Beam Propagation Method GEOMETRY OF THE BEAM PROPAGATION METHOD THE UNIVERSITY OF TEXAS AT EL PASO Pioneering 21 st Century Electromagnetics and Photonics The beam propagation method (BPM) is widely used in photonics and nonlinear optics. It propagates a beam through nonhomogeneous media and achieves its efficiency by handling the propagation problem one grid slice at a time. The basis formulation assumes one-way propagation under the paraxial approximation. Bi-directional and wide-angle formulations exist. BPM is primarily a “forward” propagating algorithm where the dominant direction of propagation is longitudinal. The grid is computed and interpreted as it is in FDFD. The algorithm and implementation looks more like the method of lines. BENEFITS • Highly efficient method • Can easily incorporate nonlinear materials properties. This is very unique for a frequency-domain method. • Simple to formulate and implement FFT-BPM is simpler to formulate and implement. BPM is commonly used to model nonlinear optical devices and waveguide circuits. DRAWBACKS Not a rigorous method Small angle approximation Ignores backward reflections FFT-BPM is slower, less stable, and less versatile than FD-BPM FORMULATION OF THE BASIC BPM ALGORITHM Step 1: Start with Maxwell’s equations. y z xx x x z yy y y x zz z E E H y z E E H z x E E H x y y z xx x x z yy y y x zz z H H E y z H H E z x H H E x y 0 0 0 x kx y ky z kz y x xx xx yy yy zz zz x y x y y x xx xx yy yy zz zz x y x y s s ss s s s s ss s s Step 2: Reduce problem to two dimensions. 0 y y xx x x z yy y y zz z E H z E E H z x E H x y xx x x z yy y y zz z H E z H H E z x H E x x z yy y y xx x y zz z H H E z x E H z E H x x z yy y y xx x y zz z E E H z x H E z H E x E-Mode H-Mode Step 3: Assume a solution using the slowly varying envelope approximation. eff eff , , , , jn z jn z Exz xze H xz xze eff eff x z x yy y y y xx x y zz z jn z x jn z x eff eff x z x yy y y y xx x y zz z jn z x jn z x E-Mode H-Mode Step 4: Write equations in matrix form. eff eff h x x x z yy y y y xx x e x y zz z jn z jn z h h Dh εe e e μh De μh eff eff e x x x z yy y y y xx x h x y zz z jn z jn z e e De μ h h h εe Dh εe E-Mode H-Mode Step 5: Derive matrix wave equation with small angle approximation. 2 2 y z e 1 2 eff eff 2 y H E xx x zz x y xx yy y jn n z e μDμDe με Ie 0 E-Mode H-Mode 2 2 y z h 1 2 eff eff 2 y e h xx x zz x y xx yy y jn n z h εDεDh εμ h 0 Step 6: Approximate the z-derivative 1 2 eff eff 2 y H E xx x zz x xx yy y j n z n e μDμD με Ie 1 2 eff eff 2 y e h xx x zz x xx yy y j n z n h ε Dε D εμ h E-Mode H-Mode 1 1 1 eff 2 2 i i i i i i y y e y e y j z n e e A e Ae 1 1 1 eff 2 2 i i i i i i y y h y h y j z n h h A h Ah 1 2 eff i i H i E i i e xx x zz x xx yy n A μD μ D με I 1 2 eff i i h i e i i h xx x zz x xx yy n A εD ε D εμ I Step 7: Solve field at i+1 E-Mode H-Mode 1 1 1 eff eff 4 4 i i i i y e e y jz jz n n e I A I A e 1 1 1 eff eff 4 4 i i i i y h h y jz jz n n h I A I A h BLOCK DIAGRAM OF THE ALGORITHM SNAPSHOTS FROM A TYPICAL MODEL