Step index planar waveguide

Transcription

1 N. Dubreuil S. Lebrun Exam without document Pocket calculator permitted Duration of the exam: 2 hours The exam takes the form of a multiple choice test. Annexes are given at the end of the text. ********************************************************************************** INSTRUCTIONS : Complete your name on the top of the answer sheet that is given with the text of the exam. For each question tick your selected answer(s) in the blank column at the right of the number of the question. When it is required, justify your answer on a separate page sheet. We apply a negative notation in case of a false response. At the end of the exam give back the 8 pages text, the answer sheet and the separate page sheet. ********************************************************************************** Step index planar waveguide Let s consider the step index symmetrical planar waveguide in figure 1 with n 1 = 1.55 ; n 2 = d = 2.5µm. d y x z n 2 n 1 n 2 Infinite along y and z z is the direction of propagation Figure 1 1. This waveguide is singlemode for (Justify your answer) : < µm > µm any wavelength d) < µm 1

2 2. Such a waveguide can always support at least one mode, whatever the operating wavelength. TRUE FALSE 3. In such a waveguide the frequency cutoffs of TM modes are the same than the frequency cutoffs of TE modes (Justify your answer). TRUE FALSE 4. Considering figure 1, the component E y is : transverse longitudinal normal d) tangential 5. Considering figure 1, the component E z is : transverse longitudinal normal d) tangential 6. Now the distance d between the two parallel planes is 3.5 µm. Let s consider an operating wavelength of 1 µm. How many symmetrical TE modes can propagate in the waveguide (see fig. 2, annexe)? d) 4 7. Give an approximation of, the longitudinal propagation constant of the fundamental TE mode. See fig. 2, annexe (Justify your answer). 2x10 4 m 1 9x10 6 m 1 4x10 6 m 1 d) 3x10 6 m 1 8. Give an approximation of, the propagation angle of the fundamental TE mode (Justify your answer) d) In such a waveguide, the effective refractive index of a guided mode : varies between 0 and 1 varies with the x coordinate is always smaller than the refractive index of the core d) decrease with V 2

3 Step index fiber We consider a two layer step index fiber. The refractive index of the core is and the refractive index of the cladding is The radius of the fiber is 4.5 µm. The operating wavelength is µm. 10. Is the weak guidance approximation correct in this case (Justify your answer)? YES NO 11. How many modes can support the fiber? See fig. 3, annexe d) Let s consider mode LP11. How is this mode for V=2.405? It is confined in the core It is similar to a plane wave d) It presents a maximum of intensity in r=0 It spreads approximately half in the core and half in the cladding 13. How is this mode for V>>2.405? It is confined in the core It is similar to a plane wave d) It presents a maximum of intensity in r=0 It spreads approximately half in the core and half in the cladding 14. We consider below intensity photos of LP modes at the output of a fiber. The white circle corresponds to the core of the fiber. White is maximum, black minimum. Give the name of each mode in the answer sheet. d) 3

4 15. We consider a two layer step index fiber. The numerical aperture of the fiber is 0.1. The radius of the fiber is a=4.5 µm. The operating wavelength is 1.5 µm. We recall the Marcuse formula that enables to calculate the waist w of the laser beam that fits at best the LP 01 mode of the fiber. What is the value of w? 12 µm 6 µm 5 µm d) 3 µm Wave equation We consider the propagation of a wave through a linear, isotropic and inhomogeneous material characterized by a refractive index profile, invariant along the direction, with a normalized function and the index parameter such as. The complex amplitude of the electric field propagating through the material is denoted. 16. The propagating wave is characterized by : A weak transverse component A purely transverse component A purely longitudinal component d) A weak longitudinal component 17. The complex amplitude of the electric field follows the equation:, with d), with 18. The wave equation can be written into the following form, (1) with a spatial differential operator and a spatial dependent operator. Select one proposition: contains a transverse Laplacian related to dispersion effect, and depicts the inhomogeneity of the material. contains a transverse Laplacian related to diffraction effect, and depicts the inhomogeneity of the material. 4 contains a transverse Laplacian related to diffraction effect, and accounts for optical nonlinearity. d) contains a transverse Laplacian related to dispersion effect, and accounts for optical nonlinearity

