Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007
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3.3. FRESNEL DIFFRACTION, FAR FIELD APPROXIMATION, AND FRAUNHOFER OBSERVATION |
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we speak of Fresnel diffraction. We use small angle approximation to show the differences in these approaches to diffraction.
3.3.1 Small Angle Approximation in Cartesian Coordinates
Since the distance from aperture to observation screen is large, we may use small angle approximation for the diffraction angle. We consider the integral in Eq. (3.4),
(A/R) exp(ikR)(1/ρ) exp(ikρ) cos θ dσ |
(3.15) |
opening of aperture
and neglect the cos θ factor. The factor (A/R) exp(ikR) is a constant and can be taken before the integral. Using only a one-dimensional approach for the Y and y directions, we have further to consider,
u(Y ) C (eikρ )/ρ dy. |
(3.16) |
opening of aperture
The coordinates are shown in Figure 3.8, using X for the distance from the aperture to the screen. Since the distance X between the observation screen and the aperture is large, we take ρ in the denominator as a constant, but not in the exponential. We develop ρ using the coordinates of Figure 3.8 and have
ρ (Y − y)2 + X2 1/2 |
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X + (1/2X)(Y − y)2. |
(3.17) |
Inserting Eq. (3.17) into (3.16) and including 1/ρ and eikX in a new constant C , we have
u(Y ) C exp{ik(1/2X)(Y − y)2} dy. |
(3.18) |
opening of aperture
FIGURE 3.8 Coordinates for small angle approximation.
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We write the exponent of Eq. (3.18) as |
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ik{+(1/2X)(y2) + (1/2X)(Y 2) − (yY )/X}. |
(3.19) |
3.3.2 Fresnel, Far Field, and Fraunhofer Diffraction
3.3.2.1 Fresnel Diffraction
If we do not neglect the quadratic terms in Eq. (3.19), we are back to Eq. (3.18) and have
u(Y ) C exp{ik(1/2X)(Y − y)2} dy. |
(3.20) |
opening of aperture
This is called Fresnel diffraction. The integral may be expressed using Fresnel’s integrals.
3.3.2.2 Far Field Diffraction
In Eq. (3.19) we neglect the quadratic term in y and consider (1/2X)(Y 2) as a constant and include it in C , we have
u(Y ) C e−ik(yY/X) dy. |
(3.21) |
opening of aperture
This is the far field approximation.
3.3.2.3 Fraunhofer Diffraction
Fraunhofer diffracton is also far field approximation, but we do not have to go so far, because we observe the pattern in the focal plane of a lens. In this case we have to find the effect on the wavefront when light is focused on the focal plane of a lens with focal length f . We obtain the result that one has the same integral for Fraunhofer diffraction as one has for far field approximation.
Small angle approximation was obtained by considering in the integral the exponent, Eq. (3.19),
ik{+(1/2X)(y2) + (1/2X)(Y 2) − (yY )/X}. |
(3.22) |
For far field approximation we neglected the quadratic term in y and considered the term in X and Y as a constant.
Now we do not neglect the quadratic term in y and show that this term is compensated by the effect of the lens (for a detailed discussion see Goodman, 1988, p. 78.) We look at the wavefront passing through the lens. The wavefront
3.4. FAR FIELD AND FRAUNHOFER DIFFRACTION |
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FIGURE 3.9 Coordinates for the calculation of the change of the wavefront by a lens.
is converging to the focal point of the lens (see Figure 3.9). Over the length y we have an increasing phase shift γ , which is calculated from
y2 + (f − γ )2 f 2. |
(3.23) |
Neglecting γ 2 in (f 2 − 2f γ + γ 2), we have for the phase shift |
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γ (y2)/2f. |
(3.24) |
This shift must be subtracted as the phase shift in our integral. We get in the exponent, that is, Eq. (3.22), with X f ,
ik{+y2/2f + (1/2f )(Y 2) − (yY )/f } − ik(y2)/2f. |
(3.25) |
The new quadratic term in y cancels the old one, which we neglected in the far field approximation. We then have to consider
u(Y ) C e−ik(yY/X) dy. |
(3.26) |
This integral is the same as obtained in the far field approximation.
3.4 FAR FIELD AND FRAUNHOFER DIFFRACTION
The far field diffraction and Fraunhofer diffraction of the Kirchhoff–Fresnel integral have the same mathematical appearance. The only difference is that in
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far field approximation the diffraction pattern is observed on a faraway screen, whereas in Fraunhofer diffraction the observation screen is placed at the focal plane of a lens and that may be closer to the aperture.
We now discuss the diffraction pattern of various geometrical shapes of apertures. From Eq. (3.4) we have
u(Y ) C (eikρ ) dσ, |
(3.27) |
where dσ is the surface element of the aperture and 1/ρ has been taken before the integral, included in C, which contains all constant terms. Note that in Eq. (3.27) we have not used small angle approximation in the exponent.
