Fast Propagation of Electromagnetic Fields through Graded-Index (GRIN) Media
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1 SPIE Paper Fast Propagation of Electromagnetic Fields through Graded-Index (GRIN) Media Huiying Zhong 1,2, Site Zhang 1,2, Rui Shi 1, Christian Hellmann 3, Frank Wyrowsk 1 1. Friedrich Schiller University Jena 2. LightTrans International UG 3. Wyrowski Photonics UG
2 Introduction: GRIN Media in Real Life Graded-Index (GRIN) meida are widely used for modeling different situations Applications multi-mode fiber optical lenses acousto-optical modulators Undesired situations stress or heating induced GRIN variations turbulence in air Pictures comes from [1-4]
3 Introduction: GRIN Media in Optical Modeling Graded-Index (GRIN) meida are widely used for modeling different situations Optical System Pictures comes from [1-4]
4 Introduction: GRIN Media in Optical Modeling Graded-Index (GRIN) meida are widely used for modeling different situations Optical System Task: How to model light propagation in GRIN media? Pictures comes from [1-4]
5 Ray and Physical Optics Optical Modeling Pierre de Fermat ( ) James Clerk Maxwell ( ) 5
6 GRIN Media: Ray Optics Optical Modeling Pierre de Fermat ( ) Ray equation for GRIN media James Clerk Maxwell ( ) 6
7 Solution of the Ray Equation Ray equation for GRIN media Runge-Kutta methods [5] A. Sharma et al. Tracing rays through graded-index media: a new method. Appl. Opt., 21(6):
8 Solution of the Ray Equation Ray equation for GRIN media Runge-Kutta methods [5] A. Sharma et al. Tracing rays through graded-index media: a new method. Appl. Opt., 21(6):
9 GRIN Media: Physical Optics Optical Modeling Pierre de Fermat ( ) James Clerk Maxwell ( ) 9
10 Solution of Maxwell Equations Mode solvers for highly symmetric structure Not valid when symmetry decreases Numerical effort is high when size of structure increase Universal Maxwell solvers: Finite element method (FEM) and Fourier modal method (FMM) + perfectly matched layers (PMLs) Numerical effort is quite high when size of structure increase Develop a fast approach to modeling electromagnetic field through GRIN media!
11 Field Representation A rigorous representation of the electromagnet fields ψψ(rr, ωω) is a common phase function, extracted from electromagnetic field. No approximation!
12 Field Representation A rigorous representation of the electromagnet fields ψψ(rr, ωω) is a common phase function, extracted from electromagnetic field. No approximation!
13 Geometric Field Zone A rigorous representation of the electromagnet fields For some EM-field, we could found a proper ψψ(rr, ωω) so that the field behaviour is dominated by the phase part. More specifically, EE 0 (rr, ωω) and HH 0 (rr, ωω) varies slowly ψψ(rr, ωω) varies much faster Geometric field zone
14 Geometric Field Equations Maxwell Eqs. Geometric field Eqs. geometric field
15 Solve the Geometric Field Eqs. for GRIN Media To solve geometric field equations for GRIN media, we get solution of how geometric field propagating through GRIN media. Geometric field Eqs. Eikonal Eq. Ray Eq. Eq. of normalized field [7] M. Born and E. Wolf, Principles of Optics, Cambridge University press (1999)
16 Solve the Geometric Field Eqs. for GRIN Media To solve geometric field equations for GRIN media, we get solution of how geometric field propagating through GRIN media. Geometric field Eqs. Eikonal Eq. Ray Eq. Eq. of normalized field ss(rr) is ray direction uu rr = EE 0 (rr)/ EE 0 (rr) [7] M. Born and E. Wolf, Principles of Optics, Cambridge University press (1999)
17 Solve the Geometric Field Eqs. for GRIN Media Rewrite field representation by using gradient theorem
18 Solve the Geometric Field Eqs. for GRIN Media Rewrite field representation by using gradient theorem Eq. of normalized field Ray Eq. Energy conservation
19 Solving Ray & Normalized Field Eq. Eq. of ray path Eq. of norm. field Runge-Kutta methods Runge-Kutta methods
20 Summary of Geom. Field Tracing in GRIN 23
21 Example: Multimode Fiber 160 µm ray propagation through a GRIN fiber electromagnetic field propagation through a GRIN fiber by a rigorous Maxwell solver, the Fourier Modal Method (FMM) with Perfectly Matched Layers (PMLs) our newly developed very fast approximated Maxwell solver 24
