《光波导理论与技术 Optical Waveguides Principles and Technologies》课程教学资源（参考文献）Waves, modes, communications, and optics - a tutoral aop-11-3-679

Tutorial Vol.11,No.3/September 2019 Advances in Optics and Photonics 679 Advances n Optics and Photonics Waves,modes,communications, and optics:a tutorial DAVID A.B.MILLER Ginzton Laboratory,Stanford University,348 Via Pueblo Mall,Stanford,California 94305-4088, USA (dabm@stanford.edu) Received April 8,2019;revised July 1,2019;accepted July 2,2019;published September26,2019(Doc.ID 364425) Modes generally provide an economical description of waves,reducing complicated wave functions to finite numbers of mode amplitudes,as in propagating fiber modes and ideal laser beams.But finding a corresponding mode description for counting the best orthogonal channels for communicating between surfaces or volumes,or for optimally describing the inputs and outputs of a complicated optical system or wave scatterer,requires a different approach.The singular-value decomposition approach we describe here gives the necessary optimal source and receiver "communication modes"pairs and device or scatterer input and output"mode-converter basis function" pairs.These define the best communication or input/output channels,allowing precise counting and straightforward calculations.Here we introduce all the mathematics and physics of this approach,which works for acoustic,radio-frequency,and optical waves. including full vector electromagnetic behavior,and is valid from nanophotonic scales to large systems.We show several general behaviors of communications modes,including various heuristic results.We also establish a new"M-gauge"for electromagnetism that clarifies the number of vector wave channels and allows a simple and general quantiza- tion.This approach also gives a new modal "M-coefficient"version of Einstein's A&B coefficient argument and revised versions of Kirchhoff's radiation laws.The article is written in a tutorial style to introduce the approach and its consequences.2019 Optical Society of America https://doi.org/10.1364/AOP.11.000679 1.ntroduction.··········…··· 683 l.l.Modes and Waves......········… 683 l.2.Idea of Modes......·...····················… 683 l.3.Modes as Pairs of Functions..··················· 684 1.3a.Communications Modes 685 1.3b.Mode-Converter Basis Sets 686 l.4.Usefulness of This Approach.···.· 686 1.4a.Using Communications Modes.. 687 1.4b.Using Mode-Converter Basis Sets.. 687 l.4c.Areas of Research and Application.............···· 688 l.5.Approach of This Paper,..·..·...··.········ 688

Waves, modes, communications, and optics: a tutorial DAVID A. B. MILLER Ginzton Laboratory, Stanford University, 348 Via Pueblo Mall, Stanford, California 94305-4088, USA (dabm@stanford.edu) Received April 8, 2019; revised July 1, 2019; accepted July 2, 2019; published September 26, 2019 (Doc. ID 364425) Modes generally provide an economical description of waves, reducing complicated wave functions to finite numbers of mode amplitudes, as in propagating fiber modes and ideal laser beams. But finding a corresponding mode description for counting the best orthogonal channels for communicating between surfaces or volumes, or for optimally describing the inputs and outputs of a complicated optical system or wave scatterer, requires a different approach. The singular-value decomposition approach we describe here gives the necessary optimal source and receiver “communication modes” pairs and device or scatterer input and output “mode-converter basis function” pairs. These define the best communication or input/output channels, allowing precise counting and straightforward calculations. Here we introduce all the mathematics and physics of this approach, which works for acoustic, radio-frequency, and optical waves, including full vector electromagnetic behavior, and is valid from nanophotonic scales to large systems. We show several general behaviors of communications modes, including various heuristic results. We also establish a new “M-gauge” for electromagnetism that clarifies the number of vector wave channels and allows a simple and general quantiza￾tion. This approach also gives a new modal “M-coefficient” version of Einstein’s A&B coefficient argument and revised versions of Kirchhoff’s radiation laws. The article is written in a tutorial style to introduce the approach and its consequences. © 2019 Optical Society of America https://doi.org/10.1364/AOP.11.000679 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 1.1. Modes and Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 1.2. Idea of Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 1.3. Modes as Pairs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 1.3a. Communications Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 1.3b. Mode-Converter Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . 