screen 2 to screen 3 is X and the coordinate on screen 3 is Y .
angles θ and θ are almost equal. We therefore approximate the propagation by assuming that the two waves move in parallel, and that the optical path difference δ is the same as it was when moving under an angle. We describe both waves in the same coordinate system and use for the intensity of their superposition I I0[A cos(2πδ/2λ)]2 (see Eq. (2.25). In Figure 2.4c we show the observation in the focal plane of a lens, giving us the opportunity to have the distance between the experimental setup and the observation screen significantly shorter.
2.4.2 Young’s Experiment
The setup of Young’s experiment is schematically shown in Figure 2.5a. The light from a source S0 travels to screen 1, which has a small hole. At the hole, a spherical wave is created and travels a large distance. The wave arrives at screen 2 as “almost” a plane wave. At two small holes on screen 2, P1 and P2, the incident plane wave is split into two spherical waves. The two spherical waves travel a large distance and arrive at screen 3 as “almost” plane waves. Their wavefronts are tilted by a small angle with respect to each other, resulting in an optical path difference and superposition results in a fringe pattern. In our
2.4. TWO-BEAM WAVEFRONT DIVIDING INTERFEROMETRY |
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model calculation, we assume that the two monochromatic waves, leaving the two openings, travel in parallel in the same direction. They have the angle θ with respect to the symmetry axis and the optical path difference δ (see Figure 2.5),
δ a sin θ. |
(2.27) |
In the small angle approximation we have δ aY/X, and obtain the intensity of the fringes on the observation screen, using Eq. 2.25
I (Y ) I0[cos{(πaY )/(Xλ)}]2. |
(2.28) |
Constructive interference is observed for |
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δ Y a/X 0, λ, 2λ, . . . . |
(2.29) |
and destructive interference for |
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δ Y a/X λ/2, 3λ/2, 5λ/2, |
(2.30) |
The graph in FileFig 2.5 shows the intensity for Young’s experiment, depending on Y , using the coordinate on the observation screen. One observes that there is a maximum at the center. In Figure 2.6, a photograph shows the interference pattern of Young’s experiment and an experimental setup for its observation.
FIGURE 2.6 (a) Photograph of interference pattern observed with Young’s experiment (from M. Cagnet, M. Francon, and J.C. Thrierr, Atlas of Optical Phenomena, Springer-Verlag, Heidelberg, 1962); (b) physical dimensions for the observation of fringes in Young’s experiment, according to R. Pohl.
92 2. INTERFERENCE
FileFig 2.5 (I5YOUNGS)
The intensity for Young’s experiment (Eq. (2.28)), is plotted for λ 0.0005 mm, a 0.4 mm, and X 4000 mm. For this choice of parameters, we see that the separation of the holes on screen 2 (see Figure 2.5), are three orders of magnitude larger than the wavelength and four orders of magnitude smaller than the distance between screens 2 and 3.
I5YOUNGS is only on the CD.
Application 2.5.
1. Change the separation of the source points a to 21 a and 2a.
2.Change the wavelength λ to 21 λ and 2λ.
3.Compare the changes of the ratio a/λ to 21 and 2 times its value.
4.There are no changes if we keep the product (a/λ)(Y/X) constant, where Y/X is the angle under which we observe the fringes.
2.4.2.1 Lloyd’s Mirror
In the experiment by Lloyd, interference fringes are produced by using one mirror. Two parts of the incident wave are superimposed. One part travels directly from the source to the observation screen, while the other part is incident on the mirror and is reflected under a grazing angle (see Figure 2.7). For the calculation of the optical path difference δ, we use a virtual source S at distance a/2 from the plane of the mirror, but on the “other side” of the mirror. Similarly, as in Young’s experiment, we have for the optical path difference δ Y a/X, and consider the two waves traveling parallel to the observation point. One has to take into account that one of the two superimposed waves has picked up a phase shift
FIGURE 2.7 Optical schematic for Lloyd’s mirror experiment drawn with exaggerated size of the angles. One wave from the source travels directly to the observation point; the other is reflected at the mirror. Since the distance of the source from the plane of the mirror is very small the first fringe is about in the plane of the reflecting surface. It is a minimum because of the phase shift.
2.4. TWO-BEAM WAVEFRONT DIVIDING INTERFEROMETRY |
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of π upon reflection on the mirror. This phase shift appears as π in cos{2π(x/λ− t/T ) + π}, and after superposition, as π/2 in the amplitude factor used for the intensity. We therefore have
I (Y ) I0[cos{π(Y a/Xλ) + π/2}]2. |
(2.31) |
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For interference one has |
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Y a/X + λ/2 0, λ, 2λ, . . . or |
δ Y a/X λ/2, 3λ/2, 5λ/2 |
(2.32) |
and for interference |
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Y a/X + λ/2 λ/2, 3λ/2, 5λ/2, . . . |
or δ Y a/X 0, λ, 2λ . . . |
(2.33) |
The graph in FileFig 2.6 shows the intensity of Lloyd’s mirror experiment depending on Y, the coordinate on the observation screen. In comparing Young’s experiment to Lloyd’s, one has a minimum at the center.
