Hybrid Propagation Method Based on Gaussian Beam Summation (GBS)

Transcription

1 Hybrid Propagation Method Based on Gaussian Beam Summation (GBS) Asaf Harel, Ehud Heyman June 19, 2014 UWA Symposium June 19, 2014 UWA Symposium 11/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

2 Outline 1 Introduction Statement of the Problem The GBSBased Hybrid Method 2 Gaussian Beam Expansion GBS Expansion of Green's Function Choice of Expansion Parameters GB in Inhomogeneous Media 3 Other Applications 4 Method Implementation Test Case I: n 2 Linear Waveguide Test Case II: Schizoidal Beams 5 Summary June 19, 2014 UWA Symposium 22/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

3 Outline 1 Introduction Statement of the Problem The GBSBased Hybrid Method 2 Gaussian Beam Expansion GBS Expansion of Green's Function Choice of Expansion Parameters GB in Inhomogeneous Media 3 Other Applications 4 Method Implementation Test Case I: n 2 Linear Waveguide Test Case II: Schizoidal Beams 5 Summary June 19, 2014 UWA Symposium 33/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

4 0 c R z June 19, 2014 UWA Symposium 44/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

5 0 c R z June 19, 2014 UWA Symposium 45/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

6 0 c R z June 19, 2014 UWA Symposium 46/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

7 0 c R z June 19, 2014 UWA Symposium 47/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

8 c R z June 19, 2014 UWA Symposium 48/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

9 c R z June 19, 2014 UWA Symposium 49/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

10 Goal Our goal is to develop a scheme for a hybrid combination of dierent solution models Utilize relative strengths of models to achieve accuracy, computational eciency and transparent interpretation of phenomena Challenges: Lack of a "common language" Optimal hybrid blend is problem specic. June 19, 2014 UWA Symposium 510/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

11 The GBSBased Hybrid Method GBS method for source of radiation pattern F (Ω): G(r, r s ) n N δωf (Ω n )B GB (r, Ω n ). ˆΩ n δω June 19, 2014 UWA Symposium 611/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

12 The GBSBased Hybrid Method GBS method for source of radiation pattern F (Ω): G(r, r s ) n N δωf (Ω n )B GB (r, Ω n ). We describe the problematic beams, of indices n N PE, with a Parabolic Equation (PE) solution: u PE (r). PE starter eld is the sum of the N PE beam elds. } NPE June 19, 2014 UWA Symposium 612/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

13 The GBSBased Hybrid Method GBS method for source of radiation pattern F (Ω): G(r, r s ) n N δωf (Ω n )B GB (r, Ω n ). We describe the problematic beams, of indices n N PE, with a Parabolic Equation (PE) solution: u PE (r). PE starter eld is the sum of the N PE beam elds. Total eld: } NPE G(r, r s ) = u PE (r) + u GB (r). June 19, 2014 UWA Symposium 613/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

14 Outline 1 Introduction Statement of the Problem The GBSBased Hybrid Method 2 Gaussian Beam Expansion GBS Expansion of Green's Function Choice of Expansion Parameters GB in Inhomogeneous Media 3 Other Applications 4 Method Implementation Test Case I: n 2 Linear Waveguide Test Case II: Schizoidal Beams 5 Summary June 19, 2014 UWA Symposium 714/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

15 The Scalar Beam Expansion Identity G(r, r s ) = e ik r rs / r r s = dωb CSB (r, r s ; b, Ω) 4π The Complex Source Beam (CSB) expansion identity: a continuous spectrum of CSBs emerging from r s in all directions Ω Paraxial approximation: CSB GB δω n B GB (r, r s ; b, Ω n ) n N Discretization: δω 1 2 πθ2 D resulting in beam overlapping. Spectral truncation: ρ 3W A.N. Norris, Complex point-source representation of real point sources and the Gaussian beam summation method, J. Opt. Soc. Am, June 19, 2014 UWA Symposium 815/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

16 Paraxial Approximation for a Collimated Beam High collimation condition: kb 1 = CSB GB GB propagation in free space: B GB (r)(r, r s ; b, Ω) = ib 1 x 2 b +x 2 1 b 2 σ ib eik[σ+ 2 ] σ ib (σ, x b1, x b2 ) - Coordinates along and transverse to the beam axis Beam width: W (σ) = b/k 1 + (σ/b) 2 Collimation angle: Θ D = 1/ kb June 19, 2014 UWA Symposium 916/ Asaf Harel, Ehud Heyman GBSBased Hybrid Propagation Method 30

