OPTI510R: Photonics. Khanh Kieu College of Optical Sciences, University of Arizona Meinel building R.626

Transcription

1 OPTI510R: Photonics Khanh Kieu College of Optical Sciences, University of Arizona Meinel building R.66 1

2 Announceents Hoework # is due today, HW#3 is assigned due Feb. 1 st No class Monday Feb. 6 Pre-record lecture Friday Feb. 3 at PM Mid-ter exa will be on Feb. 8

3 Electronics-Photonics Electrons, Integrated circuit Photonics integrated circuit Optoelectronics chip 3

4 Planar waveguides Planar irror waveguide Waveguide odes Nuber of odes Cut-off condition Dispersion Dielectric waveguide Waveguide odes Nuber of odes Cut-off condition Dispersion 4

5 Soe useful ath

6 Planar irror waveguides / n 0 0 k nk0 c c / n TEM plane wave TE: E polarized in x-direction TM: H polarized in x-direction p phase shift for each reflection (boundary conditions) Aplitude and polarization do not change (perfect irror). Not practical due to the fact that there is no perfect etal irror 6

7 Planar irror waveguides Self-consistency: The wave reflects twice and reproduces itself Therefore the phase shift in travelling fro A to B ust be equal to or differ by an integer ultiple of p fro the phase shift fro A to C Modes are fields that aintain the sae transverse distribution and polarization at all locations along the waveguide axis. 7

8 Planar irror waveguides A guided wave consists of the superposition of two plane waves in the y-z plane at angle ±q with respect to the z axis. The coponents of the ode wave vector are: k y = nk 0 sinq = p / d b = k z = k - p d 8

9 Mode field profile A guided wave consists of the superposition of two plane waves in the y-z plane at angle ±q with respect to the z axis. +,upward wave,downward wave =, is odd, is even 9

10 Mode field profile TE odes E x ( y, z) a u ( y)exp( j z) or u ( y ) d d py cos( ), d py sin( ), d 1,3,5...,4,6... Modes are orthogonal and noralized 10

11 Mode properties TE odes E x ( y, z) a u ( y)exp( j z) or u ( y ) d d py cos( ), d py sin( ), d 1,3,5...,4,6... Modes are orthogonal and noralized Orthogonal condition Noralized condition Any field distribution can be discoposed into a su of odes 11

12 Nuber of odes, Cutoff Nuber of odes sinq / d 1, M d / Reduce to nearest integer Dispersion relation / c p / d Cutoff wavelength and frequency c d, n c c / d For > c or n < n c there is no guided ode 1

13 Dispersion relation Dispersion relation. / c p / d This leads to waveguide dispersion 13

14 Group velocity Dispersion relation / c p / d Cutoff frequency = 14

15 Group velocity Is this noral or anoalous dispersion? 15

16 TE versus TM (Hoework) 16

17 Multiode fields In general, a field with arbitrary distribution and vanishing aplitude at the irror can be guided. E x M ( y, z) a u ( y)exp( j z) 0 Single-ode polyer wg Multiode polyer wg 17

18 Planar dielectric waveguide y x z Core fil sandwiched between two layers of lower refractive index Botto layer is often a substrate with n = n s Top layer is called the cover layer (n c n s ) Air can also acts as a cover (n c = 1) n c = n s in syetric waveguides 18

19 Planar dielectric waveguide Syetric waveguide Reflection due to TIR (siilar to planar irror waveguide) sinq c = n / n 1 q p -q = p c - æ n ö sin-1 ç è ø = æ n cos-1 ç è n 1 n 1 ö ø Self Consistency p d sin q r p k d p y r 19

20 Phase shift for TIR TE wave tan j r = sin q c sin q -1 TM wave tan j r = -n 1 n sin q c sin q -1 We can now arrive at an equation for the ode angles 0

21 Transcendental equation for odes pd p sin qc tan sinq 1 sin q dielectric waveguide irror waveguide p, or tan( / ) r r

22 Nuber of odes,or,where Single ode d l 0 NA <1

23 Transcendental equation for odes nk cos 1 0 q b = N eff w /c 0 3

24 Cut-off frequency Mirror waveguide Dielectric waveguide There is no gap for dielectric waveguide always one guided ode for a syetric slab (not so for asyetric) 4

25 Oscillating field coponent In the core The electric field in a syetric dielectric waveguide is haronic within the slab and exponentially decaying outside the slab. E ( y, z) a u ( y)exp( j z) x ì æ cos p l sinq ö ç y, = 0,,4,... ï è ø u (y) µ í æ sinç p, - d ï l sinq ö y d y, =1,3,5,... î ï è ø 5

26 Evanescent field coponent u ( y) exp( y), y > d / exp( y), y d / The z dependence ust be identical in order to satisfy continuity at ± d/. Signs are chosen to obtain a decaying field E x ( ( y, z) a n k 0 M 0 ) E x u ( y)exp( ( y, z) 0 j z) Extinction coefficient 0 nk nk cos 0 cos q q c 1 6

27 TE field distribution E x M ( y, z) a u ( y)exp( j z) 0 7

28 Gaussian bea and waveguide ode Gaussian bea in free space Fundaental ode in dielectric waveguide (no diffraction!) 8

29 TE versus TM (Hoework) 9