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11: Waves and Imaging Chapter 11 Waves and Imaging We now return to those thrilling days of waves to consider their effects on the per- formance of imaging systems. We first consider “interference” of two traveling waves that oscillate with the same frequency and then generalize that to the interference of many such waves, which is called “diffraction”. 11.1 Interference of Waves References: Hecht, Optics §8 Recall the identity that was derived for the sum of two oscillations with different frequencies ω1 and ω2 : y1 [t]=A cos [ω1t] y2 [t]=A cos [ω2t] ω1 + ω2 ω1 ω2 y1 [t]+y2 [t]=2A cos t cos − t 2 · 2 ∙µ ¶ ¸ ∙µ ¶ ¸ 2A cos [ωavgt] cos [ωmodt] ≡ · In words, the sum of two oscillations of different frequency is identical to the product of two oscillations: one is the slower varying modulation (at frequency ωmod)andthe other is the more rapidly oscillating average sinusoid (or carrier wave) with frequency ωavg . A perhaps familiar example of the modulation results from the excitation of two piano strings that are mistuned. A low-frequency oscillation (the beat) is heard; as one string is tuned to the other, the frequency of the beat decreases, reaching zero when the string frequencies are equal. Acoustic beats may be thought of as interference of the summed oscillations in time. We also could consider this relationship in a broader sense. If the sinusoids are considered to be functions of the independent variable (coordinate) t, the phase angles of the two component functions Φ1(t)=ω1t and Φ2(t)=ω2t are differentatthesame coordinate t. The components sometimes add (for t such that Φ1 [t] = Φ2 [t] 2nπ) ∼ ± and sometimes subtract (if Φ1(t) = Φ2(t) (2n +1)π). ∼ ± 219 220 CHAPTER 11 WAVES AND IMAGING We also derived the analogous effect for two waves traveling along the z-axis: f1 [z,t]=A cos [k1z ω1t] − f2 [z,t]=A cos [k2z ω2t] − f1 [z,t]+f2 [z,t]= 2A cos[kmodz ωmodt] cos[kavgz ωavgt] { − }· − k1 k2 kmod = − 2 ω1 ω2 ωmod = − 2 ωmod ω1 ω2 vmod = = − kmod k1 k2 k + k − k = 1 2 avg 2 ω + ω ω = 1 2 avg 2 ωavg ω1 + ω2 vavg = = kavg k1 + k2 In words, the superposition of two traveling waves with different temporal frequencies (and thus different wavelengths) generates the product of two component traveling waves, one oscillating more slowly in both time and space,i.e. a traveling modulation. Note that both the average and modulation waves move along the z-axis. In this case,k1,k2,ω1,and ω2are all positive, and so kavg and ωavg must be also. However, the modulation wavenumber and frequency may be negative. In fact, the algebraic sign of kmod maybenegativeevenifωmod is positive. In this case, the modulation wave moves in the opposite direction to the average wave. Note that if the two 1-D waves traveling in the same direction along the z-axis havethesamefrequencyω, they must have the same wavelength λ and the same 2π wavenumber k = λ . The modulation terms kmod and ωmod must be zero, and the summation wave exhibits no modulation. Recall also such waves traveling in opposite directions generate a waveform that moves but does not travel, but is a standing wave: f1 [z,t]=A cos [k1z ω1t] − f2 [z,t]=A cos [k1z + ω1t] f1 [z,t]+f2 [z,t]= 2A cos [kmodz ωmodt] cos [kavgz ωavgt] { − } − 11.1 INTERFERENCE OF WAVES 221 k1 k1 kmod = − =0 2 ω1 ( ω1) ωmod = − − = ω1 2 k + k k = 1 1 = k avg 2 1 ω1 +( ω1) ωavg = − =0 2 f1 [z,t]+f2 [z,t]=2A cos [k1z]cos[ ω1t] − =2A cos [k1z]cos[ω1t] where the symmetry of cos[θ] was used in the last step. Traveling waves also may be defined over two or three spatial dimensions; the waves have the form f[x, y, t] and f[x, y, z, t], respectively. The direction of propaga- tion of such a wave in a multidimensional space is determined by a vector analogous to k; a 3-D wavevector k has components [kx,ky,kz]. The vector may be written: k = kxxˆ + kyyˆ + kzˆz The corresponding wave travels in£ the direction of the¤ wavevector k and has wave- 2π length λ = k . In other words, the length of k is the magnitude of the wavevector: | | 2π k = k2 + k2 + k2 = . | | x y z λ q The temporal oscillation frequency ω is determined from the magnitude of the wavevec- tor through the dispersion relation: vφ ω = vφ k ν = ·| | → λ For illustration, consider a simple 2-D analogue of the 1-D traveling plane wave. Thewavetravelsinthedirectionofthe2-Dwavevectork