《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

684 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial frequency.This is a sense in which a“mode”isa“way”or“manner'”of oscillation Musical instruments offer many other examples of such modes,as in standing waves in a pipe,or resonances in the vibrations of plates or hollow bodies.Such a mode will have a specific frequency of oscillation,and the amplitude of the vibration will take a specific physical form-it can be a function of position along the string or pipe or on the surface of some plate or body. The underlying mathematical idea of modes is associated with eigenfunctions or eigenvectors in linear physical systems;in oscillating systems or resonators,the func- tion that gives the amplitude of oscillation at each position is the eigenfunction,and the frequency (or often the square of the frequency)is the eigenvalue.Indeed,we can state a useful,general definition of a mode [8-10]: A mode is an eigenfunction of an eigenproblem describing a physical system. (1) Conventional resonator and waveguide modes are each the eigenfunctions of a single eigenproblem.The fixed "shape"of this oscillation amplitude inside the resonator is often thought of as the "mode"or eigenfunction in this sense.Waveguide modes use the same mathematics,but the concept here is that the transverse shape of the mode does not change as it propagates.An analogous informal definition of a propagating mode is that everything that propagates is propagating with the same wave vector,which also implies that the(transverse)shape does not change as it propagates.That transverse shape is the eigenfunction.Though such waveguide modes may well be modes of a specific frequency that we have chosen,the eigenvalue is typically a propagation con- stant or wavevector magnitude (or,again,often the square of this quantity). Before going any further,to support these ideas of modes,we need good notations; they should be general enough to handle everything we need,but they should suppress unnecessary detail.Wherever possible,we use a Dirac"bra-ket"notation,which op- erates at just such a useful level of abstraction.We introduce this notation progres- sively (see also [9)).In this notation a function can be represented by a"ket"or"ket vector,"written as lus)or l),for example.Linear operators,such as Green's func- tions or scattering operators,are represented by a letter,and here we will mostly use "sans serif'capital letters such as G and D.Most simply,we can think of kets as column vectors of numbers and the linear operators as matrices.Dirac notation imple- ments a convenient version of linear algebra equivalent to matrix-vector operations with complex numbers,and indeed such a matrix-vector view can be the simplest way to think about Dirac notation. 1.3.Modes as Pairs of Functions To handle communications and complex optical devices,we need to go beyond just resonator or waveguide modes;fortunately,though,we can use much of the same mathematics.The key mathematical difference between resonator and waveguide modes on the one hand and our new modes on the other is that communications modes and mode-converter basis sets each result from solving a singular-value decomposition (SVD)problem,which corresponds to solving two eigenproblems. The physical reason for having two such eigenproblems is because we are defining optimum mappings between two different spaces. For example,in communications,we may have sources or transmitters in one"source" volume and resulting waves communicated into another "receiving"volume

frequency. This is a sense in which a “mode” is a “way” or “manner” of oscillation. Musical instruments offer many other examples of such modes, as in standing waves in a pipe, or resonances in the vibrations of plates or hollow bodies. Such a mode will have a specific frequency of oscillation, and the amplitude of the vibration will take a specific physical form—it can be a function of position along the string or pipe or on the surface of some plate or body. The underlying mathematical idea of modes is associated with eigenfunctions or eigenvectors in linear physical systems; in oscillating systems or resonators, the func￾tion that gives the amplitude of oscillation at each position is the eigenfunction, and the frequency (or often the square of the frequency) is the eigenvalue. Indeed, we can state a useful, general definition of a mode [8–10]: A mode is an eigenfunction of an eigenproblem describing a physical system: (1) Conventional resonator and waveguide modes are each the eigenfunctions of a single eigenproblem. The fixed “shape” of this oscillation amplitude inside the resonator is often thought of as the “mode” or eigenfunction in this sense. Waveguide modes use the same mathematics, but the concept here is that the transverse shape of the mode does not change as it propagates. An analogous informal definition of a propagating mode is that everything that propagates is propagating with the same wave vector, which also implies that the (transverse) shape does not change as it propagates. That transverse shape is the eigenfunction. Though such waveguide modes may well be modes of a specific frequency that we have chosen, the eigenvalue is typically a propagation con￾stant or wavevector magnitude (or, again, often the square of this quantity). Before going any further, to support these ideas of modes, we need good notations; they should be general enough to handle everything we need, but they should suppress unnecessary detail. Wherever possible, we use a Dirac “bra-ket” notation, which op￾erates at just such a useful level of abstraction. We introduce this notation progres￾sively (see also [9]). In this notation a function can be represented by a “ket” or “ket vector,” written as jψSi or jϕRi, for example. Linear operators, such as Green’s func￾tions or scattering operators, are represented by a letter, and here we will mostly use “sans serif” capital letters such as G and D. Most simply, we can think of kets as column vectors of numbers and the linear operators as matrices. Dirac notation imple￾ments a convenient version of linear algebra equivalent to matrix-vector operations with complex numbers, and indeed such a matrix-vector view can be the simplest way to think about Dirac notation. 