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# مساعده فى حل اسئله

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عضو جديد  مساعده فى حل اسئله

السلام عليكم ورحمه الله وبركاته
ارجوا مساعدتى فى حل مسائل الاتيه حيث انها مسائل شبتر 2 من كتاب Fundamentals of optoelectronics ولم اجد من يساعدنى وامتحاناتى قربت اى سؤال من اى عضو وجزاكم الله خيراااااااا

PROBLEMS
An asterisk indicates a problem that is best done with a computer.
1. Using the boundary conditions for a TM wave, confirm Equation 2.23 for the eigenvalue equation of an asymmetric waveguide.
2. Develop the eigenvalue equation for the TE mode in a symmetric waveguide.
3. Develop the eigenvalue equation for the TM modes in a symmetric waveguide.
*4 For an asymmetric planar waveguide with nf = 1.5, n = 1.47, and n = 1.0,
determine the allowed values of J3 for the TE modes in a waveguide with thickness h 7ILm. Assume the excitation wavelength is 1 m.
*5• For the waveguide described in Problem 4, find the cutoff wavelengths for the TE0 and TE1 modes.
*6. Which is the lowest-order mode in an asymmetric waveguide, the TE0 mode or the TM0 mode? Prove your case graphically.
7. Use the Fresnel reflection formula for TE waves to determine the attenuation
coefficient for a leaky ray in a waveguide. Assume that the waveguide is symmetric, and is 5Om thick. Assume the power lost per reflection is given by Fl = 1 — 1r21, where r is given by Equation 1.90. Express the answer in terms of nepers/meter.
Assume the guiding layer has index 1.5, the surrounding layers have index 1.45, and the ray is incident at 75°.
*8. Evaluate the size of a confined TE mode as a function of the guiding film
dimension. Consider a symmetric waveguide with guiding index of 3.5 and
surrounding indices of 3.49. The wavelength is 1gm.
a. Write an explicit description of the field for the TE0 mode.
b. Define the mode size of the field by the distance between the points where the amplitude is reduced to 1/2 of the peak intensity. Find the full width (at half maximum) of the TE0 mode for film thickness h, ranging from 0.1 to 1Om. Evaluate at least five thicknesses.
c. Approximately what film thickness leads to the smallest mode7
d. Calculate the mode confinement for h 1Om, h 1gm, and h = O.1m.
9. For the waveguide described in Example 2.1, at what distance into the substrate is the evanescent field equal to 0.00 1E0, where E0 is the peak amplitude of the mode? Determine this for each mode. At what distance into the cover is the field the same value? What does this tell you about how thick to make the cladding layers on a waveguide?
10. Consider a waveguide with a guiding film 5m thick, surrounded by thick films with index 1.45. The wavelength is 1gm.
a. If the guiding film index is 1.5, calculate the numerical aperture of the waveguide.
b. What is the mode confinement for TE0 mode in this waveguide?
c. If the guiding film is made to be 1 m, what is the mode confinement of the TE0 mode?
* 11. Consider a planar slab waveguide of infinite extent in the y and z directions. The guiding film index is 1.5, the substrate index is 1.48, and the cover index is 1.0. The thickness is h = 2gm. The waveguide is excited with a 1.3 m source. For TE modes:
a. What is the range of allowed f3 values for this waveguide?
b. What is the numerical aperture for this waveguide?
c. Numerically or graphically, determine the allowed values of 13 and K for h = 2gm.
d. How many modes will this waveguide carry if the excitation wavelength is
O.6OOm?
12. Show that the eigenvalue equation for /3 for the TE mode in an asymmetric
waveguide reduces to that of the symmetric waveguide when n. = n.
13. Show that H = -- E,, for a TE wave.
LLW
14. For an asymmetric planar waveguide with flf 1.5, n = 1.48, and a thickness of
8gm, how many TE modes for A = 1m will there be under the following conditions:
a. Covered by air (n 1).
b. Covered by water (n = 1.33).
c. Covered by another substrate (n = 1.45).
d. Explain in words why the number of modes did or did not change in these three cases.
e. If n = 1.45, what thickness waveguide is needed to increase the number of modes to ten?
f How will the number of TM modes vary in a, b, and c?
*15. Given a symmetric waveguide with flf 1.5, n 1.47, A 1.Om, and h 3ILm, determine the fraction of power carried in the cladding for the TE0 and TE1 modes.
16. Derive the mode cutoff condition for the TE mode in terms of V for the symmetric waveguide.
* 17. Repeat Example 2.1 for the TM mode case. Make plots of the allowed modes similar
to those shown in Figure 2.8.
* 18. Plot the mode profiles for the TE0 and TE1 modes in a slab waveguide with a core
index flf = 1.5, n = 1.49, and n = 1. The film thickness is l0m thick, and the
guided wavelength is 1rm.
* 19. Consider an asymmetric planar waveguide with a film index flf = 1.50, a substrate
index n = 1.495, and a cover index n = 1.40.
a. If the vacuum wavelength of the guided light is 1 m, what is the thinnest that
the guiding layer can be to support the TE0 mode?
b. Increase the thickness of the guide found in part a by 1 percent, and determine
the eigenvalue /3 of the TE mode.
c. Calculate the confinement factor for this mode which is near cutoff.
d. Plot the mode profile for this mode. What can you generalize about modes near
cutoff?
*20. A planar waveguide is made with flf = 1.48, n = 1.46, and n = 1.44. The
thickness of the guiding film is 10rm. What is the longest wavelength that can be
carried in a TE mode in this waveguide?

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