686 1.4. Usefulness of This Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 1.4a. Using Communications Modes. . . . . . . . . . . . . . . . . . . . . . . 687 1.4b. Using Mode-Converter Basis Sets. . . . . . . . . . . . . . . . . . . . . 687 1.4c. Areas of Research and Application . . . . . . . . . . . . . . . . . . . . 688 1.5. Approach of This Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 Tutorial Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics 679

680 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial 2.Organization of This Paper ............................... 689 3.Introduction to SVD and Waves-Sets of Point Sources and Receivers... 690 3.1.Scalar Wave Equation and Green's Functions................. 691 3.2.Matrix-Vector Description of the Coupling of Point Sources and Receivers..········· 691 3.3.Hermitian Adjoints and Dirac Bra-Ket Notation............... 692 3.4.Orthogonality and Inner Products...,..·..·...·.···.····· 693 3.5.Orthonormal Functions and Vectors... 694 3.6.Vector Spaces,Operators,and Hilbert Spaces................. 695 3.7.Eigenproblems and Singular-Value Decomposition............. 695 3.8.Sum Rule on Coupling Strengths...,..·..........·.······ 698 3.9.Constraint on the Choice of the Coupling Strengths of the Channels... 699 4.Introductory Example--Three Sources and Three Receivers.·..··.··. 700 4.l.Mathematical Solution........·.....·.·.·.····· 700 4.2.Physical Implementation......,........·.........·..·. 702 4.2a.Acoustic and Radio-Frequency Systems..··.············ 702 4.2b.Optical Systems..·,..····················· 704 4.2c.Larger Systems... 706 5.Scalar Wave Examples with Point Sources and Receivers.......... 706 5.l.Nine Sources and Nine Receivers in Parallel Lines.......···- 707 5.la.Channels and Coupling Strengths.·.·,..·..·..·.··· 707 5.lb.Modes and Beams.·.·...··. 707 5.2.Two-Dimensional Arrays of Sources and Receivers........ 711 5.3.Paraxial Behavior ... 713 5.3a.Behavior of Singular Values................·..·..·. 713 5.3b.Forms of the Communications Modes.......·...·.·.·.- 714 5.3c.Additional Degeneracy of Eigenvalues-Paraxial Degeneracy 717 5.3d.Paraxial Degeneracy and Paraxial Heuristic Numbers........ 718 5.3e.Use of Point Sources as Approximations to Sets of"Patches"... 726 5.4.Non-Paraxial Behavior..··......········ 727 5.4a.Longitudinal Heuristic Angle..:.·.·.·..··.······· 727 5.4b.Spherical Shell Spaces..·· 728 5.5.Deducing Sources to Give a Particular Wave......... 730 5.5a.Sources for an Arbitrary Combination of Specific Receiver Modes.......。,..··············· 730 5.5b.Sources for a Gaussian Spot-Passing the Diffraction Limit... 732 5.5c."Top-Hat"Function ........... 735 5.5d.Notes on Passing the Diffraction Limit ................ 735 6.Mathematics of Continuous Functions,Operators,and Vector Spaces.... 736 6.l.Functions,,Vectors,.Numbers,and Spaces.....·.·.··.······· 737 6.2.Inner Products.... 737 6.3.Sequences and Convergence......................... 739 6.4.Hilbert Spaces.······ 740 6.4a.Orthogonal Sets and Basis Sets in Hilbert Spaces...···.·.. 740 6.4b."Algebraic Shift"to Dirac Notation for Vectors and nner Products.,,,。。。。···········.· 741 6.5.Linear Operators.. 742 6.5a.Definition of Linear Operators..................... 742 6.5b.Operator Norms and Bounded Operators...... 742 6.5c.Matrix Representation of Linear Operators and Use of Dirac Notation.,·,··…·····…····…·… 742 6.5 d.Adjoint Operator.。..·····….·····.········· 745 6.5e.Compact Operators.. 746 6.5f.Mathematical Definition of Hilbert-Schmidt Operators.......746

