Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007
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1. GEOMETRICAL OPTICS |
matrix, the light may pass through many round trips and no light will escape. One calls such a resonator stable, and the condition for stability is where the magnitudes of the eigenvalues are equal to 1.
|λ1| |λ2| 1. |
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(1.116) |
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We may write for Eq. (1.114), |
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(2g1g2 − 1) |
+ [(2g1g2 − 1)2 − 1]1/2 |
(1.117) |
or |
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(2g1g2 − 1) |
+ i[1 − (2g1g2 − 1)2]1/2. |
(1.118) |
The real and imaginary parts of Eq. (1.118) must be on a circle of radius 1; that is,
|(2g1g2 − 1)| ≤ 1, or 0 ≤ g1g2 ≤ 1. |
(1.119) |
in agreement with the imaginary part and plotted in FileFig 1.32.
In FileFig 1.33 we show a repetition of the calculations, starting from the five matrices of the cavity in Eq. (1.111), but now in terms of r1, r2, and d. In Figure 1.33, we show schematics of the Fabry–Perot, a focal, a confocal and a spherical cavity for values of the parameters r1, r2, and d, and also of g1 and g2. For both representations one finds that the absolute values of the eigenvalues λ1 and λ2 are always 1.
FileFig 1.33 (G33RESCY)
Calculation of the eigenvalues of the cavity with two reflecting mirrors using r1, r2, and d. Numerical calculation with r1 1, r2 1, and d 2.
G33RESCY is only on the CD.
Application 1.33. Use the values of the parameters r1, r2, and d for the Fabry– Perot, focal, confocal, and spherical cavities, and find that all are stable cavities.
1.10. MATRICES FOR A REFLECTING CAVITY AND THE EIGENVALUE PROBLEM |
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FIGURE 1.33 Schematic of light path for four cavities with different values of radii of curvature and length of cavity. The corresponding values of g1 and g2 are indicated: (a) Fabry–Perot; (b) focal;
(c) confocal; (d) concentric.
C H A P T E R
Interference
2.1 INTRODUCTION
In Chapter 1 we described image formation by light, using our model which states that light propagates along straight lines and utilizes the laws of reflection and refraction. We now consider the wave nature of light. In the famous experiment by Thomas Young, one observes on a screen an interference pattern, consisting of bright and not so bright stripes of light. The interpretation of an interference pattern was done by using an analogy to water waves. However, the water wave pattern is observed as an amplitude interference pattern whereas the superposition of light waves, also generated as an amplitude pattern, is observed as an intensity pattern. Historically, Newton associated the light beams of geometrical optics with a stream of particles and some scientists attacked Young in his time, saying that he was diminishing Newton’s work. Today we know that light is an electromagnetic wave but, in a complementary way, light is also described by quantum mechanics as an assembly of particles.
In this chapter we use a model for the description of interference phenomena. We assume that there is always one incident wave when twoor more beam interferometry is discussed. After one has taken into account what happens in the experimental setup, the waves leaving the setup appear superimposed. The interference pattern is produced with finite optical path differences. The calculation of the optical path difference and the interpretation of the resulting interference pattern are the main subjects of this chapter. The process of splitting the incident wave into parts involves diffraction, which we neglect in this chapter as a secondary effect and discuss in detail in Chapter 3.
In this chapter we use for the incident wave one harmonic wave, a solution of the scalar wave equation, which is written in Cartesian coordinates as
∂2u/∂x2 + ∂2u/∂y2 + ∂2u/∂z2 (1/v)2∂2u/∂t2, |
(2.1) |
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80 2. INTERFERENCE
where v is the phase velocity of light in the medium with refractive index n, related to the speed of light c in vacuum as v c/n. The scalar wave equation follows from Maxwell’s theory. It may also be written in spherical coordinates
2u + k2u 0, |
(2.2) |
where is the differential operator in spherical coordinates, k 2π/λ, and λ is the wavelength of the light. A simple solution of this equation is a spherical wave of the type (eikr )/r, where r is the distance from the origin to the observation point. The spherical wave propagates from its origin in all directions and its intensity is attenuated by 1/r2. We consider such spherical waves only conceptually and approximate them at a large distance by plane waves.
The differential equation of the scalar wave equation is linear and superposition of solutions of the differential equation will again result in a solution. This is part of the superposition principle. In this chapter we only need the superposition of a number of monochromatic waves, each of frequency ν, to result in a monochromatic wave having the same frequency ν.
For our model description we use some results from Maxwell’s theory for quantitative expressions of the reflection and transmission coefficients of materials contained in Fresnel’s formulas. In particular, we use the results that waves pick up a phase jump of π, when reflected at an optically denser medium, and that they travel in the optically denser medium with wavelength λ/n, where n is the index of refraction. The intensity is calculated either as the time average of the square of the amplitude or the square of the absolute value of the complex representation and may be normalized with an arbitrary constant.
