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............... 7816.5g. 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