WORCESTER POLYTECHNIC INSTITUTE

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1 WORCESTER POLYTECHNIC INSTITUTE MECHANICAL ENGINEERING DEPARTMENT Optical Metrology and NDT ME-593L, C 018 Introduction: Wave Optics January 018

2 Wave optics: light waves Wave equation An optical wave -- monochromatic -- can be described mathematically by the complex wavefunction U( x, z, t) a( x, z) exp[ j( x, z)] exp[ j t] (1) where x, z are the components of the position vector r t is time ( x, z) is the optical phase a( x, z) is the amplitude is the frequency [Hz] ( = = angular frequency [rad/sec]) j is the complex quantity 1

3 Wave optics: light waves Wave equation Equation (1) can be written as U( x, z, t) U( x, z) exp[ j t] () where the time-independent term, U( x, z) a( x, z) exp[ j( x, z)] (3) is the complex amplitude of the optical wave U( x, z, t)

4 Wave optics: light waves Wave equation Function U( x, z, t) must satisfy the wave equation (in order to represent a valid wave function), therefore, U 1 U 0 (4) c t where co c for n 1 (5) n in which c o is the speed of light in free-space and the wave propagates in a medium with index of refraction n.

5 Wave optics: light waves Wave equation By substituting Eq. into the wave equation, Eq. 4, the following equation is obtained -- exercise in class/homework ( k ) U( x, z) 0 (6) which is called the Helmholtz equation, where k c c (7) in the wave number, and is the spatial wavelength. Note that: c (8)

6 Wave optics: light waves Elementary waves The two canonical solutions of the Helmholtz equation in a homogenous medium are: (1) the plane wave, and () the spherical wave. (1) The plane wave The plane wave has the complex amplitude: U( x, z) Aexp( j k r) (9) where Aexp[ j( k x x k y y k z z)] A is the amplitude, or complex envelope k k x î k y r xî yĵ ĵ k z kˆ ( k x, k y, k j is the complex quantity 1 z ) zkˆ ( x, z) is the position vector is the propagation direction vector

7 Elementary waves Plane waves For Eq. 9 to satisfy the Helmholtz equation, Eq. 6, it is necessary that k x k y k z k (10.1) that is, the magnitude of the propagation direction vector, k, is equal to the wave number, k, k k (10.)

8 Since the phase, or are k x Elementary waves Plane waves x k arg[ U( x, z)] arg[ A] k r, the wavefronts y y k z for q = integer z q arg[ A] Equation 11 describes the family of parallel planes that is perpendicular to the propagation direction vector, k. These planes are called: wavefronts. Planes are separated by the distance (11) k (1)

9 Elementary waves Plane waves If the z-axis is taken in the direction of the propagation vector, k, then using Eqs 13 and, U( z) Aexp( j k z) (13) U( z, t) A( z) exp[ j k z)] exp[ j t] (14) A( z) exp[ j( t k z)] A exp{ j( t k z arg[ A])} (15) and by separating the real component of Eq. 15, it is obtained u( z, t) Re{ U( z, y)} A cos( t k z arg[ A]) (16) A cos{ ( t ) arg[ A]} c z

10 Elementary waves Plane waves A plane wave traveling in the z-direction is a periodic function of z with spatial period and a periodic function of t with temporal period 1/.

11 Elementary waves Observations Optical phase, obtained from Eq. 16, arg[re{ U( z, t)}] ( t varies as a function of time and position ) arg[ c z A] (17) Optical intensity is determined as I U U U * (18) where * U is the complex conjugate of U

12 Wave optics: light waves Elementary waves () The spherical wave The spherical wave is another canonical solution of the Helmholtz equation. Its complex amplitude is U( r) A exp( j k r) (19) r where r is the distance from the propagation origin. k is the wave number, and I U U * A r (proportional to the square of the distance from the origin)

13 Taking arg[a] = 0, for simplicit Elementary waves Spherical waves k r k x y z q arg[ A ] (0) for q = integer Equation 0 describes the family of concentric spheres: spherical wavefronts. Spheres are separated by the distance k (1) Cross-section of the wavefronts of a spherical wave

14 Elementary waves Wavefronts Effect of adding a lens The rays of ray optics are orthogonal to the wavefronts of wave optics. Note the effect of a lens on rays and wavefronts.

15 Optical interference Interferometers Mach-Zender

16 Optical interference Interferometers Michelson

17 Optical interference Interferometers Sagnac

18 Optical interference Holographic interferometry Recording Reconstruction

19 Interference equation Consider the superposition of two monochromatic plane waves and from the same light source U U 1 U The corresponding intensity is, I U The observed intensit measured, is U 1 U U1 () U U 1 U U U 1 * if U1 A1 exp( j k1 r) A1 exp( j1), U A exp( j k r) A exp( j ) U U * 1 (3) I I 1 I I1I cos( 1) (4)

20 Interference equation, cont d I I 1 I I1I cos( 1) (5) Defining: IB IM I1 I I1I Background intensity Modulation intensity Intensity becomes: I I B I M cos( ) (6)