Waves and Imaging - RIT CIS
Chapter 10
Waves and Imaging We now return to those thrilling days of waves to consider their effects on the performance of imaging systems. We Þrst 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”.
10.1
Interference of Waves
References: Hecht, Optics §8 Recall the identity that was derived for the addition of two oscillations of different frequencies ω 1 and ω 2 :
y1 y2 y1 + y2
= A cos [ω1 t] = A cos [ω2 t] ·µ ¶ ¸ ·µ ¶ ¸ ω1 + ω2 ω1 − ω 2 = 2A cos t cos t 2 2
i.e., the sum of two oscillations of different frequency is identical to the product of two oscillations: one is the slower varying modulation (at frequency ω mod ) and the other is the more rapidly oscillating average sinusoid (or carrier wave) with frequency ω avg . 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) = ω 1 t and Φ2 (t) = ω 2 t are different at the same 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). We also derived the analogous effect for two waves traveling along the z -axis: f1 [z, t] = A cos [k1 z − ω 1 t] f2 [z, t] = A cos [k2 z − ω 2 t] f1 + f2 = {2A cos[kmod z − ω mod t]} cos[kavg z − ωavg t] 109
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CHAPTER 10. WAVES AND IMAGING
kmod
=
ωmod
=
vmod
=
kavg
=
ω avg
=
vavg
=
k1 − k2 2 ω 1 − ω2 2 ω mod ω1 − ω2 = kmod k1 − k2 k1 + k2 2 ω 1 + ω2 2 ω avg ω1 + ω 2 = 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 , ω 1 ,and ω 2 are all positive, and so kavg and ω avg must be also. However, the modulation wavenumber and frequency may be negative. In fact, the algebraic sign of kmod may be negative even if ω mod 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 have the same frequency ω, they must have the same wavelength λ and the same wavenumber k = 2π λ . The modulation terms kmod and ω mod 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 [k1 z − ω 1 t] f2 (z, t) = A cos [k1 z + ω 1 t] f1 + f2 = {2A cos [kmod z − ω mod t]} cos [kavg z − ωavg t]
kmod
=
ωmod
=
kavg
=
ω avg
=
k1 − k1 =0 2 ω 1 − (−ω1 ) =ω 2 1 k1 + k1 =k 2 1 ω 1 + (−ω1 ) =0 2
f1 [z, t] + f2 [z, t] = 2A cos [k1 z] cos [−ω1 t] = 2A cos [k1 z] cos [ω 1 t] where the symmetry of cos[θ] was used in the last step. Traveling waves also may be deÞned 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 propagation 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 = [kx x ˆ + ky y ˆ + kz ˆ z]
10.1. INTERFERENCE OF WAVES
111
The corresponding wave travels in the direction of the wavevector k and has wavelength λ = In other words, the length of k is the magnitude of the wavevector: |k| =
2π |k|
.
q 2π . kx2 + ky2 + kz2 = λ
The temporal oscillation frequency ω is determined from the magnitude of the wavevector 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. The wave travels in the direction of the 2-D wavevector k which is in the y − z plane: k = [0, ky , kz ] The points of constant phase with with phase angle φ = C radians is the set of points in the 2-D space r = [x = 0, y, z] = (r, θ) such that the scalar product k · r = C: k·r = r·k = |k||r| cos [θ] = ky y + kz z = C Therefore, the equation of a 2-D wave traveling in the direction of k with linear wavefronts is: f [x, y, t] = A cos [ky y + kz z − 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 [kx x + ky y + kz z − ωt] = A cos [k · r − ωt]
Plane wave traveling in direction k
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CHAPTER 10. WAVES AND IMAGING
Now, we will apply the equation derived when adding oscillations with different 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 [kavg · r − ω avg t] · cos [kmod •r − ωmod t] where the average and modulation wavevectors are: kavg
=
kmod
=
(kx )1 + (kx )2 (ky )1 + (ky )2 (kz )1 + (kz )2 k 1 + k2 = + + 2 2 2 2 (kx )1 − (kx )2 (ky )1 − (ky )2 (kz )1 − (kz )2 k 1 − k2 = + + 2 2 2 2
and the average and modulation angular temporal frequencies are: ω avg
=
ωmod
=
ω 1 + ω2 2 ω 1 − ω2 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: ω avg vavg = |kavg | ω mod vmod = |kmod | 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: ω 1 = ω2 ≡ ω The component waves travel in different directions so the components of the wavevectors differ: k1 = [(kx )1 , (ky )1 , (kz )1 ] 6= k2 = [(kx )2 , (ky )2 , (kz )2 ] Since the temporal frequencies are equal, so must be the wavelengths: λ1 = λ2 = λ → |k1 | = |k2 | ≡ |k|. The condition of equal ω ensures that the temporal average and modulation frequencies are: ω avg
=
ω mod
=
ω1 + ω 2 =ω 2 ω1 − ω 2 =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 − ωt) + A cos(k2 • r − ωt) = 2A cos(kavg • r − ω avg t) · cos(kmod • r − 0t) = 2A cos(kavg • r − ω avg t) · cos(kmod • r) Therefore, the superposition of two 2-D wavefronts with the same temporal frequency but traveling in different directions results in two multiplicative components: a traveling 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.
