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..·....:.····· 808Appendix 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