2. Organization of This Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 3. Introduction to SVD and Waves—Sets of Point Sources and Receivers. . . 690 3.1. Scalar Wave Equation and Green’s Functions. . . . . . . . . . . . . . . . . 691 3.2. Matrix-Vector Description of the Coupling of Point Sources and Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 3.3. Hermitian Adjoints and Dirac Bra-Ket Notation . . . . . . . . . . . . . . . 692 3.4. Orthogonality and Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . 693 3.5. Orthonormal Functions and Vectors . . . . . . . . . . . . . . . . . . . . . . . 694 3.6. Vector Spaces, Operators, and Hilbert Spaces. . . . . . . . . . . . . . . . . 695 3.7. Eigenproblems and Singular-Value Decomposition . . . . . . . . . . . . . 695 3.8. Sum Rule on Coupling Strengths . . . . . . . . . . . . . . . . . . . . . . . . . 698 3.9. Constraint on the Choice of the Coupling Strengths of the Channels . . . 699 4. Introductory Example—Three Sources and Three Receivers . . . . . . . . . . 700 4.1. Mathematical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 4.2. Physical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 4.2a. Acoustic and Radio-Frequency Systems. . . . . . . . . . . . . . . . . 702 4.2b. Optical Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 4.2c. Larger Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 5. Scalar Wave Examples with Point Sources and Receivers . . . . . . . . . . . . 706 5.1. Nine Sources and Nine Receivers in Parallel Lines . . . . . . . . . . . . . 707 5.1a. Channels and Coupling Strengths . . . . . . . . . . . . . . . . . . . . . 707 5.1b. Modes and Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 5.2. Two-Dimensional Arrays of Sources and Receivers. . . . . . . . . . . . . 711 5.3. Paraxial Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 5.3a. Behavior of Singular Values . . . . . . . . . . . . . . . . . . . . . . . . 713 5.3b. Forms of the Communications Modes . . . . . . . . . . . . . . . . . . 714 5.3c. Additional Degeneracy of Eigenvalues—Paraxial Degeneracy . . . 717 5.3d. Paraxial Degeneracy and Paraxial Heuristic Numbers. . . . . . . . 718 5.3e. Use of Point Sources as Approximations to Sets of “Patches” ... 726 5.4. Non-Paraxial Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 5.4a. Longitudinal Heuristic Angle . . . . . . . . . . . . . . . . . . . . . . . . 727 5.4b. Spherical Shell Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 5.5. Deducing Sources to Give a Particular Wave . . . . . . . . . . . . . . . . . 730 5.5a. Sources for an Arbitrary Combination of Specific Receiver Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 5.5b. Sources for a Gaussian Spot—Passing the Diffraction Limit . . . 732 5.5c. “Top-Hat” Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 5.5d. Notes on Passing the Diffraction Limit . . . . . . . . . . . . . . . . . 735 6. Mathematics of Continuous Functions, Operators, and Vector Spaces . . . . 736 6.1. Functions, Vectors, Numbers, and Spaces . . . . . . . . . . . . . . . . . . . 737 6.2. Inner Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 6.3. Sequences and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 6.4. Hilbert Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 6.4a. Orthogonal Sets and Basis Sets in Hilbert Spaces . . . . . . . . . . 740 6.4b. “Algebraic Shift” to Dirac Notation for Vectors and Inner Products . . . . . . . ......................... 741 6.5. Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 6.5a. Definition of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . 742 6.5b. Operator Norms and Bounded Operators . . . . . . . . . . . . . . . . 742 6.5c. Matrix Representation of Linear Operators and Use of Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 6.5d. Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 6.5e. Compact Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 6.5f. Mathematical Definition of Hilbert–Schmidt Operators. . . . . . . 746 680 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial

Tutorial Vol.11,No.3/September 2019 Advances in Optics and Photonics 681 6.5g.Hermitian Operators........................·..· 747 6.5h.Spectral Theorem for Compact Hermitian Operators······· 748 6.5i.Positive Operators..................··.······· 749 6.6.Inner Products Involving Operators..····...····..········· 749 6.6a.Operator-Weighted Inner Product..................... 750 6.6b.Transformed Inner Product..·...··....············ 750 6.7.Singular-Value Decomposition.......... 751 6.8.Physical Coupling Operators as Hilbert-Schmidt Operators 751 69.Diffraction Operators.....··········.···.·::······· 754 6.10.Using the Sum Rule to Validate Practical,Finite Basis Sets 755 7.Communications Modes and Common Families of Functions ........ 756 7.1.Prolate Spheroidal Functions and Relation to Hermite-Gaussian and Laguerre-Gaussian Approximations..... 756 7.2.Orbital Angular Momentum Beams and Degrees of Freedom in Communications...。.。。。