The minus sign of the first cosine means that the wave is travelling to the right of positive z. The plus sign of the second cosine implies that the wave is moving to the right of negative z.
Also, notice that the time is being multiplied by the angular frequency,
w=kv.(1.51)
Notice that
E⃗(r⃗,t)=E0cos(kz−wt)(1.52)
where we set A→E0. The last expression is the equation of the plane wave.
1.11.2 Spherical coordinates
Now let’s pay attention to the Helmholtz equation spherical coordinates. First let’s recall the Helmholtz equation,
∇2E⃗(r⃗)+k2E⃗(r⃗)=0.(1.53)
hence, in spherical coordinates the Helmholtz equation is expressed as,
1r2∂∂rr2∂E⃗(r⃗)∂r+k2E⃗(r⃗)=0.(1.54)
To solve it assume that E⃗(r⃗) has the following form,
E⃗(r⃗)=E′(r)reˆr(1.55)
where E′(r) is a function of r. Thus replacing equation (1.55) in equation (1.54),
1r2∂∂rr2−E′r2+r∂E′∂r+k2E′r=0(1.56)
expanding,
1r2−∂E′∂r+r∂2E′∂r2+∂E′∂r+k2E′r=0(1.57)
simplify,
1r∂2E′∂r2+k2E′r=0.(1.58)
Notice, that is the same equation that we solved in the previous section. Therefore, we can conclude that the solution of the wave equation in spherical coordinates has the following form,
E⃗(r⃗,t)=E0rcos(kz−wt).(1.59)
Notice that the amplitude of the wave decreases as r→∞.
As an exercise to the reader, please study the Helmholtz equation in cylindrical coordinates. The Helmholtz equation in cylindrical coordinates is the following expression,
1r∂∂rr∂E⃗(r⃗)∂r+k2E⃗(r⃗)=0.(1.60)
1.12 End notes
In this chapter, we briefly studied Maxwell’s equations, from which we find the wave equation. From the latter, we obtained some particular solutions and their spatial part—the Helmholtz equation.
The Helmholtz equation will be of great help to us because through it we will find the eikonal equation and in turn, the ray equation. These last equations lay the foundations of geometric optics. Geometric optics is the playing field of stigmatism which will be presented in-depth in chapter 5 and the following chapters; stigmatic systems will be studied in detail.
Further reading
Arfken G B and Weber H J 1999 Mathematical Methods for Physicists (New York: Academic)
Boas M L 2006 Mathematical Methods in the Physical Sciences (New York: Wiley)
Born M and Wolf E 2013 Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Amsterdam: Elsevier)
Buchdahl H A 1993 An Introduction to Hamiltonian Optics (Chelmsford, MA: Courier Corporation)
Campbell L 1882 The Life of James Clerk Maxwell (London: Macmillan)
Fleisch D 2008 A Student’s Guide to Maxwell’s equations (Cambridge: Cambridge University Press)
Goodman J W 2005 Introduction to Fourier Optics (Greenwood Village, CO: Roberts)
Griffiths D J 2005 Introduction to Electrodynamics (Cambridge: Cambridge University Press)
Hecht E 1974 Schaum’s Outline of Optics (New York: McGraw-Hill)
Hecht E 2012 Optics (Cambridge, MA: Pearson)
Jackson J D 1999 Classical Electrodynamics (Hoboken, NJ: Wiley)
Lakshminarayanan V, Ghatak A and Thyagarajan K 2002 Lagrangian Optics (Berlin: Springer)
Lax M, Louisell W H and McKnight W B 1975 From Maxwell to paraxial wave optics Phys. Rev. A 11 1365
Luneburg R K 1964 Mathematical Theory of Optics (Berkeley, CA: University of California Press)
Mahon B 2004 The Man Who Changed Everything: The Life of James Clerk Maxwell (New York: Wiley)
Maxwell J C 1990 The Scientific Letters and Papers of James Clerk Maxwell: 1846–1862 (Cambridge: Cambridge University Press)
Perko L 2013 Differential equations and Dynamical Systems vol 7 (Berlin: Springer)
Ronchi V and Barocas V 1970 The Nature of Light: An Historical Survey (Cambridge, MA: Harvard University Press)
Zill D G 2016 Differential equations with Boundary-value Problems (Boston, MA: Cengage)
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