which is in the x z plane: − k =[kx, 0,kz] The points of constant phase with with phase angle φ = C radiansisthesetofpoints in the 2-D space r =[x =0,y,z]=(r, θ) such that the scalar product k r = C: · 222 CHAPTER 11 WAVES AND IMAGING k r = r k • • = k r cos [θ] | || | = kxx + kzz = C for a point of constant phase Therefore, the equation of a 2-D wave traveling in the direction of k with linear wavefronts is: f [x, y, t]=A cos [kxx + kzz ωt] − = A cos [k r ωt] • − In three dimensions, the set of points with the same phase lie on a planar surface so that the equation of the traveling wave is: f [x, y, z, t]=f [r,t] = A cos [kxx + kyy + kzz ωt] − = A cos [k r ωt] • − Plane wave traveling in direction k Thisplanewavecouldhavebeencreatedbyapointsourceatalargedistancetothe left and below the z-axis. Now, we will apply the equation derived when adding oscillations with different 11.1 INTERFERENCE OF WAVES 223 temporal frequencies. In general, the form of the sum of two traveling waves is: f1 [x, y, z, t]+f2 [x, y, z, t]=A cos [k1 r ωt]+A cos [k2 r ωt] • − • − =2A cos k r ωavgt cos [k r ωmodt] avg · − · mod• − where the average and modulation wavevectors£ are: ¤ k + k (k ) +(k ) (k ) +(k ) (k ) +(k ) k = 1 2 = x 1 x 2 xˆ + y 1 y 2 yˆ + z 1 z 2 ˆz avg 2 2 2 2 k k (kx)1 (kx)2 (ky)1 (ky)2 (kz)1 (kz)2 k = 1 − 2 = − xˆ + − yˆ + − ˆz mod 2 2 2 2 and the average and modulation angular temporal frequencies are: ω + ω ω = 1 2 avg 2 ω1 ω2 ωmod = − 2 Note that the average and modulation wavevectors kavg and kmod point in different directions, in general, and thus the corresponding waves move in different directions at velocities determined from: ωavg vavg = kavg |ω | v = mod mod k | mod| Because the phase of the multidimensional traveling wave is a function of two parameters (the wavevector k and the angular temporal frequency ω), the phases of two traveling waves usually differ even if the temporal frequencies are equal. Consider the superposition of two such waves: ω1 = ω2 ω ≡ Thecomponentwavestravelindifferent directions so the components of the wavevec- tors differ: k =[(kx)1, (ky)1, (kz)1] = k =[(kx)2, (ky)2, (kz)2] 1 6 2 Since the temporal frequencies are equal, so must be the wavelengths: λ1 = λ2 = λ k = k k . → | 1| | 2| ≡ | | The condition of equal ω ensures that the temporal average and modulation frequen- 224 CHAPTER 11 WAVES AND IMAGING cies are: ω + ω ω = 1 2 = ω avg 2 0 ω1 ω2 ωmod = − =0 2 The summation of the two traveling waves with identical magnitudes may be expressed as: f1[x, y, z, t]+f21[x, y, z, t]=A cos(k1 r ω0t)+A cos(k r ω0t) • − 2• − =2A cos(k r ωavgt) cos(k r 0 t) avg • − · mod • − · =2A cos(k r ωavgt) cos(k r) avg • − · mod • Therefore, the superposition of two 2-D wavefronts with the same temporal frequency but traveling in different directions results in two multiplicative components: a trav- eling wave in the direction of kavg, and a wave in space along the direction of kmod that does not move. This second stationary wave is analogous to the phenomenon of beats, and is called interference in optics. 11.1.1 Superposition of Two Plane Waves of the Same Fre- quency Consider the superposition of two plane waves: f1 [x, y, z, t]=A cos [k r ω0t] 1 • − f2 [x, y, z, t]=A cos [k r ω0t] 2 • − k1 =[kx,ky =0,kz] k =[ kx, 0,kz] 2 − i.e., the wavevectors differ only in the x-component, and there only by a sign. There- fore the two wavevectors have the same “length”: 2π k = k = | 1| | 2| λ = λ1 = λ2 λ. ⇒ ≡ Also note that: 2π kz = k cos [θ]= cos [θ] | | λ 2π k = sin [θ] x λ 11.1 INTERFERENCE OF WAVES 225 k + k [k , 0,k ]+[k , 0,k ] 2π k = 1 2 = x z x z =[0, 0,k ]=ˆz cos [θ] avg 2 2 z λ k1 k2 [kx, 0,kz]+[ kx, 0,kz] 2π k = − = − =[kx, 0, 0] = xˆ sin [θ] avg 2 2 λ ω + ω ω = 1 2 = ω avg 2 0 ω1 ω2 ωmod = − =0 2 The wavevectors of two interfering plane waves with the same wavelength. These two waves could have been generated at point sources located above and be- low the z-axis a large distance to the left. This is the classic “Young’s double slit” experiment, where light from a single source is split into to waves (spherical waves in this case) and propagate a large distance to the observation plane: 226 CHAPTER 11 WAVES AND IMAGING How two “tilted” plane waves are generated in the Young double-aperture experiment.
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