2.2 HARMONIC WAVES
The solution of the scalar wave equation, (Eq. (2.1)), is a function, depending on the space coordinates x, y, z and the time t. In addition, there may be an arbitrary phase factor. We consider harmonic waves in vacuum and in an isotropic and nonconducting medium of index n. However, in most cases, we only need waves depending on one space coordinate and time. We describe the transverse waves by vibrating in the u direction and moving in the x direction, having wavelength λ and time period T .
u A cos[2π(x/λ − t/T + φ)]. |
(2.3) |
The amplitude u of the wave varies in the x direction, A is the magnitude of the wave, and φ is a phase constant. The first graph of FileFig 2.1 shows the amplitude u, depending on the space coordinate x for three time instances t and three phase constants. The second graph shows the dependence on time for three points in space and three phase constants. The magnitudes A1 to A3 and B1 to
2.2. HARMONIC WAVES |
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FIGURE 2.1 Change of wavelength as the wave enters and leaves a dielectric medium.
B3 have been assumed to have the same value, and the three phase constants φ1 to φ3 and 1 to 3 are assumed to be different. Comparing the graphs, one observes equivalence of the dependence of the cosine function on x/λ and t/T . Changing the range of variable from x to t, the family of curves depending on x is similar to the one depending on t.
We may modify x/λ and t/T in such a way that they contain phase constants. Then, in the “net” expression cos[2π(x/λ−t/T +φ)] we can not distinguish if φ belongs to the space part or the time part. We show below that for our discussions on interference we do not need the time dependence and it is eliminated.
The product of the frequency ν and wavelength λ/n is equal to the phase velocity v ω/k of the wave propagating in the medium of refractive index n. The angular frequency ω 2πν, and the wave vector k 2πn/λ, where λ is the wavelength in vacuum. We may then write Eq. (2.3) as
u A cos(kx − ωt) |
(2.4) |
or
u A cos k(x − (ω/k)t) A cos k(x − vt).
The phase velocity in vacuum is c, and in an isotropic medium with refractive index n it is c/n. The wavelength of “free space” λ is reduced in the medium to λ/n (see Figure 2.1).
FileFig 2.1 (I1COSWS)
Cosine functions depending on space and time coordinate and one additional phase constant. Graphs are shown for cosine functions depending on the space coordinates for three time instances. This may be interpreted as graphs of the
82 2. INTERFERENCE
same wave at three consecutive snapshots. Graphs are shown for cosine functions depending on the time coordinates for three points in space.
I1COSWS is only on the CD.
Application 2.1.
1.One may change the phase φ and the space coordinate and choose both so there is no resulting change in the graph. Choose φ 2, 4, 6.
2.One may change the phase φ and the time coordinate and choose both so there is no resulting change in the graph. Choose φ 2, 4, 6.
3.Change φ in such a way that there is a shift to smaller values of the position coordinate.
4.Change φ in such a way that there is a shift to larger values of the time coordinate.
2.3SUPERPOSITION OF HARMONIC WAVES
2.3.1Superposition of Two Waves Depending on Space and Time Coordinates
We describe the interference of two waves in a simple way, using the superposition of two harmonic waves u1 and u2. Both waves will propagate in the x direction and vibrate in the y direction.
u1 A cos 2π[x/λ − t/T ] u2 A cos 2π[(x − δ)/λ − t/T ]. (2.5)
We assume that the two waves have an optical path difference δ. At time instance t 0, the wave u1 has its first maximum at x 0, and u2 at x δ (Figure 2.2). Adding u1 and u2 we have
u u1 + u2 A cos 2π[x/λ − t/T ] + A cos 2π[(x − δ)/λ − t/T ]. (2.6)
Using
cos(α) + cos(β) 2 cos{(α − β)/2} cos (α + β)/2 |
(2.7) |
we get
u [2A cos{2π(δ/2)/λ}][cos{2π(x/λ − t/T ) − 2π(δ/2)/λ}]. |
(2.8) |
In FileFig 2.2 we show graphs of the square of Eq. (2.8) for the same time instant t1 and wavelength λ. We choose a number of optical path differences δ1 0, δ2 0.1, δ3 0.2, δ4 0.3, δ5 0.4, δ6 0.5, corresponding to the ratios of the optical path difference to the wavelength between 0 and 21 . One observes that the height of the maxima decreases with increasing δ1 to δ6, and shifts to larger values of x.
2.3. SUPERPOSITION OF HARMONIC WAVES |
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FIGURE 2.2 Two waves with magnitude A and wavelength λ. We have u1 A for x 0 and u2 A for x δ.