17 Spectral Truncation Figure: Expansion error as function of the truncation ρ/w. ρ is the distance from observation point to axis of the last beam included in summation ρ/w Conclusion: use ρ/w 3 June 19, 2014 UWA Symposium 17 10

18 Numerical Test of the Beam Expansion Figure: Error of GB expansion Figure: Error of CSB expansion At large R: dominant parabolic phase error of the GB, goes as R/b 2 k; Only truncation error for CSB expansion. GBs still preferable as are readily extended to complex media Conclusion: Choose b such that R/b < bk June 19, 2014 UWA Symposium 18 11

19 The beam lattice Expansion of the unit sphere to triangular mesh: nodes = beam directions Discretization angle: δω n = area of triangle associated with node n Discretization condition: δω π kb Collimation condition: b R For large R ne discretization For smaller R coarse discretization June 19, 2014 UWA Symposium 19 12

20 GB in Inhomogeneous Media Beam axis follows a ray trajectory Σ(σ) Method of Propagation in continuous dierentiable medium: solution of Hamilton-Jacobi equations for tracking beam width and phase front curvature dynamics Sources of error: 1 Parabolic phase error 2 Fast medium variance relative to beam width Propgation trajectory Beam width Σ(σ) xb 2 nb x b1 ϑ ϑ n σ W June 19, 2014 UWA Symposium 20 13

21 Transmission of GB Through Interfaces At discontinuity interfaces, a GB is reected and transmitted along ray trajectories satisfying Snell's law. The collimation of the reected and transmitted beams is found by phase matching on discontinuity interface S(y) to parabolic order. Source of error: Phase matching error due to higher orders; aggravated for wide beam imprint on S, i.e, for: 1 Wide beams 2 Low grazing angles 3 Near critical angle Σi S(y) Σr Σt June 19, 2014 UWA Symposium 21 14

22 Outline 1 Introduction Statement of the Problem The GBSBased Hybrid Method 2 Gaussian Beam Expansion GBS Expansion of Green's Function Choice of Expansion Parameters GB in Inhomogeneous Media 3 Other Applications 4 Method Implementation Test Case I: n 2 Linear Waveguide Test Case II: Schizoidal Beams 5 Summary June 19, 2014 UWA Symposium 22 15

23 GBS Expansion for Extended Source Distribution Time harmonic eld- GB propagator dependent eld- Pulsed beam (PB) propagator June 19, 2014 UWA Symposium 23 16

24 GBS-Imaging: Back-Propagation, Time-Reversal and MUSIC Algorithms z z The conguration Regular imaging Beam domain local imaging ρ ρ June 19, 2014 UWA Symposium 24 17

25 Fast GBS Algorithm for Underwater Propagation Time harmonic eld- GB propagator dependent eld- Pulsed beam (PB) propagator 50 ray solution GB solution z/b 0 z/b *log 10 field c/c ρ/b ρ/b June 19, 2014 UWA Symposium 25 18

26 Other Complex Media Problems Rough surface waveguide Volume random heterogeneities Edge diraction of a beam hitting arbitrarily close to the edge x z B i ( r) B (1) ( r) µ 1 z 1 =0 θ 0 r T z r R (2) B µ 2 z 2 =H ( r) June 19, 2014 UWA Symposium 26 19

27 Outline 1 Introduction Statement of the Problem The GBSBased Hybrid Method 2 Gaussian Beam Expansion GBS Expansion of Green's Function Choice of Expansion Parameters GB in Inhomogeneous Media 3 Other Applications 4 Method Implementation Test Case I: n 2 Linear Waveguide Test Case II: Schizoidal Beams 5 Summary June 19, 2014 UWA Symposium 27 20

28 z n 2 Linear Waveguide 0 c R Wavespeed: c(z) = c 0 1 2αz, α > 0 Canonical case, characteristic of mixed layer conditions or winter prole. Analytical exact normal modes solutions: Airy functions. overlapping caustics at low takeo angles cause breakdown of ray models; accordingly, GBs lose collimation. June 19, 2014 UWA Symposium 28 21