3.4.1 Diffraction on a Slit
The diffraction on a slit is important because it is a simple one-dimensional diffraction problem and appears in the diffraction pattern of all types of gratings and in other diffraction-related phenomena. The coordinates for the calculation of the diffraction on a slit are shown in Figure 3.10. We divide the opening into N intervals y and sum up all waves traveling in direction θ. In the first step it is assumed that these waves are generated at the limits of all intervals and all adjacent waves have the same optical path difference δ. This is similar to the discussion in Chapter 2 for interference on an array (Eq. (2.107)). The optical path difference between waves of finite steps is y sin θ and replacing qδ in the sum of Eq. (2.107), we have
e−ik( y sin θ). |
(3.28) |
Making the step y infinitesimally small, one gets the integral |
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u(Y ) C e−ik(y sin θ) dy, |
(3.29) |
where C includes all constant terms. The integration is from −d/2 to d/2, and we have to calculate
u(Y ) C |
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e−ik(y sin θ) dy, |
(3.30) |
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y1 −d/2 |
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or in small angle approximation with sin θ Y/X, |
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u(Y ) C |
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e−ik(yY/X) dy. |
(3.31) |
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y1 −d/2 |
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The result of the integration of Eq. (3.31) is |
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u Cd sin(πd sin θ/λ)/{(πd sin θ)/λ}. |
(3.32) |
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The normalized intensity is written |
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I Io{sin(πd sin θ/λ)/{(πd sin θ/λ)}2. |
(3.33) |
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3.4. FAR FIELD AND FRAUNHOFER DIFFRACTION |
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FIGURE 3.10 Coordinates for the calculation of the diffraction on a slit: (a) phase difference of all wavelets with respect to the center one used for the summation process; (b) path difference between ρα and ρα+1.
Or in small angle approximation, |
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I Io[{sin(πdY/Xλ)}/{πdY/Xλ}]2. |
(3.34) |
The diffraction pattern of a slit has a periodic appearance with decreasing intensity of the maxima. The graph in FileFig 3.2 shows three diffraction patterns. The wider ones are for the smaller slit openings. In Figure 3.11 we show a photograph of a diffraction pattern of a slit. In Application FF2 the width of the diffraction pattern with respect to changes in λ and d is studied.
FIGURE 3.11 Diffraction pattern formed by a single slit (from M. Cagnet, M. Francon, J.C. Thrierr, Atlas of Optical Phenomena, Springer-Verlag, Heidelberg, 1962).
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The main maxima is at Y 0. At that point we have sin 0/0, and a similar discussion to that presented in Chapter 2 results in I Io. The angle from the center of the slit to the first minimum of the diffraction pattern is called the diffraction angle θ λ/d, and is used when discussing resolution or the Fresnel number for characterizing the losses of a laser cavity. The side maxima are approximately at Y/X (m + 1/2)λ/d, which is approximately centered between the minima. An exact determination is done using FileFig 3.3 and Application FF3.
FileFig 3.2 |
(D2FASLITS) |
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A graph of the intensity of the diffraction pattern on a slit. By changing the width d, we see that the width of the diffraction pattern is inversely proportional to the width of the slit. By changing λ the width of the diffraction pattern is proportional to the wavelength. This is a general property one observes for diffraction patterns. The minima are at mλ/d.
D2FASLITS
Diffraction on a Slit of Width d at Wavelength λ
X is distance; slit-screen, Y is coordinate on screen. For small angles, Y/X is proportional to the diffraction angle θ. MCAD notice the singularity at 0. For the graph we get around it using the range Y −100.1, −99.1 to 100.1. All lengths are in mm.
Three slits with different widths d1, d2, and d3:
d1 ≡ .08 |
d2 ≡ .12 |
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X : 4000 |
λ ≡ .0005 |
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d3 ≡ .16 |
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θ ≡ −2, −1.99 . . . 1.
3.4. FAR FIELD AND FRAUNHOFER DIFFRACTION |
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q
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Application 3.2. The dependence of the width of the diffraction pattern on values of λ and d may be studied by changing λ to λ/2 and 2λ and d to d/2 and 4d and considering the ratio of d/λ.
FileFig 3.3 (D3FASLITEXS)
Expanded graph of the intensity of the side maxima and minima for the diffraction pattern on a slit, Y 18, 19 . . . 150, X 4000, and λ 0.0005. Numerical determination of the values of the side maxima and minima.
D3FASLITEXS is only on the CD.
Application 3.3.
1.Give the position of the first five minima.
2.Determine the maxima. The values of the secondary maxima are obtained by differentiation of I IO [sin{πdY/Xλ}/{πdY/Xλ}]2 with respect to Y and setting the resulting expression equal to 0. One may perform the differentiation with a symbolic computer calculation program. One obtains the transcendental equation πyd/λ tan(πyd/λ), where y Y/X. The solution of the transcendental equation may be obtained by plotting πyd/λ and tan(πyd/λ) on the same graph and using the intersections. Determine the values of the first five intersections, corresponding to the first five side maxima, and compare with the values read from the graph in FF3 and with the approximate formula Y/X (m + 1/2)λ/d.