We now discuss the two factors of Eq. (2.8). The first factor 2A cos{2π(δ/2)/λ} depends on δ and λ, but not on x and t. One obtains for δ equal to 0 or a multiple integer of the wavelength
[2A cos{2π(δ/2)/λ}]2 |
is 4A2 |
(2.9) |
and for δ equal to a multiple of half a wavelength |
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[2A cos{2π(δ/2)/λ}]2 |
is 0. |
(2.10) |
The first factor in Eq. (2.8) may be called the amplitude factor and is used for characterization of the interference maxima and minima.
One has
maxima for δ mλ, where m is 0 or an integer |
(2.11) |
minima for δ mλ, where m is 21 plus an integer |
(2.12) |
and m is called the order of interference.
The second factor is a time-dependent cosine wave with a phase constant depending on δ and λ. For the description of the interference pattern this factor is averaged over time and results in a constant, which may be factored out and included in the normalization constant (see below).
In Figure 2.3 we show schematically the interference of two water waves with a fixed phase relation. When the interference factor is zero one has minima, indicated by white strips. They do not depend on time. The maxima oscillate and appear and disappear along the line in the observable direction.
Maxima and minima are shown in FileFig 2.3 as 3-D graphs. The maxima are shown for δ λ, and in the second graph, for δ λ/2, there is just one minimum. The maxima show the time dependence of the second factor for each of the space coordinates. One can estimate that a time average will result in half the maximum value. The minimum is zero. It is zero for all time.
84 2. INTERFERENCE
FIGURE 2.3 Schematic of the interference pattern produced by two sources vibrating in phase. At the crossing of the lines, the amplitudes of the waves of both sources are the same and adding. Taking the time dependence into account, the magnitude changes between maximum and minimum. These are the maxima when considering light. Between the maxima we indicate the two lines corresponding to the minima. Along these lines the amplitude of the two waves compensate each other; their sum is zero for all times.
FileFig 2.2 |
(I2COSSUPS) |
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Graphs of the superposition of two cosine waves with wavelength λ 1, for a number of optical path differences δ1 0, δ2 0.1, δ3 0.2, δ4 0.3, δ5 0.4, δ6 0.5 corresponding to ratios of the optical path difference to the wavelength between 0 and 21 .
I2COSSUPS is only on the CD.
Application 2.2.
1.Extend the range of the optical path differences of the six graphs from 21 wavelength to 1 wavelength, and then from 1 wavelength to 23 wavelength and indicate in a list when there is repetition.
2.Make a graph of y cos{2π(δ/2)/λ} for fixed λ as function of δ and make a list of the δ values for minima and maxima. Compare with a list of δ/λ values.
FileFig 2.3 (I3COSGRA)
3-D demonstration of the superposition of two waves for δ/λ 1 corresponding to a maximum, and δ/λ 0.5 corresponding to a minimum. In the graph of the maximum, the amplitude changes in time for a specific spot in space between
2.3. SUPERPOSITION OF HARMONIC WAVES |
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0 and (2A)2, and one can estimate that the time average will be half of it. The graph of the minimum is zero for all time and space values.
I3COSGRA
Superposition of Two Cosine Waves
One wave has optical path difference δ with respect to the other. The sum is squared to result in the intensity. We are looking at them time dependence; the graphs are plots in space x and time t. Period T , path difference δ, wavelength λ. 1. Graph for optical path difference corresponding to a maximum
λ : 1 |
A : 1 |
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t1j : −.2 + .05 · j |
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uc(x, t1) : |
2 · A · cos 2 · π · |
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t1 ≡ .1. |
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2. Graph for optical path difference corresponding to a minimum
N : 40
i : 0 . . . N j : 0 . . . N
xxi : −.2 + .04 · i t1j : −.2 + .02 · j δ2 ≡ .5
86 2. INTERFERENCE
ud(xx, t1) : |
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Application 2.3. One may change the wavelength λ such that for δ1/λ one gets minima, and for δ2/λ one gets maxima.
2.3.2 Intensities
The interference pattern of water waves is an amplitude pattern. We may observe minima and maxima with respect to the level of the undisturbed water surface. The interference pattern of light shows intensity minima as dark spots in space and maxima as bright spots. An amplitude pattern shows negative amplitudes, but an intensity pattern has only positive or zero values. The amplitude pattern has to be considered first; it produces interference. Then we have to obtain the intensity pattern. We compare the intensity pattern to observations.
For the square of the amplitude of equation (2.8) we have
u2 [2A cos{2π(δ/2)/λ}]2[cos{2π(x/λ − t/T ) − 2π(δ/2)/λ}]2 |
(2.13) |
In Section 2.1 we mentioned that for the intensity we use either the time average of the square of the amplitude or the square of the absolute value, when using complex notation. In this section we compare these two calculations.