Title: Analysis of instantaneous poynting vector and application

Abstract

We derive analytical expressions for the spatial evolution of the instantaneous Poynting vector (PV) for optical waveguides and propose new formulae for the propagation length and the penetration depth of the ‘instantaneous’ PV. These are different from their conventional formulae defined for ‘average’ PV [1]. Starting from Maxwell’s equations [2], we obtain following equations for the electric and magnetic fields of symmetric bound TM modes of a planar symmetric Dielectric-metal-dielectric (DMD) waveguide of core thickness d along the x- axis, infinite in extent in y-axis, and direction of propagation along the z-axis, at an instant t=0 : H_y (x,z)={█(cos⁡〖(κx)〗 cos⁡〖〖(β〗_r z〗)e^(-β_i z) ;|x|≤d/2@ cos(κd/2) e^(γd/2) e^(-γ|x| ) cos⁡〖〖(β〗_r z〗)e^(-β_i z) ;|x|≥d/2)} (1) E_x (x,z)={█(β/(ωε_o 〖n_m〗^2 ) cos⁡〖(κx)〗 cos⁡〖〖(β〗_r z〗)e^(-β_i z) ;|x|≤d/2@β/(ωε_o 〖n_d〗^2 ) cos(κd/2) e^(γd/2) e^(-γ|x| ) cos⁡〖〖(β〗_r z〗)e^(-β_i z) ;|x|≥d/2)} (2) E_z (x,z)={█(κ/(ωε_o 〖n_m〗^2 ) sin⁡〖(κx)〗 sin⁡〖〖(β〗_r z)〗 e^(-β_i z) ;|x|≤d/2@ γ/(ωε_o 〖n_d〗^2 ) x/|x| cos(κd/2) e^(γd/2) e^(-γ|x| ) sin⁡〖〖(β〗_r z)〗 e^(-β_i z) ;|x|≥d/2)} (3) Where the symbols have their usual meanings. The PV associated with an electromagnetic wave is given by: S=E×H (4) Substituting Eqns. (1)-(3) in Eqn. (4), we get the expressions for x- (transverse) and z- (longitudinal) components of the PV: S_x (x,z)={█((-κ)/(ωε_o 〖n_m〗^2 ) cos⁡〖(κx)〗 sin⁡〖(κx)〗 sin⁡〖(β_r z)〗 cos⁡〖〖(β〗_r z〗) e^(-2β_i z) ;|x|≤d/2@ (-γ)/(ωε_o 〖n_d〗^2 ) x/|x| 〖cos〗^2 (κd/2) e^γd e^(-2γ|x| ) sin⁡〖(β_r z)〗 cos⁡〖〖(β〗_r z)〗 e^(-2β_i z) ;|x|≥d/2)} (5) S_z (x,z)={█(β/(ωε_o 〖n_m〗^2 ) 〖cos〗^2 (κx) 〖cos〗^2 (β_r z)e^(-2β_i z) ;|x|≤d/2@ β/(ωε_o 〖n_d〗^2 ) 〖cos〗^2 (κd/2) e^γd e^(-2γ|x| ) 〖cos〗^2 (β_r z) e^(-2β_i z) ;|x|≥d/2)} (6) Combining Eqns. (5) and (6) we obtain the equations for instantaneous PV due to S_x and S_z [3]. {█( sin⁡κx 〖(sec⁡〖β_r z)〗〗^((κ^2⁄(β.β_r )) )=C_1 ;|x|≤d/2@x/|x| e^(((-β.β_r.|x|)⁄γ)) cos⁡〖β_r z〗=C_2 ;|x|≥d/2)} (7) C_1and C_2 are the constants whose absolute values determine the strength of the flux lines. The penetration depth for instantaneous PV is |γ/β^2 | due to the presence of the term e^(((-β.β_r.|x|)⁄γ)) and the propagation length is 1/(〖β_r〗^2/β)_i ⋍ 1/β_i (obtained by simplifying Eqn. (7)). The corresponding expressions for the ‘average’ PV are |1/2γ| and 1/〖2β〗_i respectively. We plot the spatial evolution of the instantaneous PV for silica-gold-silica waveguide and show that inside the metal core, it consists of broken flux lines signifying the optical absorption of electromagnetic waves propagating through the metal in order to excite the surface plasmons in a resonant manner in the metal film at the interface [2]. We use the proposed formula for the penetration depth of instantaneous PV to calculate the optimum thickness of the high index dielectric layer to be used as affinity layer or for the enhancement of sensitivity of a surface plasmon resonance-based sensor. Our results match very well with the already reported experimental results [4].

Biography

Himanshu Kushwah received the B.Sc. (Hons.) degree in Electronics from the Hansraj College University of Delhi, Delhi in 2008 and the M.Sc. degree in Electronics Science from the University of Delhi South Campus Delhi in 2010. Since 2011, he is teaching Electronics Science to undergraduate students at the Department of Electronics, Keshav Mahavidyalaya, University of Delhi. He is also pursuing Ph.D degree in Electronics at the University of Delhi, South Campus, Delhi India. In addition, he has worked as Design Engineer at STMicroelectronics Inc. from June 2010 to Dec. 2010.