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.626

2 Announceents HW#3 is due next Wednesday, Feb. 21 st No class Monday Feb. 26 Pre-record lecture Friday Feb. 23 at 2PM Mid-ter exa will be on Feb. 28

3 Planar waveguides-continue Review planar waveguides Dielectric waveguide Waveguide odes Nuber of odes Cut-off condition Dispersion Dielectric waveguide ode calculation using the Maxwell s equations

4 Planar irror waveguides ) )exp( ( ), ( z j y u a z y E x 2,4,6... ), sin( 2 1,3,5... ), cos( 2 ) ( d y d d y d u y TE odes or

5 Nuber of odes, Cutoff Nuber of odes sin / 2d 1, M 2d / Reduce to nearest integer Dispersion relation / c / d Cutoff wavelength and frequency c 2d, n c c / 2d For > c or n < n c there is no guided ode

6 Dispersion relation Dispersion relation / c / d This leads to waveguide dispersion

7 Group velocity Waveguide provides anoalous dispersion without an atoic resonance

8 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

9 Transcendental equation for odes TE ode (syetric) 2 d sin c tan sin sin dielectric waveguide irror waveguide, or tan( / 2) r r

10 Nuber of odes,or,where Single ode 2d l 0 NA <1

11 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 2p l sinq ö ç y, = 0,2,4,... ï è ø u (y) µ í æ sinç 2p, - d ï l sinq ö 2 y d 2 y, =1,3,5,... î ï è ø

12 Evanescent field coponent u ( y) exp( y), y > d / 2 exp( y), y d / 2 The z dependence ust be identical in order to satisfy continuity at ± d/2. Signs are chosen to obtain a decaying field E x ( ( y, z) a 2 n 2 k 2 0 M 0 ) E x u ( y)exp( ( y, z) 0 j z) Extinction coefficient nk nk cos cos c 1

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

14 Dispersion relation Self Consistency 2 2 d sin 2 r 2 2k d 2 2 y r TE wave tan j r 2 = sin2 q c sin 2 q -1

15 Dispersion relation Dispersion relation in paraetric for: (n: effective refractive index)

16 Dispersion relation n 1, n 2 are constant

17 Dispersion relation Noral or anoalous dispersion?

18 What is the sallest waveguide? Mirror waveguide Dielectric waveguide There is no gap for dielectric waveguide always one guided ode for a syetric slab (not so for asyetric) Can we then ake an infinitely sall dielectric waveguide?

19 Confineent factor Confineent factor: = P core /P total = P core /(P core +P clad ) Core u Cladding ( y) A A 2 cos sin y, 0,2,4,..., 2 sin sin y, 1,3,5,... d 2 d y 2 nk nk cos cos c 1

20 Confineent factor At the boundary: (Total power in the core) (Total power in the cladding)

21 Confineent factor Confineent factor:,and

22 What is the sallest waveguide? In single-ode operation the evanescent field becoes larger as d is reduced Difficult to confine light to nanoeter scale! The confineent factor quantifies how uch light is in the waveguide core vs. the cladding More highly confining waveguides can be bent ore draatically without loss However, ore highly confining waveguides are often difficult to couple light into

23 Using Maxwell s equations y d -d 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

24 Using Maxwell s equations An optical ode is a solution of the Maxwell's equations satisfying all boundary conditions Its spatial distribution does not change with propagation Modes are obtained by solving the curl equations These six equations are solved in each layer of the waveguide Boundary condition: Tangential coponent of E and H be continuous across both interfaces Waveguide odes are obtained by iposing the boundary conditions

25 Using Maxwell s equations Assue waveguide is infinitely wide along the x axis E and H are then x-independent For any ode, all coponents vary with z as exp(iz). Thus,

26 Using Maxwell s equations These equations have two distinct sets of linearly polarized solutions For Transverse-Electric (TE) odes, E z = 0 and E y = 0 TE odes are obtained by solving: Magnetic field coponents are related to E x as:

27 Using Maxwell s equations G. Agrawal

28 Using Maxwell s equations G. Agrawal

29 Using Maxwell s equations G. Agrawal

30 TE ode for syetric waveguide G. Agrawal

31 TE ode for syetric waveguide G. Agrawal

32 Modes of asyetric waveguide G. Agrawal

33 Modes of asyetric waveguide G. Agrawal

34 Universal dispersion curve G. Agrawal

35 Literature