1.3. Modes as Pairs of Functions To handle communications and complex optical devices, we need to go beyond just resonator or waveguide modes; fortunately, though, we can use much of the same mathematics. The key mathematical difference between resonator and waveguide modes on the one hand and our new modes on the other is that communications modes and mode-converter basis sets each result from solving a singular-value decomposition (SVD) problem, which corresponds to solving two eigenproblems. The physical reason for having two such eigenproblems is because we are defining optimum mappings between two different spaces. For example, in communications, we may have sources or transmitters in one “source” volume and resulting waves communicated into another “receiving” volume 684 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial

Tutorial Vol.11,No.3/September 2019/Advances in Optics and Photonics 685 [Fig.1(a)].The solutions to our problem are then the set of optimum source functions in the source or input volume that couple,one by one,to the resulting optimal waves in the receiving or output volume;SVD solves for both of those sets of functions,and it is these two sets of functions that are the communications modes.So,a given com- munications mode is not one function but two. We can also view a communications mode as defining a communications "channel." A simple view of a "channel"is that,when we put an input in one end,the corre- sponding output comes of the other end,without "leaking"into any other such "channel,"as in the literal meaning of a channel as carrying a stream of water, separately from other such streams or channels.In the case of the communications modes,modulating the "source"function leads to an amplitude in the corre- sponding receiving wave;the "separateness"here is defined by some mathematical "orthogonality"of all the source functions and all the receiving functions,and we clarify this idea below.We will be able to have separate channels for information flow even if the actual waves are mixed in the space between the source and receiver.When we use the term "channels"we mean such independent "ways"for sending informa- tion from source to receiver.In this sense,a communications mode describes the physical carrier for such an information "channel." In practice we may only need to solve one of these two SVD eigenproblems,and we can then deduce the solutions to the other.But because we can view this through two eigenproblems,each of these sets of functions,one in the source space and one in the receiving space,therefore has all the useful mathematical properties of eigenfunctions, including this idea of"orthogonality";such properties have profound consequences for the physical interpretation and the mathematics that follows. 1.3a.Communications Modes Note immediately that,in this view, the communications mode is not the propagating wave (or what we will call the beam)between the source volume and receiver volume. Figure 1 (a)Source or input Receiving or output volume or space volume or space V g v) lx） Hs Hg (b)Source or input Receiving or output volume or space volume or space Device Vs 1)》 Hs Hg Conceptual view for (a)communications modes and (b)mode-converter basis sets.In both cases a source function lus)in a source or input volume Vs,or more generally in a mathematical (Hilbert)space Hs,results in a wave function )in a receiving or output volume VR,or more generally in a mathematical (Hilbert)space HR.In the commu- nications mode case(a)the coupling is through a Green's function operator Gsk as appropriate for the intervening medium between the spaces.In the mode-converter case (b),the coupling is through the action of a device (or scattering)operator D

[Fig. 1(a)]. The solutions to our problem are then the set of optimum source functions in the source or input volume that couple, one by one, to the resulting optimal waves in the receiving or output volume; SVD solves for both of those sets of functions, and it is these two sets of functions that are the communications modes. So, a given com￾munications mode is not one function but two. We can also view a communications mode as defining a communications “channel.” A simple view of a “channel” is that, when we put an input in one end, the corre￾sponding output comes of the other end, without “leaking” into any other such “channel,” as in the literal meaning of a channel as carrying a stream of water, separately from other such streams or channels. In the case of the communications modes, modulating the “source” function leads to an amplitude in the corre￾sponding receiving wave; the “separateness” here is defined by some mathematical “orthogonality” of all the source functions and all the receiving functions, and we clarify this idea below. We will be able to have separate channels for information flow even if the actual waves are mixed in the space between the source and receiver. When we use the term “channels” we mean such independent “ways” for sending informa￾tion from source to receiver. In this sense, a communications mode describes the physical carrier for such an information “channel.” In practice we may only need to solve one of these two SVD eigenproblems, and we can then deduce the solutions to the other. But because we can view this through two eigenproblems, each of these sets of functions, one in the source space and one in the receiving space, therefore has all the useful mathematical properties of eigenfunctions, including this idea of “orthogonality”; such properties have profound consequences for the physical interpretation and the mathematics that follows. 