5 In the following, the refractive index profile is characterized by. 19. In order to observe guiding effect, the refractive index profile should be set equal to d) 20. Give the expression for the operator d) We next simulate the propagation of a Gaussian beam defined at by. One could then rewrite the wave equation (1) into a transformed form, (2) with and two length parameters (length unit), and. 21. The length parameter is equal to (no demonstration is needed): d) 22. The length parameter is equal to (Justify your answer by referring to the previous questions): a c b d 23. The parameter is set equal to 4 µm, and. For, calculate the waist for which the initial shape of the Gaussian beam will be conserved during its propagation through the material (Justify your answer by referring to the previous questions) 2 µm 14 µm 7 µm d) 4 µm 5

6 Coupling effects 24. In the case of a directional coupler, the envelop amplitudes follow the coupled equations: (3) Show that the energy conservation of the coupling effect (assuming no extra loss) imposes the relation (Justify your answer): d) is purely imaginary 25. Setting the boundary conditions and, the envelop at the output coupler of length is given by (Justify your answer): d) One seeks to engrave a Bragg grating reflector inside an optical fibre. The characteristics of the fibre are: core radius, refractive index of the cladding, and core index defined by. The wavelength of operation is set at. 26. Select one proposal (Justify your answer) The fibre operates in a singlemode regime. The fibre operates in multimode regime with 3 guided modes. The fibre operates in multimode regime with 2 guided modes. d) No guided modes can be supported at. 27. In order to realize a mirror, select an order of magnitude for the Bragg grating period (Justify your answer): 268 nm 1.07 µm 535 nm d) 745 nm One can show that the reflectivity of the Bragg grating takes the relation: 6

7 , with, and the phase mismatch term. The coupling coefficient is taken equal to, with the modulation depth of the refractive index grating. 28. For, calculate the length of the grating in order to reach a reflectivity at (Justify your answer) mm 3.41 mm d) 4.31 mm 29. The spectral bandwidth can be approximated by the wavelengths at which the reflectivity coefficient vanishes. For this filter, it is equal to (Justify your answer): e) f) 0.53 nm g) 0.26 nm h) 0.13 nm Dispersion effect A 10 ps pulse duration at propagates through a 10 mm long waveguide made in silicon (whose refractive index is equal to ). Using an interferometric measurement (not detailed), the dispersion properties of waveguide are characterized. Hereafter, the coefficients and related to the Taylor s expansion for, the propagation constant of waveguide mode, are introduced. 30. The group index of the waveguide is measured equal to, coinciding with : ns/m km/ps km/ns d) ns/km 31. The second order dispersion coefficient is measured equal to. Select the unity: ps 2 /m m/ps m/ps 2 d) ps 2 /m One can conclude that the second order dispersion effect can (Justify your answer) be neglected cannot be neglected 7

8 ANNEXES Figure 2. Dotted line : symmetrical TE modes. Dashed line : antisymmetrical TE modes. Figure 3

9 NAME : FIRST NAME : Answer sheet Q1 Q6 Q11 Q16 Q21 Q26 Q31 a a a a a a a b b b b b b b c c c c c c c d d d d d d d Q2 Q7 Q12 Q17 Q22 Q27 Q32 a a a a a a a b b b b b b b c c c c c d d d d d Q3 Q8 Q13 Q18 Q23 Q28 a a a a a a b b b b b b c c c c c d d d d d Q4 Q9 Q14 Q19 Q24 Q29 a a a a a a b b b b b b c c c c c c d d d d d d Q5 Q10 Q15 Q20 Q25 Q30 a a a a a a b b b b b b c c c c c d d d d d