1. Consider a plane wave of wavelength 6 × 10-5 cm incident normally
on a circular aperture of radius 0.01 cm. Calculate the positions of
the brightest and the darkest points on the axis. Get solution
2. What would happen if the circular aperture in Problem 20.1 is replaced by a circular disc of the same radius? Get solution
3. A plane wave (λ = 6 × 10-5 cm) is incident normally on a circular aperture of radius a.(a) Assume a = 1 mm. Calculate the values of z (on the axis) for which maximum intensity will occur. Plot the intensity as a function of z and interpret physically. Repeat the calculations for λ = 5 × 10-5 cm and discuss chromatic aberration of a zone plate.(b) Assume z = 50 cm. Calculate the values of a for which minimum intensity will occur on the axial point. Plot the intensity variation as a function of a and interpret physically. Get solution
4. Consider a circular aperture of diameter 2 mm illuminated by a plane wave. The most intense point on the axis is at a distance of 200 cm from the aperture. Calculate the wavelength. [Ans: 5 × 10-5 cm] Get solution
5. If a zone-plate has to have a principle focal length of 50 cm corresponding to λ = 6 × 10-5 cm, obtain an expression for the radii of different zones. What would be its principle focal length for λ = 5 × 10-5 cm? [...mm, 60 cm] Get solution
6. In a zone-plate, the second, fourth, sixth…zones are blackened; what would happen if instead the 1st, 3rd, 5th, etc., zones were blackened? Get solution
7. (a) A plane wave is incident normally on a straight edge (see Fig. 20.24). Show that the field at an arbitrary point P is given by...where ....(b) Assume λ0 = 5000 Å and d = 100 cm. Write approximately the values of I/I0 at the points O, P (y = 0.5 mm), Q (y = 1 mm) and R (y = -1 mm) where O is at the edge of the geometrical shadow.... Get solution
8. Consider a straight edge being illuminated by a parallel beam of light with λ = 6 × 10-5 cm. Calculate the positions of the first two maxima and minima on a screen at a distance of 50 cm from the edge. Get solution
9. In a straight edge diffraction pattern, one observes that the most intense maximum occurs at a distance of 1 mm from the edge of the geometrical shadow. Calculate the wavelength of light, if the distance between the screen and the straight edge is 300 cm. [Ans. ≈ 4480 Å] Get solution
10. In a straight edge diffraction pattern, if the wavelength of the light used is 6000 Å and if the distance between the screen and the straight edge is 100 cm, calculate the distance between the most intense maximum and the next maximum. Find approximately the distance in centimeters inside the geometrical shadow where I /I0 = 0.1.[Ans.y ≈ 0.027 cm] Get solution
11. Consider a plane wave falling normally on a narrow slit of width 0.5 mm. If the wavelength of light is 6 × 10-5 cm, calculate the distance between the slit and the screen so that the value of v1 would be 0.5, 1.0, 1.5 and 5.0 (see Fig. 20.19 – 20.22). Discuss the transition to the Fraunhofer region. Get solution
12. Consider the Fresnel diffraction pattern produced by a plane wave incident normally on a slit of width b. Assume λ = 5 × 10-5 cm, d = 100 cm. Using Table 20.1, approximately calculate the intensity values (for b = 0.1 cm) at y = 0, ± 0.05 cm, ± 0.1 cm. Repeat theanalysis for b = 5 cm. Get solution
13. In Sec. 19.7 we obtained the diffraction pattern of a circular aperture of radius a. Obtain the diffraction pattern of an annular aperture bounded by circles of radii a1 and a2 (> a1). [This Problem is already given as Problem 19.5]. Get solution
14. Consider a rectangular aperture of dimensions 0.2 mm × 0.3 mm. Obtain the positions of the first few maxima and minima in the Fraunhofer diffraction pattern along directions parallel to the length and breadth of the rectangle. Assume λ = 5 × 10-5 cm and that the diffraction pattern is produced at the focal plane of a lens of focal length 20 cm. Get solution
15. The Fraunhofer diffraction pattern of a circular aperture (of radius 0.5 mm) is observed on the focal plane of a convex lens of focal length 20 cm. Calculate the radii of the first and the second dark rings. Assume λ = 5.5 × 10-5 cm.[Ans. 0.13 mm, 0.18 mm] Get solution
16. In the above problem, calculate the area of the patch (on focal plane) which will contain 95% of the total energy. Get solution
17. (a) The output of a He-Ne laser (λ = 6328 Å) can be assumed to be Gaussian with plane phase front. For w0 = 1 mm and w0 = 0.2 mm, calculate the beam diameter at z = 20 m.(b) Repeat the calculation for λ = 5000 Å and interpret the results physically. Get solution
18. A Gaussian beam is coming out of a laser. Assume λ = 6000 Å and that at z = 0, the beam width is 1 mm and the phase front is plane. After traversing 10 m through vacuum what will be (a) the beam width and (b) the radius of curvature of the phase front.... Get solution
19. A plane wave of intensity I0 is incident normally on a circular aperture as shown in Fig. 20.25. What will be the intensity on the axial point P? [Hint: You may use Eq. (25)] Get solution
20. Show that a phase variation of the type ...represents a diverging spherical wave of radius R. Get solution
21. Consider a resonator consisting of a plane mirror and a concave mirror of radius of curvature R (see Fig. 20.26). Assume λ = 1 μm, R = 100 cm and the distance between the 2 mirrors to be 50 cm. Calculate the spot size of the Gaussian beam.... Get solution
22. The output of a semiconductor laser can be approximated described by a Gaussian function with two different widths along the transverse (wT) and lateral (wL) directions as...where x and y represent axes parallel and perpendicular to the junction plane. Typically wT ≈ 0.5 μm and wL = 2 μm. Discuss the far field of this beam (see Fig. 20.27).... Get solution