1.3a. Communications Modes Note immediately that, in this view, the communications mode is not the propagating wave (or what we will call the beam) between the source volume and receiver volume. Figure 1 Conceptual view for (a) communications modes and (b) mode-converter basis sets. In both cases a source function jψSi in a source or input volume VS, or more generally in a mathematical (Hilbert) space HS, results in a wave function jϕRi in a receiving or output volume VR, or more generally in a mathematical (Hilbert) space HR. In the commu￾nications mode case (a) the coupling is through a Green’s function operator GSR as appropriate for the intervening medium between the spaces. In the mode-converter case (b), the coupling is through the action of a device (or scattering) operator D. Tutorial Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics 685

686 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial Indeed,in general the beam will change shape as it propagates,and it is not itself the "eigenfunction"of the mathematical problem (though it is easily deduced from the ac- tual eigenfunctions in simple communication problems).In this SVD way of looking at communications,the jth communications mode is a pair of functions-s)in the source or input space,and in the receiving or output space.Explicitly,therefore, communications modes are pairs of functions-one in the source space and one in the receiving space. They are a set of communications mode pairs of functions-a pair ls)and o),a pair ls2)and lg2),and so on.To find these functions,we perform the SVD of the coupling operator Gsg between the volumes or spaces.For the communications prob- lems we consider first,this Gsk is effectively the free-space Green's function for our wave equation. 1.3b.Mode-Converter Basis Sets When we change from thinking just about waves in free space to trying to describe a linear optical device,we can consider how it scatters input waves to output waves [Fig.1(b)].By analyzing this also as an SVD problem,in this case of a device (or scattering)operator D,we can similarly deduce a set [11]of input source functions si))that couple one by one to a set of output wave functions{));these two sets of functions are the mode-converter basis sets. In this second case,we want to describe the device as one that converts from a specific input mode lus to the corresponding output mode l),and so on,for all such mode pairs;again,as in the case of communications modes,we think in terms of pairs of functions here,one in the source or input space,and one in the receiving or output space.We can consider these as mode-converter pairs-a pair si)and Ri),a pair lus2)and oR2),and so on,just as in the communications modes.In this way of look- ing at a linear optical device [6], any linear optical device can be viewed as a mode converter,converting from specific sets of functions in the input space one by one to specific corresponding functions in the output space,giving the mode-converter pairs of functions. The device converts input mode lusi)to output mode R),input mode lus2)to output mode 2),and so on.In this case,though the mathematics is similar to the communications modes,this is more a way of describing the device,whereas the communications modes are a way of describing the communications channels from sources to receivers.For the device case,we may not have anything like a simple beam between the sources and receivers,but we do have these well-defined functions or "modes"inside the source space or volume and inside the receiving space or vol- ume.We could also view the mode-converter basis sets as describing the communi- cations modes“through”the device. In an actual physical problem for a device,there are ways in principle in which we could deduce the mode-converter pairs of functions by experiment [7,12]without ever knowing exactly what the wave field is inside the device.Then we could know the mode-converter pairs as eigenfunctions without knowing the "beam";this point em- phasizes that it can be more useful and meaningful to use the pairs of functions in the source and receiving spaces as the modes of the system rather than attempting to use the beam through the whole system as the way to describe it. 1.4.Usefulness of This Approach There are several practical and fundamental reasons why these pairs of functions are useful

Indeed, in general the beam will change shape as it propagates, and it is not itself the “eigenfunction” of the mathematical problem (though it is easily deduced from the ac￾tual eigenfunctions in simple communication problems). In this SVD way of looking at communications, the jth communications mode is a pair of functions—jψSji in the source or input space, and jϕRji in the receiving or output space. Explicitly, therefore, communications modes are pairs of functions—one in the source space and one in the receiving space. They are a set of communications mode pairs of functions—a pair jψS1i and jϕR1i, a pair jψS2i and jϕR2i, and so on. To find these functions, we perform the SVD of the coupling operator GSR between the volumes or spaces. For the communications prob￾lems we consider first, this GSR is effectively the free-space Green’s function for our wave equation. 