1 Waves 9

Lecture 9Lecture 9

Wave propagation.Wave propagation.

Aims:Aims: Huygen’s Principle:

Reflection and refraction. Problems

Huygen’s-Fresnel principle

Fraunhofer diffraction(waves in the “far field”).

Young’s double slits Three slits N slits and diffraction gratings A single broad slit General formula - Fourier transform.

This lectureThis lecture

2 Waves 9

Huygens’ PrincipleHuygens’ Principle

Remember the concept of wavefront - a surface of constant phase.

1690 1690 “Treatise on light”“Treatise on light”, Huygens., Huygens. “Every point of a primary wavefront behaves as

the source of spherical, secondary wavelets, such that the primary wavefront at a later time is the envelope of these wavelets; the wavelets have the same frequency and velocity as the incoming wave”

Rectilinear propagation

Spherical propagation

3 Waves 9

Reflection and refractionReflection and refraction

rr = = ii

Result follows from the 2 right-angled triangles with same hypotenuse, both having one side of length vt. Thus r = i.

Snell’s Law Snell’s Law

/ACsin

/ACsin

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11

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ncv

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2

2

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2

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4 Waves 9

Huygen’s-Fresnel principleHuygen’s-Fresnel principle

Shortcomings Shortcomings It is easy to criticise Huygens: No theoretical basis; Why neglect parts of the wavelet other than

those forming the envelope; Why don’t wavelets propogate backwards; It is no help in predicting amplitudes; etc...

None detract from its historical significnce and the fact that it works.

Fresnel (1818) Fresnel (1818) (See handout).(See handout). He built in Young’s concept of interference.

“Every unobstructed point of a wavefront … serves as a source of spherical secondary wavelets … The amplitude of the optical field at any point beyond is the superposition of all these wavelets (considering their amplitudes and relative phases)”

Note backward travelling wavelets tend to interfere destructively

Kirchoff (1824-1887)Kirchoff (1824-1887) Provided theoretical foundation by connecting

the wave equation to a surface integral of spherical wavelets.

See Optics course, next term.