22 Results: 3D System Ray Tracing dot diagram 25
23 Results: Our Fast Approach vs FMM ~17 s ~40 h Deviation between results of both approaches is < 1% 26
24 Example: PSF of GRIN Lens x z 5 mm 1 mm 4 mm Fast physical optics Tearing: diffractive integral -> our fast approach -> diffractive integral 27
25 Example: PSF of GRIN Lens PSF x z 5 mm 1 mm 4 mm EE xx EE yy EE zz Amplitude of the image [V/m] 28
26 Example: PSF of GRIN Lens PSF x z 5 mm 1 mm 4 mm EE xx EE yy EE zz How to model such case that the input or output field is not geometric field? Amplitude of the image [V/m] 29
27 Problem Statement 138 µm 30
28 Pieces of the Solution Cut the GRIN component into slabs Δzz
29 Pieces of the Solution Cut the GRIN component into slabs For one slab, Simplest geometric field ideal plane wave κκκ Ideal plane wave κκκ Fast approach output geometric field Δzz output geometric field Fourier transform weighted ideal plane wave κκ In linear optics, Field can be decomposed into ideal plane waves Field can be reconstructed by add all sub-fields up
30 Concept of B-operator Input plane, define a grid of κκκ For each κκκ Define input plane waves EE 1 in (ρρ ) = EE 2 in (ρρ ) = 1 0 zz zz 2 e iiκκ ρρ e iiκκ ρρ with ρρ = (xx, yy ) Calculate the output field EE 1 out (ρρ)and EE 2 out (ρρ) Calculate the spectrum EE 1 out (κκ) and EE 2 out (κκ) Input plane Output plane Δzz
31 B-Operator For each κκκ, we have EE 1 out (κκ) and EE 2 out (κκ) kkk kk EE out 1xx (κκ) EE out 2xx (κκ) κκ EE out 1yy (κκ) κκκ EE out 2yy (κκ)
32 B-Operator For each κκκ, we have EE 1 out (κκ) and EE 2 out (κκ) kkk kk EE out 1xx (κκ) EE out 2xx (κκ) κκ EE out 1yy (κκ) κκκ EE out 2yy (κκ)
33 B-Operator For each κκκ, we have EE 1 out (κκ) and EE 2 out (κκ) EE out 1xx (κκ) EE out 2xx (κκ) BB xxxx kk, kkk BB xxyy kk, kkk BB yyxx kk, kkk BB yyyy kk, kkk = BB kk, kkk EE out 1yy (κκ) EE out 2yy (κκ) Apply BB κκ, κκ onto spectrum of input field EE out κκ = 1 2ππ BB κκ, κκ EE in κκ ddκκ
34 Problem Statement 138 µm 37
35 Results: Output Field Geometric approach ~1.5 h+~1 min B-operator FMM ~40 h Deviation < 1%
36 Apply B-Operator for Slabs For each κκκ, we have EE 1 out (κκ) and EE 2 out κκ Δzz Apply BB κκ, κκ onto spectrum of input field EE out κκ = 1 2ππ BB NN κκ, κκ EE in κκ ddκκ
37 B-Operator: Split-Step Method Review of split-step method Thin element approximation Same BB κκ, κκ for all κκ, so the slab should be quite thin. Linear operation in spatial domain. zz = zz ii z zz = zz ii+1 Elementary B-operator is TEA 40 [6] M. D. Feit and J. A. Fleck, Light propagation in graded-index optical fibers, Appl. Opt. 17, (1978).
38 B-Operator: Wave Propagation Method Review of wave propagation method EE out κκ = 1 2ππ EE in κκ ee iikk zz κκ Δz δδ(κκ κκ) ddκκ Elementary B-operator is Parabasal TEA Neglect the curvature of each ray, so the slab should be quite thin! 41 [9] Brenner, K.-H. & Singer, W. Light propagation through microlenses: a new simulation method Appl. Opt., OSA, 1993, 32,
39 Conclusion and Outlook Field propagation through GRIN media Elementary B-operator: geometric field propagating B-operator concept How to calculate B-operator What is the elementary B-operator of split-step method and wave propagation method Calculation of B-operator need most time. Just calculate BB κκ, κκ for a few κκ 00 BB κκ, κκ =: BB κκ, κκ 00 for κκ κκ 00 < δδ BB κκ, κκ =: BB κκ, κκ 00 + δδκκ withδδκκ = κκ κκ 00
40 Implementation All algorithms are implemented in the physical optics simulation and design software VirtualLab Fusion VirtualLab Fusion is developed, following the field tracing concept, by Wyrowski Photonics UG, Jena, Germany Visit German for more infomation 45
41 Reference [1] [2] [3] [4] slideplayer.com/slide/ / [5] A. Sharma et al. Tracing rays through graded-index media: a new method. Appl. Opt., 21(6): (1982) [6] F. Wyrowski and C. Hellmann. The geometric Fourier Transform, in Proc. DGaO, vol. 118, p2 (2017) [7] M. Born and E. Wolf, Principles of Optics, Cambridge University press (1999)
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