29 Choice of Parameters We must choose: 1 Initial collimation length of beams: b 0. 2 division of the lattice. It can be shown that the collimation length of a beam after m hop cycles goes as: b m b0/m 2. Thus for a given range, the collimation length deteriorates faster for lower takeo angles. We describe these using the PE method. Dividing angle: θ H c R ΘH 300 z Asaf Harel, Ehud Heyman GBSBased = blue, Hybrid Propagation PE = green Method June 19, 2014 UWA Symposium / 30

30 Choice of Initial Collimation Length For a given beam, we nd the optimal choice of initial collimation length b 0 = half the hop distance θ. We set b0 as the optimal length for the beam of lowest angle = θ H June 19, 2014 UWA Symposium 30 23

31 Optimal Angular Division PE is accurate for small takeo angles, but produces phase error for larger takeo angles. By equating the terms for the GB and PE errors as a function of takeo angle, the optimal division angle θ H is found. June 19, 2014 UWA Symposium 31 24

32 Results Ray trace; GB = blue, PE = green c R z Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 5000m, θ H = 2.9 deg June 19, 2014 UWA Symposium 32 25

33 Results GB solution Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 5000m, θ H = 2.9 deg June 19, 2014 UWA Symposium 33 25

34 Results PE solution Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 5000m, θ H = 2.9 deg June 19, 2014 UWA Symposium 34 25

35 Results GB + PE solution Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 5000m, θ H = 2.9 deg June 19, 2014 UWA Symposium 35 25

36 Results Normal Modes reference solution Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 5000m, θ H = 2.9 deg June 19, 2014 UWA Symposium 36 25

37 Results Error Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 5000m, θ H = 2.9 deg June 19, 2014 UWA Symposium 37 25

38 Results Relative error as a function of range Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 5000m, θ H = 2.9 deg June 19, 2014 UWA Symposium 38 25

39 Results Error at R=30km as a function of the division angle: θ H Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 5000m, θ H = 2.9 deg June 19, 2014 UWA Symposium 39 25

40 Schizoidal Beams c R z Beams grazing a wavespeed maximum cannot maintain Gaussian structure, either ripping into two parts or losing collimation quickly. Congruent with leaky modes, at edge of the mixed layer, resulting in non-localized eld. As a canonical case, we take a bi-linear wavespeed prole, presented above. June 19, 2014 UWA Symposium 40 26

41 Suggested Solution Using our hybrid approach, the problem can be circumvented. We replace beams that turn near the maximum point by PE solutions. Note: We lack a reference solution for this problem, and by focusing on low takeo angle beams, we will consider the PE solution as accurate. June 19, 2014 UWA Symposium 41 27

42 Results Ray trace; GB = blue, PE = green c R z Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 4000m, θ H = 2.4 deg June 19, 2014 UWA Symposium 42 28

43 Results GB solution Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 4000m, θ H = 2.4 deg June 19, 2014 UWA Symposium 43 28

44 Results PE solution Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 4000m, θ H = 2.4 deg June 19, 2014 UWA Symposium 44 28

45 Results GB + PE solution Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 4000m, θ H = 2.4 deg June 19, 2014 UWA Symposium 45 28

46 Results Solution by regular GBS approach Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 4000m, θ H = 2.4 deg June 19, 2014 UWA Symposium 46 28

47 Results Dierence between hybrid and regular GBS Problem parameters: f = 8kHz, source depth= 50m, c 0 = 1500m/s, α = 1/60/1500[1/m], b 0 = 4000m, θ H = 2.4 deg June 19, 2014 UWA Symposium 47 28

48 Outline 1 Introduction Statement of the Problem The GBSBased Hybrid Method 2 Gaussian Beam Expansion GBS Expansion of Green's Function Choice of Expansion Parameters GB in Inhomogeneous Media 3 Other Applications 4 Method Implementation Test Case I: n 2 Linear Waveguide Test Case II: Schizoidal Beams 5 Summary June 19, 2014 UWA Symposium 48 29

49 Summary 1 We have introduced a general framework for hybrid solution of propagation problems based on the GBS. 2 The method allows exploitation of relative strengths of dierent model solutions, avoiding problematic features in the media and fundamental weaknesses of the models. 3 We demonstrated how this scheme may be enacted to tackle specic problems of underwater propagation. 4 This framework can be readily extended to aperture radiation or edge diraction problems. June 19, 2014 UWA Symposium 49 30