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

FIGURE 4.4 Michelson stellar interferometer: (a) light waves from two stars, I and II, forming an angle φ when they arrive at a double slit of width a. The waves from each star produce a fringe pattern on a screen at distance X, described by the coordinate YI and YI I ; (b) four mirrors are added, M1 to M4, to produce a new optical path difference hφ for the incident light, where h is adjustable. The angle θ is the diffraction angle.

The difference for h2 − h1 λ/2φ and φ is obtained since h2 − h1 can be measured. This type of modified interferometer was applied to measure the angular diameter of Betelgeuse in the Orion constellation. At the Mt. Wilson observatory an interferometer was used with a distance h of the two mirrors of 302 cm (121 in.). The angle was determined to be 22.6 × 10−8 rad. The distance to the star was known from parallax measurement and the diameter was determined to be about 300 times that of the sun. A simulation with a numerical example is given in FileFig 4.5. The graph shows a plot of the interference pattern for assumed values of h and φ. We determine the two values of h for observance and disap-

pearance of fringes. Application of h2 − h1 λ/2φ results in the angle and that must be the angle we had to assume for the simulation.

FileFig 4.5 (C5MICHSTS)

Graphs are shown of the resulting intensity interference pattern of uI and uI I , depending on values of h over the range Y −20 to 20. The distance from source to interferometer is X 4000, wavelength λ 0.0005, separation in the interferometer a 0.5. For the simulation we have to use a value for the angle we actually want to determine and choose φ 0.00005. We then can determine the h values of the minima and maxima and they must satisfy the relation h2 − h1 λ/2φ.

C5MICHSTS

Michelson’s Stellar Interferometer

Diffraction angle is Y/X, wavelength λ, and angle to be determined is . Interferometer distance of mirrors M1 and M4 is h. In the real setup we change h to go from fringe pattern to no fringe pattern. From the difference of these two values we calculate the angle . In this simulation we choose an angle and show that the fringe pattern changes for the two values of h we determine. Example h equals 100 and 95.

This is an indication of the presence or absence of fringes.

+

202 4. COHERENCE

C7COHTEMS is only on the CD.

Application 4.7. Change the wavelength interval to smaller values and approach

in 1: the corresponding case of the monochromatic wave;

in 2: the corresponding monochromatic case for destructive interference; in 3: the corresponding monochromatic case for constructive interference.

FileFig 4.8 (C8COHINTS)

Intensity pattern of superposition of two wavetrains. Graphs 1 to 3: Integration of waves over wavelength range from λ 1.85 to 2.15, having optical path differences of δ 0, 21 λm, and λm, respectively.

C8COHINTS is only on the CD.

Application 4.8. Change the wavelength interval to smaller values and approach

in 1: the corresponding case of the monochromatic wave;

in 2: the corresponding monochromatic case for destructive interference; in 3: the corresponding monochromatic case for constructive interference.

4.2.3 Length of Wavetrains

The length of finite wavetrains (see Fig. 4.5) may be determined with a Michelson interferometer. The incident light is divided at the beam splitter into two parts traveling to M1 and M2, respectively (see Fig. 4.6a). Each part is reflected by a mirror and travels back to the beamsplitter (Fig. 4.6b). At the beam splitter, the reflected part of the light from M1 and the transmitted part of the light from M2 are superimposed and travel to the detector (Fig. 4.6c). When the mirror in the Michelson interferometer is displaced, the light traveling to the detector shows a superposition pattern of two wavetrains of finite length. First, the wellknown pattern of the superposition of two monochromatic waves appears. At a certain large distance this pattern disappears. The wavetrain from one arm of the Michelson interferometer will “miss” the other wavetrain because of the finite length of the wavetrains (see Fig. 4.6d). As an example we may look at the emission of light from 86Kr at 6056.16 A˚ . Since the wavetrain has a length of about 1 m and displaces the one mirror by 1/2 meter because of reflection, the interference is not observable. For this reason one calls the length of the wavetrain the coherence length. For most atomic emission processes the coherence length is much smaller whereas resonance in laser cavities may produce a much longer coherence length of the order of 105 m.

