LAB1 2D and 3D step-index waveguides. TE and TM modes.

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Edwin Underwood
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1 LAB1 2D and 3D tep-index waveguide. T and TM mode. 1. Getting tarted 1.1. The purpoe o thi laboratory are: - T/TM mode propagation in 2D (lab waveguide) tep-index waveguide a a unction o guide peciic parameter and coupling-radiation into the guide - Hybrid mode pq ditribution in tranveral ection o a 3D (channel waveguide) tep-index waveguide a a unction o phyical tructure 2. Theory a. 2D waveguide General tructure o a waveguide conit o three layer: core, ubtrate and cladding, tanding conventional a in igure 1. Optical waveguide i called 2D or lab i the light coninement take place only in the x-direction. The light i guided i n >n >n c and T> critical thickne, where nreractive index o the core, n r reractive index o the ubtrate and nc- reractive index o the clad and T-thickne o the core.. a. b. c. Fig 1. 2D optical waveguide tructure: a. general 3D-tructure, b. tranveral ection, c. T ield (T i on Oz direction) There are two dierent type o 2D waveguide: tep-index- in which the index change abruptly along the depth and graded-index in which the index change gradually along the depth. Conider an incident coherent light, with an angle theta between the wave normal and the normal to the interace in the tep-index lab waveguide. There are deined two critical angle, at both the upper and lower interace o the core, deined a: nc θ c = arcin n 1

2D waveguide General tructure o a waveguide conit o three layer: core, ubtrate and cladding, tanding conventional a in igure 1.

2 n θ = arcin n that point the condition to the guided mode, uch a: i n>nc than θ > θ c and i incident light ha an angle uch a 90> θ> θ there are guided mode θc< θ< θ - ubtrate radiation θ < θc - ubtrate and clad radiation a. b. Fig 2. a. tep-index proile, b. graded-index proile Conidering the Maxwell equation, in iothropic, loele dielectric medium: H = µ 0 t 2 H = ε 0n t with the deinition o electric and magnetic ield repectively: j( ωt βz) j( ωt βz) 2πc = ( x, y) e and H = H ( x, y) e,and ω = = k c, λ I deined β = k N = k n inθ - plane wave propagation contant. 0 0 With the condition or T mode, where vector i on Oz direction (direction o guided light propagation), that mean ield component y, Hx, Hz it i obtained: 2 y 2 + ( k0 n β ) = 0 2 y x β H x = y ωµ 0 0 2

graded-index proile Conidering the Maxwell equation, in iothropic, loele dielectric medium: H = µ 0 t 2 H = ε 0n t with the deinition o electric and magnetic ield repectively: j( ωt βz) j( ωt

3 1 y H z = jωµ 0 x OBS. For TM mode, the condition are ymmetric. Normalized requency i deined a: V = k T n n 0 n nc N n With a = - aymmetry grade, and b = n n n n Applying the boundary condition at X=0 and X=-T, i obtain: V 1 b = ( m + 1) π arctg 1 b b arctgarctg 1 b b + a Maximal mode condition: N=n, b =0 From monomode condition: V 0 = arctg a i obtained cutt-o wavelength. Multimode equation i Vm = arctg a + mπ, where m = number o mode T: T0, T1, Tm. The eective waveguide Thickne i deined a Gooth-Hanchen eect: 1 1 T e = T + +, where γ = k0 N n, γ c = k0 N nc, that mean the γ γ c 1 guide mode penetrate to depth and in the ubtrate and cover. γ1 γ c Fig3. Te repreentation The power carried by a guided mode i conidered a: P = + ( x) H ( x) dx = y x 1 2 H T e 3

arctgarctg 1 b b + a Maximal mode condition: N=n, b =0 From monomode condition: V 0 = arctg a i obtained cutt-o wavelength.

4 β where H =, and, y = co( k x x + ct) -T<x<0 ωµ 0 Thereore the wave ront o T mode propagation are hown in Figure 3. Fig 4. Optical electric ield ditribution o T guided mode The accepted incident prim or numerical aperture look like: b. 3D waveguide Optical waveguide i called 3D or channel i the light coninement take place in the x-direction and y-direction. There are our dierent type: buried type, ridge type, loaded type, voltage induced type. [1]. Fig5. Baic 3D waveguide type 4

3D waveguide Optical waveguide i called 3D or channel i the light coninement take place in the x-direction and

5 ective Index Method wa apply to analyze thi type o waveguide (ig 6). The 3D waveguide i divided in two 2D wave: one with Ox direction conination o light and the econd, which i alo ymmetric) with Oy conination. pq = Tp + TMq Fig 6. ective index method pq are compoed TM mode, p T and p TM. Fig. 7. An ymmetric and aymmetric mode repreentation. It reult a pq combination i i predominant. The aim to deign uch a 3D waveguide i to obtain a monomode regime. From normalized requency equation: V = arctg a mπ or irt 2D guide and I + V I = mπ, or the econd one, where a=0 i obtain the wavelength interval or monomode working regime. OPTIONAL c. Optical iber i a particular waveguide: ymmetric and circular. 5

ective index method pq are compoed TM mode, p T and p TM. Fig. 7. An ymmetric and aymmetric mode repreentation. It reult a pq combination i i predominant.

6 Figure 8. T and TM mode repreentation into optical iber 6

7 Figure 9. Intenity pattern in LP01, LP11, LP21 mode. Fig. 10. Intenity pattern o mode Laguerre-Gauian LG21 7

8 3. Activitie The uer interace i preented in igure 8. Fig 10. Uer interace or 2D waveguide Input data: - n, n, nc - value in [1 4] interval - T value in [0,1 5] µm - λ nm Figure 11. Uer interace or 3D waveguide 8

9 Input data: - n, n, nc - value in [1 4] interval - T value in [0,1 5] µm - W value in [1..10] µm - λ nm - two combo-boxi or pq viualiation 3. Reult a. Output data: - m number o T mode - λ cut-o or monomode regime - θ propagation angle - Te eective depth o the waveguide - P optical coupled power From combo-box (right ide) you can elect the correpondent repreentation or Ti mode (let ide). On the let ide alo it i repreented index-proile o the 2D tep-index waveguide. Make a report with the ollowing tak: 1. Obervation about how reractive index, T and λ inluence the T mode number. - m=(nc,n,n, T, λ); 2. How the mode number inluence the θ, Te, P (θ, Te, P =(m)); 3. xercie deign a 2D waveguide with the ollowing parameter: T=2.5 and λ=1550nm (third communication window) work a monomod with maximum coupling power (Pmax). b. Output data - p mode (number o mode on Ox direction) - q mode (number o mode on Oy direction) - Te, We eective depth o the waveguide - P optical coupling power - λ interval or which 3D waveguide work in monomode regime. - pq- lectrical ield proile a a graphical repreentation Make a report with the ollowing tak: 1. Obervation about how reractive index inluence the mode number, monomode working λ interval. 2. Obervation about how T and W inluence the mode number and repectively monomode working λ interval. 9

correpondent repreentation or Ti mode (let ide). On the let ide alo it i repreented index-proile o the 2D tep-index waveguide. Make a report with the ollowing tak: 1.

6. Friction, Experiment and Theory

6. Friction, Experiment and Theory The lab thi wee invetigate the rictional orce and the phyical interpretation o the coeicient o riction. We will mae ue o the concept o the orce o gravity, the normal