2. What would happen if the circular aperture in Problem 20.1 is replaced by a circular disc of the same radius? Get solution
3. A plane wave (λ = 6 × 10-5 cm) is incident normally on a circular aperture of radius a.(a) Assume a = 1 mm. Calculate the values of z (on the axis) for which maximum intensity will occur. Plot the intensity as a function of z and interpret physically. Repeat the calculations for λ = 5 × 10-5 cm and discuss chromatic aberration of a zone plate.(b) Assume z = 50 cm. Calculate the values of a for which minimum intensity will occur on the axial point. Plot the intensity variation as a function of a and interpret physically. Get solution
4. Consider a circular aperture of diameter 2 mm illuminated by a plane wave. The most intense point on the axis is at a distance of 200 cm from the aperture. Calculate the wavelength. [Ans: 5 × 10-5 cm] Get solution
5. If a zone-plate has to have a principle focal length of 50 cm corresponding to λ = 6 × 10-5 cm, obtain an expression for the radii of different zones. What would be its principle focal length for λ = 5 × 10-5 cm? [...mm, 60 cm] Get solution
6. In a zone-plate, the second, fourth, sixth…zones are blackened; what would happen if instead the 1st, 3rd, 5th, etc., zones were blackened? Get solution
7. (a) A plane wave is incident normally on a straight edge (see Fig. 20.24). Show that the field at an arbitrary point P is given by...where ....(b) Assume λ0 = 5000 Å and d = 100 cm. Write approximately the values of I/I0 at the points O, P (y = 0.5 mm), Q (y = 1 mm) and R (y = -1 mm) where O is at the edge of the geometrical shadow.... Get solution
8. Consider a straight edge being illuminated by a parallel beam of light with λ = 6 × 10-5 cm. Calculate the positions of the first two maxima and minima on a screen at a distance of 50 cm from the edge. Get solution
9. In a straight edge diffraction pattern, one observes that the most intense maximum occurs at a distance of 1 mm from the edge of the geometrical shadow. Calculate the wavelength of light, if the distance between the screen and the straight edge is 300 cm. [Ans. ≈ 4480 Å] Get solution
10. In a straight edge diffraction pattern, if the wavelength of the light used is 6000 Å and if the distance between the screen and the straight edge is 100 cm, calculate the distance between the most intense maximum and the next maximum. Find approximately the distance in centimeters inside the geometrical shadow where I /I0 = 0.1.[Ans.y ≈ 0.027 cm] Get solution
11. Consider a plane wave falling normally on a narrow slit of width 0.5 mm. If the wavelength of light is 6 × 10-5 cm, calculate the distance between the slit and the screen so that the value of v1 would be 0.5, 1.0, 1.5 and 5.0 (see Fig. 20.19 – 20.22). Discuss the transition to the Fraunhofer region. Get solution
12. Consider the Fresnel diffraction pattern produced by a plane wave incident normally on a slit of width b. Assume λ = 5 × 10-5 cm, d = 100 cm. Using Table 20.1, approximately calculate the intensity values (for b = 0.1 cm) at y = 0, ± 0.05 cm, ± 0.1 cm. Repeat theanalysis for b = 5 cm. Get solution
13. In Sec. 19.7 we obtained the diffraction pattern of a circular aperture of radius a. Obtain the diffraction pattern of an annular aperture bounded by circles of radii a1 and a2 (> a1). [This Problem is already given as Problem 19.5]. Get solution
14. Consider a rectangular aperture of dimensions 0.2 mm × 0.3 mm. Obtain the positions of the first few maxima and minima in the Fraunhofer diffraction pattern along directions parallel to the length and breadth of the rectangle. Assume λ = 5 × 10-5 cm and that the diffraction pattern is produced at the focal plane of a lens of focal length 20 cm. Get solution
15. The Fraunhofer diffraction pattern of a circular aperture (of radius 0.5 mm) is observed on the focal plane of a convex lens of focal length 20 cm. Calculate the radii of the first and the second dark rings. Assume λ = 5.5 × 10-5 cm.[Ans. 0.13 mm, 0.18 mm] Get solution
16. In the above problem, calculate the area of the patch (on focal plane) which will contain 95% of the total energy. Get solution
17. (a) The output of a He-Ne laser (λ = 6328 Å) can be assumed to be Gaussian with plane phase front. For w0 = 1 mm and w0 = 0.2 mm, calculate the beam diameter at z = 20 m.(b) Repeat the calculation for λ = 5000 Å and interpret the results physically. Get solution
18. A Gaussian beam is coming out of a laser. Assume λ = 6000 Å and that at z = 0, the beam width is 1 mm and the phase front is plane. After traversing 10 m through vacuum what will be (a) the beam width and (b) the radius of curvature of the phase front.... Get solution
19. A plane wave of intensity I0 is incident normally on a circular aperture as shown in Fig. 20.25. What will be the intensity on the axial point P? [Hint: You may use Eq. (25)] Get solution
20. Show that a phase variation of the type ...represents a diverging spherical wave of radius R. Get solution
21. Consider a resonator consisting of a plane mirror and a concave mirror of radius of curvature R (see Fig. 20.26). Assume λ = 1 μm, R = 100 cm and the distance between the 2 mirrors to be 50 cm. Calculate the spot size of the Gaussian beam.... Get solution
22. The output of a semiconductor laser can be approximated described by a Gaussian function with two different widths along the transverse (wT) and lateral (wL) directions as...where x and y represent axes parallel and perpendicular to the junction plane. Typically wT ≈ 0.5 μm and wL = 2 μm. Discuss the far field of this beam (see Fig. 20.27).... Get solution
