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

have

The sample is assumed to be on the mirror in the movable arm of the interferometer, and we assume an amplitude reflection coefficient r and a phase change ϕ. We call y the displacement from the equal path length position and have

Introduction of R for r2 and for the frequency ν 1/λ, one has for the intensity

I ,

For very large values of y the cosine term oscillates so fast that the average will be zero. We call

The difference in phase change for reflection on the sample (π − ϕ) may also be written as (ϕ − π). The sign is of no importance. We can then factor out exp i(ϕ − π) and extend the frequency range formally to negative frequencies (as done above in Section 2.3).

−∞

This is the Fourier transform integral in the frequency domain for the function {A2r(ν) exp i(ϕ − π)}, and the corresponding Fourier transform integral in coordinate space is

−∞

Equation (A9.11) is the Fourier transformation of the data {I (y) − I (∞)}. Performing the Fourier transformation, we obtain for the integral in Eq. (A9.11) a function of complex numbers depending on ν, and call this function P (ν)−iQ(ν),

−∞

The background interferogram is obtained with a mirror in each arm of the Michelson interferometer, and one has for the transformation

{A2r exp i(−π)}

∞

∫ {IB (y) − IB (∞)} exp −i(2πν)dν PB (ν) − iQB (ν), (A9.13)

−∞

where r is assumed to be set to 1 because of the background. Dividing the transformations of the sample and background interferogram we have

where the sign of ϕ, that is, its relation to 2π, has to be chosen.

The optical constants are obtained by using Fresnel’s formulas for normal incidence and complex refractive index

where the sign of ϕ is important in Eq. (A9.21).

We have obtained from the Fourier transformation the complex numbers P (ν) and Q(ν) and could use them to calculate r and ϕ. From Fresnel’s formulas we calculate n and K.

See also on the CD

PF1. Fourier Transformation of the Gauss-Function. (see p. 332–333) PF2. Fourier Transformation of the Functions 1/(1 + x2). (see p. 333) PF3. Numerical Fourier Transformation. (see p. 334)

PF4. Step Functions.(see p.335)

PF6. Example of a Convolution Integral. (see p. 342)

PF7. Interferogram of Michelson interferometer. (see p. 350) PF8. Folding of the Spectrum. (see p. 351)

PF9. Apodization. (see p. 359)

C H A P T E R

Imaging

Using Wave Theory

10.1 INTRODUCTION

In geometrical optics we used the thin lens equation to find the image point of an object point when using a thin lens of focal length f . Using wave theory we assume that Huygens’ wavelets emerge from each point of the object and travel to the lens. The lens produces the diffraction pattern of the object in its focal plane, which may be seen as the Fourier transformation of the object pattern. The lens also produces the image of the object by refraction. A second Fourier transformation performed on the diffraction pattern results in a pattern having the shape of the object. Since the light travels forward, we associate it with the image. This is schematically shown in Figure 10.1.

The model we use for the description of image formation by a lens is, that one Fourier transformation is applied to the object pattern to obtain the diffraction pattern of the object and a second Fourier transformation is applied to the diffraction pattern to obtain the image pattern. Since we found that the Fourier transform of a Fourier transform is the original, but know that experimentally the image is not exactly the same as the object, we may ask the question, “Where is the perturbation entering the process?” It has been one of the great discoveries of optics (E. Abbe in Born and Wolf, 1964, xxi) that any perturbation of the diffraction pattern modifies the image. In Figure 10.2 we show how changes in the diffraction pattern, such as blocking off certain parts, change the image. We may also introduce phase shifts at some spots and not at others and obtain an image with much stronger contrast (F. Zernicke in Born and Wolf, 1964, xxi).

375

FIGURE 10.1 (a) The geometrical optical imaging process; (b) imaging process using wave theory. The Huygens’ wavelets of the object generate the diffraction pattern, and the Huygens’ wavelets of the diffraction pattern generate the image.

10.2SPATIAL WAVES AND BLACKENING CURVES, SPATIAL FREQUENCIES, AND FOURIER TRANSFORMATION

Using scalar diffraction theory, the Kirchhoff–Fresnel integral uses monochromatic light to describe the diffraction pattern of the light emerging from the object. A lens is used in Fraunhofer diffraction to have the diffraction pattern observed in the focal plane of the lens. This same integral may be written as a Fourier transform integral, as done in Chapter 3 for the diffraction on a slit. The coordinates of the object, the slit, are length coordinates in the length domain. The coordinates of the Fourier transformation, the diffraction pattern of the slit, are coordinates in the spatial frequency domain and have 1/length dimensions. We note that we deal with an amplitude diffraction pattern in the frequency domain, which contains phase information, even if we started with a real function in the object plane. After one applies a second Fourier transformation on the diffraction pattern (in the spatial frequency domain), the result is a geometrical image pattern similar to the original object and appearing in the space domain.

In our model description we use the first Fourier transformation from the geometrical space domain into the spatial frequency domain. The object is described by geometrical spatial waves and the Fourier transformation describes the fre-

FIGURE 10.2 (a) Schematic of image formation by geometrical optics, the focal plane is indicated;

(b) the object is a grid, the diffraction pattern is shown, and the image is again a grid. When masks 1 and 2 are placed at the focal plane of the lens, certain parts of the diffraction pattern are blocked off. The resulting images are shown on the right.

quency spectrum of these spatial waves. The object function h(y) is interpreted as a superposition of spatial waves and the result of the superposition is recorded as the blackening of a photographic plate. The maxima of a spatial wave corresponds to black, the minima corresponds to white, and the gray level indicates zero. By doing so, we attribute phase information to these waves. Superposition of a black maxima (+A) and a white minima (−A) results in gray (0). In the section on holography, we present more details on the presentation of phase information with the black, white, and gray of the photographic plate. In FileFig 10.1 we show this concept in mathematical terms, considering the Fourier series

with frequencies fn (2n + 1)/2a, where a is a certain length constant in the spatial domain.

The spatial waves with wavelengths 0 2a, 1 2a/3, 2 2a/5 correspond to spatial frequencies f0, f1, f2 and are shown in FileFig 10.1. The superposition of these three waves is also shown and one observes that the su-

perposition of a large number of such spatial waves results in a rectangular shape.

Fourier series of cosine functions for the composition of a rectangular-shaped object. Different numbers of elements of the sum are plotted separately and as a sum for comparison.

W1FTSERIS

Fourier Series of Spatial Wavelength λ for Interval from −1 to 1 (Shown to 2)

For N 0 the only term is a sine wave from −1 to 1, of wavelength λ 2. For N 1 a sine term with 1/3 of λ and smaller amplitude is added. For N 2 a term with 1/5 of λ and smaller amplitude is added, and so on. If N is large, we see a perfect step function. For smaller N (in the 20th), we see Gibb’s phenomenon, the corners are not round, and there is overshooting. For large N it disappears.

For larger and larger N one can see how more and more waves with shorter and shorter wavelengths are used to build the step function.

W2FTCFTS

Example of Real fft and Complex cfft on a Real Object Function

1.The real FT fft

The Object: i : 0, 1 . . . 255

The real Fourier transformation

c : ff t(y) Nc : last(c) Nc 128 j : 0 . . . Nc − 1.

The inverse Fourier transformation

x : iff t((c))

Nx : last(x)

Nx 255

k : 0 . . . Nx − 1.