FileFig 2.6 |
(I6LOYDS) |
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Intensity of Lloyd’s experiment for λ 0.0005 mm, a .4 mm, and X 4000 mm. The “first” fringe is dark, that is, a minimum for Y 0, because of the phase shift upon reflection of one of the two waves. The dependence of the fringes on a and λ are the same as in Young’s experiment.
I6LOYDS is only on the CD.
2.4.2.2 Fresnel’s Double Mirror Experiment
Fresnel’s mirror experiment was originally performed to prove the wavelike character of light. Even today it is of some interest since the fringes depend on the tilting angle of the mirrors and may be applied to wavelengths as short as Xrays. The wavefront of the spherical wave, emerging from the source, is divided by two mirrors, which are tilted by a very small angle β (Figure 2.8a). Each part of the incident wave is reflected, and the two wavefronts are tilted by the angle β. The superposition produces the fringe pattern. In our model description we consider the waves at the faraway observation screen as parallel. The optical path difference is the same as calculated when the two waves travel under the small angle β. Making these assumptions, the optical path difference is obtained using the two virtual source points S1 and S2 at distance a (see Figure 2.8a). It is similarly done as for Young’s experiment (Figure 2.8b). The distance from the virtual sources to the observation screen is c + f X, and the optical path difference δ a(Y/X) of the light reflected by the two Fresnel mirrors is
δ (Y 2b sin β)/(c + f ). |
(2.34) |
94 2. INTERFERENCE
FIGURE 2.8 (a) and (b) are for Fresnel’s mirror experiment. S is the source, S1 and S2 are virtual source points. They are separated by a and have a tilting angle β. The coordinate on the observation screen is Y . The relation a 2b sin β is obtained by associating “one b” with one mirror and the “other b” with the other mirror, and remembering that light is reflected by 2β if the mirror is tilted by β; (c) approximate treatment of Fresnel’s mirror experiment similar to Young’s experiment.
The intensity is |
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I (Y ) I0[cos{(πY 2b sin β)/[(c + f )λ]}]2, |
(2.35) |
and for constructive interference |
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δ Y 2b sin β/(c + f ) 0, λ, 2λ, . . . |
(2.36) |
and destructive interference |
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δ Y 2b sin β/(c + f ) λ/2, 3λ/2, 5λ/2, . . . |
(2.37) |
The graph in FileFig 2.7 shows the intensity of Fresnel’s double mirror experiment with a maximum at the center, since both waves are reflected on a mirror
2.4. TWO-BEAM WAVEFRONT DIVIDING INTERFEROMETRY |
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and pick up a phase shift π. In FileFig 8 we compare the intensities of Young’s, Lloyd’s, and Fresnel’s double mirror experiments and see the role the phase shift plays.
FileFig 2.7 |
(I7FREMIRS) |
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Intensity of Fresnel’s double mirror experiment, Eq. (2.35) for λ 0.0005 mm, b 1000 mm, f 5000 mm, and for the angle β 0.0002.
I7FREMIRS is only on the CD.
Application 2.7.
1.How is the pattern changing when changing λ. How much must λ be changed to double the separation of the maxima or to make them one half.
2.How is the pattern changing when changing β. How much must β be changed to double the separation of the maxima or to make them one half.
3.What angle β has to be chosen to have the maxima separated by 500 microns
˚ −10
when using Xrays of 500 Angstrom (1 A is 10 m).
FileFig 2.8 (I8FRYOLOS)
Comparison of experiments by Young, Lloyd, and Fresnel. For the choice of parameters, we see that the separation of the “sources” are three orders of magnitude larger than the wavelength, and four orders of magnitude smaller than the distance X between experimental setup and observation screen.
I8FRYOLOS
Fresnel’s Mirror, Young’s Double Slit, and Lloyd’s Mirror
1.Fresnel’s mirror
Y , c, f , b, and λ in mm, β in rad; c is about b for calculation of X c + f . All lengths in mm.
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Con : 1 |
λ defined above |
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b : 1000 |
f : 5000 |
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Y : −10, −9.99 . . . 10 |
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c : b cos(β) |
β ≡ .0002 |
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Y · 2 |
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IF(Y ) : Con · cos |
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λ
96 2. INTERFERENCE
2. Young’s experiment
a ≡ .4 |
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≡ 4000 |
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I Y (Y ) : |
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λ ≡ .0005 |
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3.Lloyd’s mirror
Same as Young, phase term is added.
I L(Y ) : |
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We see that at Y 0 Young has a maximum, Lloyd a minimum. For Young and Lloyd: the position of maxima are changed by changing d and λ; X is considered fixed. For Fresnel, a, b are fixed; position of fringes changes with
βand λ.
2.5TWO-BEAM AMPLITUDE DIVIDING INTERFEROMETRY
2.5.1 Model Description for Amplitude Division
We again assume a monochromatic wave incident on the experimental setup, propagating in the x direction, vibrating in the y direction, and having a large lateral extension in the z direction.
The incident wave is incident on a beam splitter. One part is reflected and the other part is transmitted. One or both parts are manipulated in the experimental setup to pass through a beamsplitter a second time. Fractions of each part are superimposed and travel parallel to the observation screen.