3.Calculate the intensity ratio of the first, second, and third maxima to the zeroth maximum and compare with the theoretical values from the intensity formula for the diffraction on a slit.
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3.4.2 Diffraction on a Slit and Fourier Transformation
The integral for the calculation of the diffraction on a slit in small angle approximation (Eq. (3.31)), is
u(Y ) C |
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y2 d/2 |
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exp −i2π(y/λ)(Y/X) dy, |
(3.35) |
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y1−d/2 |
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where we used k 2π/λ.We do the following substitutions |
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x Y/X, |
a d/2λ |
(3.36) |
and have |
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u(x) C |
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exp[−i2π(ν)(x)]dν. |
(3.37) |
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To write the integral with integration limits from −∞ to ∞ we define the function Q(ν) as
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Q(v) 1 for x between − a and a |
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Q(v) 0 otherwise. |
(3.38) |
We then have |
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u(x) C |
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Q(ν) exp −i2π(ν)(x) dν. |
(3.39) |
−∞
The integral u(x) in Eq. (3.39) is the Fourier transform of Q(ν). We may integrate and obtain
u(x) |
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(sin 2πax)/(2πax) |
(3.40) |
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similar to that obtained in Eq. (3.40). We have the result that the Fourier transform of the slit function Q(v) with opening width a is the function (sin 2πax)/(2πax) which is sometimes called a sinc-function. When Q(ν) is not the slit function, but given as a numerical function or a complicated analytical function, one can not obtain an analytical expression for u(x) but one can calculate the numerical Fourier transformation. Most computational programs offer Fourier transformation. In FileFig 3.4 we write a step function for xi with i 0 to 255, assuming that xi 1 for 0 to d and otherwise 0, and plot xi as a function of i/255. Here we consider only half of the slit and do the Fourier transformation cj , shown as the graph of cj depending on j/255, plotted from 0 to 0.5. Since we cover with the input data only half of the slit, we get only half of the diffraction pattern. However, because the Fourier transformation is real, and we have used the fast Fourier transformation (FFT), the Fourier transformation cj shows only N 128 points. The inverse Fourier transformation results again in 256 points. More details on this subject are given in Chapter 9 on Fourier transformations.
3.4. FAR FIELD AND FRAUNHOFER DIFFRACTION |
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FileFig 3.4 |
(D4FASLITFT) |
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Fourier transformation of a step function. The step function has been defined for a width of d 20. The number of points to be used is 2n−1. The real and imaginary parts of the Fourier transformation are shown. The Fourier transformation of the Fourier transformation is also calculated and the real part is again a step function, and the imaginary part is 0.
D4FASLITFT is only on the CD.
Application 3.4. For several widths of the step, read off the value of the first zero of the transform. A formula for the value of the first zero may be obtained from Eq. (3.40). Compare with the value of the graph of the Fourier transformation and with Application FF3.
3.4.3 Rectangular Aperture
The diffraction pattern of the rectangular aperture is easy to calculate as an extension from one dimension, as used for the slit, to two dimensions. In the next section we do it in small angle approximation and show that the integral is useful for the calculation of far field diffraction on a round aperture, important for all optical devices and instruments with circular symmetry.
For the calculation of the diffraction pattern of a rectangular aperture with dimensions d in the y direction and a in the z direction, we go back to the integral of Eq. (3.31).
y2 d/2
u(Y ) e−ik(yY/X) dy.
y1−d/2
Integration over the y direction will be extended to include the z direction; see Figure 3.12. As a result, the diffraction pattern is described by a product of two integrals of the type of Eq. (3.31), one over the opening d in the y direction and the other over the opening a in the z direction.
u(Y, Z) |
y2 d/2 |
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exp −i2π(y/λ)(Y/X) dy |
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exp −i2π(z/λ)(Z/X) dz. |
(3.41) |
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We obtain |
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u(Y, Z) C{d sin(πdY/Xλ)/(πdY/Xλ){a sin(πaZ/Xλ)/(πaZ/Xλ)} |
(3.42) |
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and we have for the normalized intensity |
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I Io[{sin(πdY/λX)}/{πdY/λX}]2[{sin(πaZ/λX)}/{πaZ/λX}]2. |
(3.43) |
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FIGURE 3.12 Coordinates for the calculation of the diffraction pattern of a rectangular aperture.
FIGURE 3.13 Far field diffraction pattern of a rectangular aperture (from M. Cagnet, M. Francon, and J.C. Thrierr, Atlas of Optical Phenomena, Springer-Verlag, Heidelberg, 1962).
In FileFig 3.5 we have a 3-D graph of the calculated intensity diffraction pattern with maxima and minima in the y and z directions. The diffraction angle, that is, the angle from the center of the aperture to the first minimum, is in the y direction Y/X λ/d, and in the z direction Z/X λ/d. A photograph of the diffraction pattern is shown in Figure 3.13.