·,·。··················· 757 7.3.Paraxial Degeneracy,Sets of Functions,and Fourier Optics....... 758 8.Extending to Electromagnetic Waves.,.,...·.·..····..··.··· 758 8.1.How Many Independent Fields?........ 758 8.2.Vector Wave Equation for Electromagnetic Fields............ 759 8.3.Green's Functions for Electromagnetic Waves 759 8.4.Inner Products for Electromagnetic Quantities and Fields. 761 8.4a.Cartesian Inner Product for Sets of Sources or Receivers..··· 761 8.4b.Cartesian Inner Product for Vector Fields.............. 762 8.4c.Electromagnetic Mode Example...................... 762 8.4d.Energy Inner Product for the Electromagnetic Field......... 764 8.5.Energy-Orthogonal Modes for Arbitrary Volumes.........···.. 765 8.6.Sum Rule and Communications Modes for Electromagnetic Fields 767 9.Quantizing the Electromagnetic Field Using the M-Gauge........... 767 10.Linear Scatterers and Optical Devices........................ 768 10.1.Existence of Orthogonal Functions and Channels............. 769 10.2.Establishing the Orthogonal Channels through Any Linear Scatterer... 769 10.3.Bounding the Dimensionalities of the Spaces....... 769 10.4.Emulating an Arbitrary Linear Optical Device and Proving Any Such Device Is Possible-Arbitrary Matrix-Vector Multiplication 771 11.Mode-Converter Basis Sets as Fundamental Optical Descriptions...... 772 1l.l.Radiation Laws...........·.····· 772 11.2.Modal "A&B Coefficient"Argument-the M Coefficient for Emission and Absorption..... 774 11.3.Mode-Converter Basis Sets as Physical Properties of a System.... 774 l2.Conclusions.....。。.·。······ 775 Appendix A:History and Literature Review of Communications Modes and Related Concepts...·.·。.···········::··· 775 A.l.Early History of Degrees of Freedom in Optics and Waves..···. 775 A.2.Eigenfunctions for Wave Problems with Regular Apertures.·.··· 776 A.3.Emergence of Communications Modes.··...··.·..···.···. 777 A.3a.Wireless Communications..................··.·..·. 777 A.3b.Electromagnetic Scattering and Imaging.·.·····.······· 777 778 A.4.Complex Optics,Matrix Representations,and Mode-Converter Basis Sets....·.····· 778 Appendix B:Approximating Uniform Line or Patch Sources with Point Sources......... 779 Appendix C:Longitudinal Heuristic Angle....................... 780 Appendix D:Spherical Heuristic Number............... 781

6.5g. Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 6.5h. Spectral Theorem for Compact Hermitian Operators . . . . . . . . 748 6.5i. Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749 6.6. Inner Products Involving Operators. . . . . . . . . . . . . . . . . . . . . . . . 749 6.6a. Operator-Weighted Inner Product . . . . . . . . . . . . . . . . . . . . . 750 6.6b. Transformed Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . 750 6.7. Singular-Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 6.8. Physical Coupling Operators as Hilbert–Schmidt Operators . . . . . . . 751 6.9. Diffraction Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 6.10. Using the Sum Rule to Validate Practical, Finite Basis Sets . . . . . . 755 7. Communications Modes and Common Families of Functions . . . . . . . . . 756 7.1. Prolate Spheroidal Functions and Relation to Hermite–Gaussian and Laguerre–Gaussian Approximations . . . . . . . . . . . . . . . . . . . . 756 7.2. Orbital Angular Momentum Beams and Degrees of Freedom in Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757 7.3. Paraxial Degeneracy, Sets of Functions, and Fourier Optics . . . . . . . 758 8. Extending to Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 758 8.1. How Many Independent Fields?. . . . . . . . . . . . . . . . . . . . . . . . . . 758 8.2. Vector Wave Equation for Electromagnetic Fields . . . . . . . . . . . . . . 759 8.3. Green’s Functions for Electromagnetic Waves . . . . . . . . . . . . . . . . 759 8.4. Inner Products for Electromagnetic Quantities and Fields . . . . . . . . . 761 8.4a. Cartesian Inner Product for Sets of Sources or Receivers . . . . . 761 8.4b. Cartesian Inner Product for Vector Fields. . . . . . . . . . . . . . . . 762 8.4c. Electromagnetic Mode Example . . . . . . . . . . . . . . . . . . . . . . 762 8.4d. Energy Inner Product for the Electromagnetic Field. . . . . . . . . 764 8.5. Energy-Orthogonal Modes for Arbitrary Volumes . . . . . . . . . . . . . . 765 8.6. Sum Rule and Communications Modes for Electromagnetic Fields . . 767 9. Quantizing the Electromagnetic Field Using the M-Gauge . . . . . . . . . . . 