1.3b. Mode-Converter Basis Sets When we change from thinking just about waves in free space to trying to describe a linear optical device, we can consider how it scatters input waves to output waves [Fig. 1(b)]. By analyzing this also as an SVD problem, in this case of a device (or scattering) operator D, we can similarly deduce a set [11] of input source functions fjψSjig that couple one by one to a set of output wave functions fjϕRjig; these two sets of functions are the mode-converter basis sets. In this second case, we want to describe the device as one that converts from a specific input mode jψSji to the corresponding output mode jϕRji, and so on, for all such mode pairs; again, as in the case of communications modes, we think in terms of pairs of functions here, one in the source or input space, and one in the receiving or output space. We can consider these as mode-converter pairs—a pair jψS1i and jϕR1i, a pair jψS2i and jϕR2i, and so on, just as in the communications modes. In this way of look￾ing at a linear optical device [6], any linear optical device can be viewed as a mode converter, converting from specific sets of functions in the input space one by one to specific corresponding functions in the output space, giving the mode-converter pairs of functions. The device converts input mode jψS1i to output mode jϕR1i, input mode jψS2i to output mode jϕR2i, and so on. In this case, though the mathematics is similar to the communications modes, this is more a way of describing the device, whereas the communications modes are a way of describing the communications channels from sources to receivers. For the device case, we may not have anything like a simple beam between the sources and receivers, but we do have these well-defined functions or “modes” inside the source space or volume and inside the receiving space or vol￾ume. We could also view the mode-converter basis sets as describing the communi￾cations modes “through” the device. In an actual physical problem for a device, there are ways in principle in which we could deduce the mode-converter pairs of functions by experiment [7,12] without ever knowing exactly what the wave field is inside the device. Then we could know the mode-converter pairs as eigenfunctions without knowing the “beam”; this point em￾phasizes that it can be more useful and meaningful to use the pairs of functions in the source and receiving spaces as the modes of the system rather than attempting to use the beam through the whole system as the way to describe it. 1.4. Usefulness of This Approach There are several practical and fundamental reasons why these pairs of functions are useful. 686 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial

Tutorial Vol.11,No.3/September 2019/Advances in Optics and Photonics 687 1.4a.Using Communications Modes In communications,we continually want larger amounts of useful bandwidth.This need is strong for wireless radio-frequency transmission [13],for optical signals in fibers [14-17]or free space [17-20],and even for acoustic information transmission [21-23].Recent progress in novel optical ways to separate different [16]and even arbitrary modes [24-29],including automatic methods [24-29],gives additional mo- tivation to consider the use of different modes (or "spatial degrees of freedom")in communications. Increasingly,therefore,we need to understand the spatial degrees of freedom in such communications and the limits in their use;a natural way to describe and quantify those is in terms of communications modes.Specifically, we can understand how to count the number of useful available spatial channels. Essentially,this can also be viewed as a generalization of the ideas of diffraction lim- its,and we will develop these ideas below.A key novel result is that this SVD approach gives a sum rule that bounds the number and strength of those channels. As we solve the problem this way,we can also unambiguously establish just exactly what the best channels are;we do not need to presume any particular form of these modes to start with.So,specifically,we do not need to analyze in terms of plane-wave “modes,.”Hermite--Gaussian or Laguerre-.Gaussian beams,optical“orbital”angular momentum(OAM)[19,20,30-32]"modes,"prolate spheroidals [33],arrays of spots, or any other specific family of functions;specifically, the SVD solution will tell us the best answers for the transmitting and receiving functions-the communications modes-and those will in general be none of the standard mathematical families of functions or beams. 1.4b.Using Mode-Converter Basis Sets In analyzing linear optical devices or scatterers, if we establish the mode-converter basis sets by solving the SVD problem,we will have the most economical and complete description of a device or scatterer. Essentially,we establish the "best"functions to use here,starting with the most im- portant and progressing to those of decreasing importance.An incidental and univer- sal consequence of this approach is that we realize that there is a set of independent channels through any linear scatterer (which are the mode-converter basis sets),and that we can describe the device com- pletely using those.The implications of the mode-converter basis sets go beyond sim- ple mathematical economy: Mode-converter basis functions have basic physical meaning and implications, giving fundamental results that can be economically and uniquely expressed using them. They allow us,for example,to write new versions and extensions of Kirchhoff's radi- ation laws [7],including ones that apply specifically and only to the mode-converter pairs,and to derive a novel modal version of Einstein's"A&B"coefficient argument on spontaneous and stimulated emission(Subsection 11.2).Such results suggest that this mode-converter basis set approach is deeply meaningful as a way to describe optical