FIGURE 4.6 Splitting of the incident light in a Michelson interferometer: (a) incident light;

(b) reflected light traveling to beam splitter; (c) two waves traveling to the detector; (d) for large displacements of M1, a finite wavetrain “misses” recombination with its “counterpart.”

APPENDIX 4.1

A4.1.1 Fourier Transform Spectrometer and Blackbody

Radiation

Blackbody radiation contains a large band of wavelengths and has a coherence length with respect to the medium wavelength λm, of only a few wavelength’s. If we consider a series of filters with smaller and smaller band width, the coherence length of the light passing these filters has increasingly larger and larger values. Michelson’s interferometer may be used for Fourier transform spectroscopy, as discussed in Chapter 9. When using blackbody radiation, the total bandwidth of

the incident light has to be limited to the interval from 0 to a highest frequency, determined by the sampling theorem. The Fourier transform process analyzes the light and determines the intensity of the resolution width ν, which is equal to 1/2 L where L is the length of the interferogram in meters. This length L may be increased to larger values and consequently ν is decreased. Therefore the length of the corresponding wavetrain lc is increasing. This process comes to a halt when the signatures of the interferogram are obscured by noise.

In comparison, the size of the coherence length of the atomic emission of 86Kr has its limitation in a time-limited emission process whereas when using blackbody radiation in Fourier transform spectroscopy, the coherence length is limited by the available signal-to-noise level.

See also on CD

PC1. Two Source Points (see p. 185). PC2. Extended Sourc.(see p. 190). PC3. Visibility (see p. 194).

PC4. Caparison of Visibilities (see p. 194).

PC5. Calculation of the Visibility for Fresnel’s Mirror Interferometer (see p. 193 and 93).

PC6. Michelson Stellar Interferometer (see p. 195). PC7. Quasimonochromatic Light (see p. 198).

PC8. Quasimonochromatic Light and Interferogram (see p. 199).

bearing his name. Today we write Maxwell’s equations in vector notation and call B the magnetic field vector. This point is well explained by Feynman in his Lecture Notes, Volume II, pp. 32–34.

The four Maxwell’s equations may be written as

where E is the electrical field vector, B the magnetic field vector, j the current density vector, ρ the charge density, and ε0 8.854×10−12 F/m the permittivity of vacuum. The mathematical form of the differential vector operator and its scalar “square" 2 is given in Appendix 5.1.

For light propagating in a vacuum, we have j 0 and ρ 0 and Maxwell’s equations are reduced to

From this set of equations we arrive at the wave equations for the vectors E and B as shown in Appendix 5.1.

5.2HARMONIC PLANE WAVES AND THE SUPERPOSITION PRINCIPLE

5.2.1 Plane Waves

We consider a nondispersive medium.A plane wave solution of the wave equation for the electrical field components vibrating in the y direction and propagating in x direction may be written as

where Eyo is the magnitude of the electrical field, λ the wavelength, t the time, and T the period of vibration. Equation (5.5) may be rewritten, introducing the wave vector k 2π/λ, and the angular velocity ω 2π/T . Using exponential notation we have

f

q

FIGURE 5.1 Coordinate system for a wave with wave vector kn 2π/λn, traveling in x, y, z space. Using k here must be distinguished from k, the unit vector in the z direction. The vector r points from 0 to a point in space with the coordinates (x, y, z).

Introduction of Equation (5.6) into the wave equation results in

where ω/k is the phase velocity.

To extend the propagation to any direction in x, y, z space we use vector notation. For the description of the plane waves we choose in the x, y, z space the coordinate axes xi, yj, and zk, where i, j, k are unit vectors and a point in x, y, z space is given as r xi + yj + zk. (Note that k is used for the wave vector and for the unit vector in z direction.) When solving Eq. (5.4) by “separation of variables" one obtains for the wave vector in the direction of propagation kn kx i + ky j + kzk, where n is a unit vector and k 2π/λ (Figure 5.1). We need to find the components kx , ky , and kz for a wave moving in x, y, z space in direction n. To do this we evaluate the dot product k(n · i), k(n · j), and k(n · k) and obtain for kx , ky , and kz (Figure 5.1),

For the special case of the Ey component moving in the x direction we have θ 0◦ and φ 90◦, and get (Chapter 2),