767 10. Linear Scatterers and Optical Devices . . . . . . . . . . . . . . . . . . . . . . . . 768 10.1. Existence of Orthogonal Functions and Channels . . . . . . . . . . . . . 769 10.2. Establishing the Orthogonal Channels through Any Linear Scatterer . . . 769 10.3. Bounding the Dimensionalities of the Spaces . . . . . . . . . . . . . . . . 769 10.4. Emulating an Arbitrary Linear Optical Device and Proving Any Such Device Is Possible—Arbitrary Matrix-Vector Multiplication . . . . . . 771 11. Mode-Converter Basis Sets as Fundamental Optical Descriptions . . . . . . 772 11.1. Radiation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 11.2. Modal “A&B Coefficient” Argument—the M Coefficient for Emission and Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 11.3. Mode-Converter Basis Sets as Physical Properties of a System . . . . 774 12. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775 Appendix A: History and Literature Review of Communications Modes and Related Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775 A.1. Early History of Degrees of Freedom in Optics and Waves . . . . . . . 775 A.2. Eigenfunctions for Wave Problems with Regular Apertures . . . . . . . 776 A.3. Emergence of Communications Modes . . . . . . . . . . . . . . . . . . . . . 777 A.3a. Wireless Communications . . . . . . . . . . . . . . . . . . . . . . . . . . 777 A.3b. Electromagnetic Scattering and Imaging . . . . . . . . . . . . . . . . 777 A.3c. Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778 A.4. Complex Optics, Matrix Representations, and Mode-Converter Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778 Appendix B: Approximating Uniform Line or Patch Sources with Point Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 Appendix C: Longitudinal Heuristic Angle . . . . . . . . . . . . . . . . . . . . . . . 780 Appendix D: Spherical Heuristic Number . . . . . . . . . . . . . . . . . . . . . . . . 781 Tutorial Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics 681

682 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial Appendix E:Singular-Value Decomposition of Compact Operators....... 781 Appendix F:Hilbert-Schmidt Operators with Weighted Inner Products..... 784 Appendix G:Electromagnetic Gauge,Green's Functions,and Energy Inner 785 G.l.Background Electromagnetism.··.··· 785 G.2.Choosing a Gauge for Communications Problems............. 787 G.2a.Gauge for Communications-the M-Gauge.............. 788 G.2b.Wave Equations in the M-Gauge..................... 789 G.3.Dyadic Green's Function for the Vector Potential in the M-Gauge..... 790 G.3a.Derivation of General Form for Monochromatic Waves...... 790 G.3b.Explicit Form for the Dyadic Green's Function for Monochromatic Waves............................ 791 G.3c.Green's Functions for General Time-Dependent Waves...... 793 G.3d.Green's Functions for the Electric and Magnetic Fields 794 G.4.Energy Inner Product for the Vector Potential........... 794 G.4a.Expressions for Energy Density in Electromagnetic Fields.... 794 G.4b.nner-Product Form..·............·..·.······ 795 Appendix H:Divergence of the Vector Potential in the M-Gauge.......。.······…····· 797 Appendix I:Dyadic Notation and Useful Identities for Green's functions 798 Il.Vector Calculus Extended to Dyadics...,...........······· 799 L.2.Useful Derivatives for Dyadics and Green's Functions.·.··.·.·.. 801 Appendix J:Quantization of the Electromagnetic Field in the M-Gauge...······· 802 Appendix K:Modal“A&B”Coefficient Argument 804 Appendix L:Novel Results in this Work......................... 806 L.1.Minor Extensions of Prior Work and Introduction of New Terminology.。。。·。·…····…······ 806 L.2.Novel Observations..·.,··.·······················… 807 L.3.Substantial New Concepts and Results..............·····.· 808 L.3a.Introduction of the M-Gauge for Electromagnetism.... 808 L.3b.Novel Quantization of the Electromagnetic Field .... 808 L.3c.Novel "M-Coefficient"Modal Alternate to Einstein's "A&B" Coefficient Argument.,...·......·.··...··.···· 808 Funding·。·…·…···…·……·……·… 808 Acknowledgment.·..。··..···················· 808 References and Notes..·....:.····· 808