1.4a. Using Communications Modes In communications, we continually want larger amounts of useful bandwidth. This need is strong for wireless radio-frequency transmission [13], for optical signals in fibers [14–17] or free space [17–20], and even for acoustic information transmission [21–23]. Recent progress in novel optical ways to separate different [16] and even arbitrary modes [24–29], including automatic methods [24–29], gives additional mo￾tivation to consider the use of different modes (or “spatial degrees of freedom”) in communications. Increasingly, therefore, we need to understand the spatial degrees of freedom in such communications and the limits in their use; a natural way to describe and quantify those is in terms of communications modes. Specifically, we can understand how to count the number of useful available spatial channels. Essentially, this can also be viewed as a generalization of the ideas of diffraction lim￾its, and we will develop these ideas below. A key novel result is that this SVD approach gives a sum rule that bounds the number and strength of those channels. As we solve the problem this way, we can also unambiguously establish just exactly what the best channels are; we do not need to presume any particular form of these modes to start with. So, specifically, we do not need to analyze in terms of plane-wave “modes,” Hermite–Gaussian or Laguerre–Gaussian beams, optical “orbital” angular momentum (OAM) [19,20,30–32] “modes,” prolate spheroidals [33], arrays of spots, or any other specific family of functions; specifically, the SVD solution will tell us the best answers for the transmitting and receiving functions—the communications modes—and those will in general be none of the standard mathematical families of functions or beams. 1.4b. Using Mode-Converter Basis Sets In analyzing linear optical devices or scatterers, if we establish the mode-converter basis sets by solving the SVD problem, we will have the most economical and complete description of a device or scatterer. Essentially, we establish the “best” functions to use here, starting with the most im￾portant and progressing to those of decreasing importance. An incidental and univer￾sal consequence of this approach is that we realize that there is a set of independent channels through any linear scatterer (which are the mode-converter basis sets), and that we can describe the device com￾pletely using those. The implications of the mode-converter basis sets go beyond sim￾ple mathematical economy: Mode-converter basis functions have basic physical meaning and implications, giving fundamental results that can be economically and uniquely expressed using them. They allow us, for example, to write new versions and extensions of Kirchhoff’s radi￾ation laws [7], including ones that apply specifically and only to the mode-converter pairs, and to derive a novel modal version of Einstein’s “A&B” coefficient argument on spontaneous and stimulated emission (Subsection 11.2). Such results suggest that this mode-converter basis set approach is deeply meaningful as a way to describe optical Tutorial Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics 687

688 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial systems.These mode-converter basis functions can also be identified in principle for a given linear object through physical experiments [7],independent of the mathematics. 1.4c.Areas of Research and Application This approach to waves,though not yet very widely known,has a history that goes back some decades,and already has many applications.The earliest,and very successful, application of eigenfunction approaches in waves is for laser resonators [34-36]with some related work in imaging [33].Such applications are special cases of the present approach in which the"source"and"receiver"functions are essentially mathematically the same.Following the introduction of the full SVD approach [5,37,38],there has been a broadening range of applications in wireless communications [13,39-51],where space-division multiplexing is increasingly an important option,r.f.imaging [52,53], electromagnetic scattering [54-64],optical systems [65-81],acoustic wave communi- cations [23],finding strong channels though strong scatterers [82-98](which is related to earlier work on electron transport though disordered media [97]),multiple-mode op- tical fibers for communications [81,99]and imaging [100-105],and free-space com- munications [18].This approach can also resolve paradoxes and confusions in counting available communications channels generally,such as whether OAM leads to more channels(and we discuss this below).The growing availability of optical systems that can generate complex and controllable devices [12,24-29,106-119]also means this SVD approach is practically accessible for more applications because we can generate sets of sources and can separate sets of waves,and SVD is also a good way to describe and even design those devices themselves [25].We give an extended discussion of this history and the wider literature in Appendix A. Aspects of this field have developed somewhat independently,and different authors therefore refer to similar concepts with different terminology.Our "device operator" or Green's function coupling operator between spaces is similar to the channel matrix in wireless communications [13].and the communications modes there are referred to [l3]as“eigenmodes of the channel”or“eigenchannels.”n optics,the“optical ei- genmodes"of [76-80]are similar to our communications modes,or,for more com- plex optical systems,the mode-converter basis sets.In work in channels through strong scatterers [81-92,96-98],the coupling operator (the "device"or "scattering" operator in our notation)is often called a "transmission"matrix (see,e.g.,[97])(with our mode-converter basis sets or communications modes through a scatterer known in that work as“optical eigenchannels'”or“transmission eigenchannels'”[97])or,some times,a"transfer matrix"[81].For consistency in this paper,we will use our notation, but the link to this other independent work and terminology is important to clarify. 