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.......7462. 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