Appendix E: Singular-Value Decomposition of Compact Operators . . . . . . . 781 Appendix F: Hilbert–Schmidt Operators with Weighted Inner Products. . . . . 784 Appendix G: Electromagnetic Gauge, Green’s Functions, and Energy Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 G.1. Background Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . 785 G.2. Choosing a Gauge for Communications Problems . . . . . . . . . . . . . 787 G.2a. Gauge for Communications—the M-Gauge . . . . . . . . . . . . . . 788 G.2b. Wave Equations in the M-Gauge . . . . . . . . . . . . . . . . . . . . . 789 G.3. Dyadic Green’s Function for the Vector Potential in the M-Gauge. . . . . 790 G.3a. Derivation of General Form for Monochromatic Waves . . . . . . 790 G.3b. Explicit Form for the Dyadic Green’s Function for Monochromatic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 G.3c. Green’s Functions for General Time-Dependent Waves . . . . . . 793 G.3d. Green’s Functions for the Electric and Magnetic Fields . . . . . . 794 G.4. Energy Inner Product for the Vector Potential . . . . . . . . . . . . . . . . 794 G.4a. Expressions for Energy Density in Electromagnetic Fields . . . . 794 G.4b. Inner-Product Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 Appendix H: Divergence of the Vector Potential in the M-Gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 Appendix I: Dyadic Notation and Useful Identities for Green’s functions . . . 798 I.1. Vector Calculus Extended to Dyadics . . . . . . . . . . . . . . . . . . . . . . 799 I.2. Useful Derivatives for Dyadics and Green’s Functions . . . . . . . . . . . 801 Appendix J: Quantization of the Electromagnetic Field in the M-Gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802 Appendix K: Modal “A&B” Coefficient Argument . . . . . . . . . . . . . . . . . . 804 Appendix L: Novel Results in this Work . . . . . . . . . . . . . . . . . . . . . . . . . 806 L.1. Minor Extensions of Prior Work and Introduction of New Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 L.2. Novel Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 L.3. Substantial New Concepts and Results . . . . . . . . . . . . . . . . . . . . . 808 L.3a. Introduction of the M-Gauge for Electromagnetism . . . . . . . . . 808 L.3b. Novel Quantization of the Electromagnetic Field . . . . . . . . . . 808 L.3c. Novel “M-Coefficient” Modal Alternate to Einstein’s “A&B” Coefficient Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 Funding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 682 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial

Tutorial Vol.11,No.3/September 2019/Advances in Optics and Photonics 683 Waves,modes,communications, and optics:a tutorial DAVID A.B.MILLER 1.INTRODUCTION The idea of modes is common in the world of waves,especially in optics.Modes are very useful in simplifying many problems.But,there is much confusion about them. Are modes“resonances'”?Are they“beams"?Do they have to stay the same“shape"? Are they“communication channels'”?How do we“count'”modes?Are they properties of space or of objects such as scatterers?Just what is the definition of a mode?The purpose of this paper is to sort out the answers to questions like these,and to clarify and extend the idea of "modes."In particular,we want to use them for describing waves in communications and in describing sophisticated optical devices.Such ap- plications are increasingly important:communications may require mode-or space- division multiplexing to increase capacity,and we are able to fabricate progressively more complex optical devices with modern micro-and nano-fabrication. 1.1.Modes and Waves At their simplest,modes can be different shapes of waves.Some such modes arise naturally in waveguides and resonators;these modes are well understood and are taught in standard texts (see,e.g.,[1-4]).A key benefit of modes is that,when we choose the right ones,problems simplify;instead of describing waves directly as their values at each of a large number of points,we can just use the amplitudes of some relatively small number of modes.But when we want to use modes to under- stand communications with waves more generally,or when we want to describe some linear optical device or object economically using modes,we need to move beyond the ideas of just resonator or waveguide modes.Specifically,we can introduce the ideas of communications modes in communicating with waves [5]and mode-converter basis sets [6,7]in describing devices.These modes are not yet part of standard texts,nor is there even any broad and deep introduction to them.Further,many of their details and applications are not yet discussed in the literature. The reason for writing this paper is to provide exactly such an introduction.As well as sorting out the ideas of modes generally,we explain the physics of these additional forms of modes,which brings clearer answers to our opening questions above. We show how these ideas are supported by powerful and ultimately straightforward mathematics.We introduce novel,useful,and fundamental results that follow.This approach resolves many confusions.It reveals powerful concepts and methods,gen- eral limits,new physical laws,and some simple and even surprising results.It works over a broad range of waves,from acoustics,through classical microwave electromag- netism,to quantum-mechanical descriptions of light. 