1.5.Approach of This Paper Because the ideas here go beyond conventional textbook discussions,and because we are combining concepts and techniques that cross several different fields,the approach of this article is quite tutorial.Most algebra steps are written explicitly,and many "toy" examples illustrate the key steps and points.I have tried to write the main text so that it is readable,and with a progressive flow of ideas.I introduce core mathematical ideas in the main text,but relegate most other derivations and mathematics to appendices. This article has been written to be accessible to readers with a good basic undergraduate knowledge of mathematics and some physical science,such as would be acquired in a subject such as electrical engineering or physics or a discipline such as optics(for specific presumed background,see [120)),but I explicitly introduce all other required advanced mathematics and electromagnetism.Wherever possible,I take a direct approach in der- ivations,working from fundamental results,such as Maxwell's equations or core math- ematical definitions and principles,without invoking intermediate results or methods

systems. These mode-converter basis functions can also be identified in principle for a given linear object through physical experiments [7], independent of the mathematics. 1.4c. Areas of Research and Application This approach to waves, though not yet very widely known, has a history that goes back some decades, and already has many applications. The earliest, and very successful, application of eigenfunction approaches in waves is for laser resonators [34–36] with some related work in imaging [33]. Such applications are special cases of the present approach in which the “source” and “receiver” functions are essentially mathematically the same. Following the introduction of the full SVD approach [5,37,38], there has been a broadening range of applications in wireless communications [13,39–51], where space-division multiplexing is increasingly an important option, r.f. imaging [52,53], electromagnetic scattering [54–64], optical systems [65–81], acoustic wave communi￾cations [23], finding strong channels though strong scatterers [82–98] (which is related to earlier work on electron transport though disordered media [97]), multiple-mode op￾tical fibers for communications [81,99] and imaging [100–105], and free-space com￾munications [18]. This approach can also resolve paradoxes and confusions in counting available communications channels generally, such as whether OAM leads to more channels (and we discuss this below). The growing availability of optical systems that can generate complex and controllable devices [12,24–29,106–119] also means this SVD approach is practically accessible for more applications because we can generate sets of sources and can separate sets of waves, and SVD is also a good way to describe and even design those devices themselves [25]. We give an extended discussion of this history and the wider literature in Appendix A. Aspects of this field have developed somewhat independently, and different authors therefore refer to similar concepts with different terminology. Our “device operator” or Green’s function coupling operator between spaces is similar to the channel matrix in wireless communications [13], and the communications modes there are referred to [13] as “eigenmodes of the channel” or “eigenchannels.” In optics, the “optical ei￾genmodes” of [76–80] are similar to our communications modes, or, for more com￾plex optical systems, the mode-converter basis sets. In work in channels through strong scatterers [81–92,96–98], the coupling operator (the “device” or “scattering” operator in our notation) is often called a “transmission” matrix (see, e.g., [97]) (with our mode-converter basis sets or communications modes through a scatterer known in that work as “optical eigenchannels” or “transmission eigenchannels” [97]) or, some￾times, a “transfer matrix” [81]. For consistency in this paper, we will use our notation, but the link to this other independent work and terminology is important to clarify. 1.5. Approach of This Paper Because the ideas here go beyond conventional textbook discussions, and because we are combining concepts and techniques that cross several different fields, the approach of this article is quite tutorial. Most algebra steps are written explicitly, and many “toy” examples illustrate the key steps and points. I have tried to write the main text so that it is readable, and with a progressive flow of ideas. I introduce core mathematical ideas in the main text, but relegate most other derivations and mathematics to appendices. This article has been written to be accessible to readers with a good basic undergraduate knowledge of mathematics and some physical science, such as would be acquired in a subject such as electrical engineering or physics or a discipline such as optics (for specific presumed background, see [120]), but I explicitly introduce all other required advanced mathematics and electromagnetism. Wherever possible, I take a direct approach in der￾ivations, working from fundamental results, such as Maxwell’s equations or core math￾ematical definitions and principles, without invoking intermediate results or methods. 688 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial