- Figure 1.1 shows a snapshot of a harmonic wave that propagates in the z-direction. - The argument of the cosine function is termed the phase and δ the phase constant. - ν = the frequency (the number of waves per unit time) k = 2π/λ the wave number. - The relation between the frequency and the wavelength is given by. - Figure 1.1 Harmonic wave. - Table 1.1 The electromagnetic spectrum (From Young (1968)). - The ratio of the speed c of an electromagnetic wave in vacuum to the speed v in a medium is known as the absolute index of refraction n of that medium. - THE PLANE WAVE. - 1.3 THE PLANE WAVE. - Electromagnetic waves are not two dimensional as in Figure 1.1, but rather three-dimen- sional waves. - The simplest example of such waves is given in Figure 1.2 where a plane wave that propagates in the direction of the k-vector is sketched. - Points of equal phase lie on parallel planes that are perpendicular to the propagation direction. - In the figure, only some of the infinite number of phase planes are drawn. - Equation (1.1) describes a plane wave that propagates in the z-direction. - planes that are normal to the z-direction.) In the general case where a plane wave propagates in the direction of a unit vector n, the expression describing the field at an arbitrary point with radius vector r = (x, y, z) is given by. - U cos[kn · r − 2π νt + δ] (1.4) That the scalar product fulfilling the condition n · r = constant describes a plane which is perpendicular to n is shown in the two-dimensional case in Figure 1.3. - That this is correct also in the three-dimensional case is easily proved.. - k Figure 1.2 The plane wave. - They are directed lines that are everywhere perpendicular to the phase planes. - This is illustrated in Figure 1.4 where the cross-section of a rather complicated wavefront is sketched and where some of the light rays perpen- dicular to the wavefront are drawn.. - Let us for a moment turn back to the plane wave described by Equation (1.1). - φ = φ 1 − φ 2 = k(z 1 − z 2 ) (1.5) Hence, we see that the phase difference between two points along the propagation direction of a plane wave is equal to the geometrical path-length difference multiplied by the wave number. - When the light passes a medium different from air (vacuum), we have to multiply by the refractive index n of the medium, such that. - OBLIQUE INCIDENCE OF A PLANE WAVE 5. - In the description of wave phenomena, the notation of Equation (1.6) is commonly adopted and ‘Re’ is omitted because it is silently understood that the field is described by the real part.. - One advantage of such complex representation of the field is that the spatial and temporal parts factorize:. - U e iφ e −i2π vt (1.7) In optical metrology (and in other branches of optics) one is most often interested in the spatial distribution of the field. - Figure 1.5(a, b) shows examples of a cylindrical wave and a spherical wave, while in Figure 1.5(c) a more complicated wavefront resulting from reflection from a rough surface is sketched. - Note that far away from the point source in Figure 1.5(b), the spherical wave is nearly a plane wave over a small area. - A point source at infinity, represents a plane wave.. - 1.6 OBLIQUE INCIDENCE OF A PLANE WAVE. - In optics, one is often interested in the amplitude and phase distribution of a wave over fixed planes in space. - Let us consider the simple case sketched in Figure 1.6 where a plane wave falls obliquely on to a plane parallel to the xy-plane a distance z from it. - The wave propagates along the unit vector n which is lying in the xz-plane (defined as the plane of incidence) and makes an angle θ to the z-axis. - The components of the n- and r-vectors are therefore. - Figure 1.5 ((a) and (b) from Hecht &. - These expressions put into Equation (1.6) (Re and temporal part omitted) give. - A spherical wave, illustrated in Figure 1.5(b), is a wave emitted by a point source. - It should be easily realized that the complex amplitude representing a spherical wave must be of the form. - r e ikr (1.10). - where r is the radial distance from the point source. - We see that the phase of this wave is constant for r = constant, i.e. - the phase fronts are spheres centred at the point source. - The r in the denominator of Equation (1.10) expresses the fact that the amplitude decreases as the inverse of the distance from the point source.. - Consider Figure 1.7 where a point source is lying in the x 0 , y 0 -plane at a point of coordinates x 0 , y 0 . - The field amplitude in a plane parallel to the x 0 y 0 -plane at a distance z then will be given by Equation (1.10) with. - r = z 2 + (x − x 0 ) 2 + (y − y where x, y are the coordinates of the illuminated plane. - One therefore usually makes some approximations, the first of which is to replace z for r in the denominator of Equation (1.10). - A convenient means for approximation of the phase is offered by a binomial expansion of the square root, viz.. - where r is approximated by the two first terms of the expansion.. - The complex field amplitude in the xy-plane resulting from a point source at x 0 , y 0 in the x 0 y 0 -plane is therefore given by. - With regard to the registration of light, we are faced with the fact that media for direct recording of the field amplitude do not exist. - effect per unit area) which is proportional to the field amplitude absolutely squared:. - We mention that the correct relation between U 2 and the irradiance is given by I = εv. - where v is the wave velocity and ε is known as the electric permittivity of the medium.. - For completeness, we refer to the three laws of geometrical optics:. - On reflection from a mirror, the angle of reflection is equal to the angle of incidence (see Figure 1.8). - Figure 1.8 The law of reflection. - Figure 1.9 Scattering from a rough surface. - n 1 sin θ 1 = n 2 sin θ 2 (1.16) where θ 1 is the angle of incidence and θ 2 is the angle of emergence (see Figure 1.10).. - From Equation (1.16) we see that when n 1 >. - This occurs for an angle of incidence called the critical angle given by. - Finally, we also mention that for light reflected at the interface in Figure 1.10, when n 1 <. - n 2 , the phase is changed by π. - Figure 1.10 The law of refraction. - We shall here not go into the general theory of lenses, but just mention some of the more important properties of a simple, convex, ideal lens. - Figure 1.11 illustrates the imaging property of the lens. - Figure 1.12 shows three of them. - The distance b from the lens to the image plane is given by the lens formula. - In Figure 1.13(a), the case of a point source lying on the optical axis forming a spherical diverging wave that is converted to a converging wave and focuses onto a point on the optical axis is illustrated. - In Figure 1.13(b) the point source is lying on-axis at a distance. - Figure 1.11. - Figure 1.12. - A PLANE-WAVE SET-UP 11. - Figure 1.13. - from the lens equal to the focal length f . - We then get a plane wave that propagates along the optical axis. - In Figure 1.13(c) the point source is displaced along the focal plane a distance h from the optical axis. - We then get a plane wave propagating in a direction that makes an angle θ to the optical axis where. - 1.11 A PLANE-WAVE SET-UP. - Finally, we refer to Figure 1.14 which shows a commonly applied set-up to form a uniform, expanded plane wave from a laser beam. - The laser beam is a plane wave with a small cross-section, typically 1 mm. - A lens L 2 of greater diameter and longer focal length f 2 is placed as shown in the figure. - In the focal point of L 1 a small opening (a pinhole) of diameter typically 10 µ m is placed. - Figure 1.14 A plane wave set-up. - way via other optical elements (like mirrors, beamsplitters, etc.) and it causes the beam not to be a perfect plane wave.. - 1.1 How many ‘yellow’ light waves (λ = 550 nm) will fit into a distance in space equal to the thickness of a piece of paper (0.1 mm)? How far will the same number of microwaves (ν = 10 10 Hz, i.e 10 GHz, and v m/s) extend?. - 1.3 Consider the plane electromagnetic wave (in SI units) given by the expressions U x = 0, U y = exp i[2π × 10 14 (t − x/c. - What is the frequency, wavelength, direction of propagation, amplitude and phase constant of the wave?. - Find (a) the frequency of the light,. - (c) the index of refraction of the glass.. - The second term on the right may be interpreted as the field arising from the oscil- lating molecules in the glass plate.. - PROBLEMS 13 1.6 Show that the optical path, defined as the sum of the products of the various indices times the thicknesses of media traversed by a beam, that is, i n i x i , is equivalent to the length of the path in vacuum which would take the same time for that beam to travel.. - 1.7 Write down an equation describing a sinusoidal plane wave in three dimensions with wavelength λ, velocity v, propagating in the following directions:. - (c) Perpendicular to the planes x + y + z = const.. - 1.8 Show that the rays from a point source S that are reflected by a plane mirror appear to be coming from the image point S . - It is given by δ = δ 1 + δ 2 . - 1.11 (a) Starting with Snell’s law prove that the vector refraction equation has the form n 2 k 2 − n 1 k 1 = (n 2 cos θ 2 − n 1 cos θ 1 )u n. - where k 1 , k 2 are unit propagation vectors and u n is the surface normal pointing from the incident to the transmitting medium.. - (b) In the same way, derive a vector expression equivalent to the law of reflection.
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