1.2.Idea of Modes One subtle point about modes is that it can be difficult to find a definition or even a clear statement of what they are.We should clarify this now. Modes are particularly common in describing oscillations of physical objects and sys- tems.Simple examples include a mass on a spring,or waves on a string,especially one with fixed ends.In these cases,an informal definition of an oscillating mode is that it is a way of oscillating in which everything that is oscillating is oscillating at the same

Waves, modes, communications, and optics: a tutorial DAVID A. B. MILLER 1. INTRODUCTION The idea of modes is common in the world of waves, especially in optics. Modes are very useful in simplifying many problems. But, there is much confusion about them. Are modes “resonances”? Are they “beams”? Do they have to stay the same “shape”? Are they “communication channels”? How do we “count” modes? Are they properties of space or of objects such as scatterers? Just what is the definition of a mode? The purpose of this paper is to sort out the answers to questions like these, and to clarify and extend the idea of “modes.” In particular, we want to use them for describing waves in communications and in describing sophisticated optical devices. Such ap￾plications are increasingly important: communications may require mode- or space￾division multiplexing to increase capacity, and we are able to fabricate progressively more complex optical devices with modern micro- and nano-fabrication. 1.1. Modes and Waves At their simplest, modes can be different shapes of waves. Some such modes arise naturally in waveguides and resonators; these modes are well understood and are taught in standard texts (see, e.g., [1–4]). A key benefit of modes is that, when we choose the right ones, problems simplify; instead of describing waves directly as their values at each of a large number of points, we can just use the amplitudes of some relatively small number of modes. But when we want to use modes to under￾stand communications with waves more generally, or when we want to describe some linear optical device or object economically using modes, we need to move beyond the ideas of just resonator or waveguide modes. Specifically, we can introduce the ideas of communications modes in communicating with waves [5] and mode-converter basis sets [6,7] in describing devices. These modes are not yet part of standard texts, nor is there even any broad and deep introduction to them. Further, many of their details and applications are not yet discussed in the literature. The reason for writing this paper is to provide exactly such an introduction. As well as sorting out the ideas of modes generally, we explain the physics of these additional forms of modes, which brings clearer answers to our opening questions above. We show how these ideas are supported by powerful and ultimately straightforward mathematics. We introduce novel, useful, and fundamental results that follow. This approach resolves many confusions. It reveals powerful concepts and methods, gen￾eral limits, new physical laws, and some simple and even surprising results. It works over a broad range of waves, from acoustics, through classical microwave electromag￾netism, to quantum-mechanical descriptions of light. 1.2. Idea of Modes One subtle point about modes is that it can be difficult to find a definition or even a clear statement of what they are. We should clarify this now. Modes are particularly common in describing oscillations of physical objects and sys￾tems. Simple examples include a mass on a spring, or waves on a string, especially one with fixed ends. In these cases, an informal definition of an oscillating mode is that it is a way of oscillating in which everything that is oscillating is oscillating at the same Tutorial Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics 683