10.1. INTERFERENCE OF WAVES
10.1.1
113
Superposition of Two Plane Waves of the Same Frequency
Consider the superposition of two plane waves: f1 [x, y, z, t] f2 [x, y, z, t] k1 k2
= = = =
A cos [k1 · r − ωt] A cos [k2 · r − ωt] [0, ky , kz ] [0, −ky , kz ]
i.e., the wavevectors differ only in the y-component, and there only by a sign. Therefore the two wavevectors have the same “length”: |k1 |
2π λ =⇒ λ1 = λ2 ≡ λ. =
|k2 | =
Also note that:
kavg
=
kavg
=
ω avg
=
ωmod
=
k 1 + k2 2 k 1 − k2 2 ω 1 + ω2 2 ω 1 − ω2 2
kz
= |k| cos [θ] =
ky
=
2π cos [θ] λ
2π sin [θ] λ
[0, ky , kz ] + [0, −ky , kz ] 2π = [0, 0, kz ] = ˆ z cos [θ] 2 λ [0, ky , kz ] − [0, −ky , kz ] 2π = = [0, ky , 0] = y sin [θ] ˆ 2 λ =
=ω =0
The wavevectors of two interfering plane waves with the same wavelength. The superposition of the two electric Þelds is:
f[x, y, z, t] = f1 [x, y, z, t] + f2 [x, y, z, t] = 2A cos [kavg • r − ω avg t] · cos [kmod • r] h z i h y i = 2A cos 2π cos [θ] − ωt cos 2π sin [θ] λ λ
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CHAPTER 10. WAVES AND IMAGING
The second term (with no dependence on time) is a spatial wave along y, while the second term is a traveling wave in the direction deÞned by k = [0, 0, kz ]. The amplitude variation in the y direction is: i h y y ´ 2A cos 2π sin [θ] = 2A cos 2π ³ λ λ sin[θ]
which has a period of
λ sin[θ] .
The intensity of the superposition is:
· ¸ h z i 2πy sin [θ] |f [x, y, z, t] |2 = 4A2 cos2 2π cos [θ] − ωt cos2 λ λ The cosine terms can be rewritten using: 1 cos2 [θ] = (1 + cos [2θ]) 2 As before, the Þrst term varies rapidly due to the angular frequency term ω ' 1014 Hz. Therefore, just the average value is detected: · ¸ ® 2πy sin [θ] 1 |f [x, y, z, t] |2 = 4A2 cos2 · λ 2 · µ · ¸¶¸ 1 4πy sin [θ] = 2A2 1 + cos 2 λ y ´ = A2 1 + cos 2π ³ λ sin[θ]
Note that this derivation may be applied to Þnd the irradiance of one of the individual component waves: ® I1 = |f1 [x, y, z, t] |2 ® I2 = |f2 [x, y, z, t] |2 ® ® ® I0 = |f1 [x, y, z, t] |2 = |A cos [k1 · r − ωt] |2 = A2 cos2 [k1 · r − ωt] 1 = A2 · 2
So the irradiance of the sum of the two waves can be rewritten in terms of the irradiance of a single wave: · ¸ ® 2πy sin [θ] = 4I0 cos2 |f [x, y, z, t] |2 λ µ ¶ cos [2πy · 2 · sin [θ]] = 2I0 1 + λ y ´ = 2I0 1 + cos 2π ³ λ 2 sin[θ]
λ The irradiance exhibits a sinusoidal modulation of period Y = 2 sin[θ] and its irradiance oscillates between 0 and 2I0 · (2) = 4I0 , so that the average irradiance is 2I0 . The period varies directly with λ and inversely with sin(θ); for small θ, the period of the sinusoid is large, for θ = 0, there is no modulation of the irradiance. The alternating bright and dark regions of this time-stationary sinusoidal intensity pattern often are called interference fringes. The shape, separation, and orientation of the interference fringes are determined by the incident wavefronts, and thus provide information about them. The argument of the cosine function is the optical phase difference of the two waves.
10.1. INTERFERENCE OF WAVES
115
At locations where the optical phase difference is an even multiple of π, the cosine evaluates to unity and a maximum of the interference pattern results. This is an example of constructive interference. If the optical phase difference is an odd multiple of π, the cosine evaluates to -1 and the irradiance is zero; this is destructive interference. Again, the traveling wave in the images of the amplitude and intensity of the superposed images moves in the z-direction (to the right), thus blurring out the oscillations in the z-direction. The oscillations in the y-direction are preserved as the interference pattern, which is plotted as a function of y below. Note that the spatial frequency of the intensity fringes is twice as large as that of the amplitude fringes.
10.1.2
SUPERPOSITION of TWO PLANE WAVES with DIFFERENT FREQUENCIES
For further illustration, consider the case the two waves travel in the same directions, so that k1 6= k2 , but with different temporal frequencies ω1 6= ω 2 . This means that |k1 | 6= |k2 |. The average and modulation wavevectors are found as before, but the modulation wave now travels because both kmod 6= 0 and ω mod 6= 0. Consider the example of two component waves: f1 [r,t] directed at an angle θ1 = +40◦ ' 23 radian with λ1 = 8 units and ω 1 = 18 radians/second, and f2 [r, t] directed 1 at θ2 = −40◦ ' − 23 radian with λ2 = 12 units and ω 2 = 12 radians/second. The corresponding average and modulation frequencies are: ωavg
=
ω mod
=
µ ¶ 2π 1 1 ω1 + ω2 = + = 2 2 8 12 µ ¶ 2π 1 1 ω1 − ω2 = − = 2 2 8 12
5π 2π = radians/second 9.6 24 2π radians/second 48
and the average and modulation wavevectors are: kavg
= =
kmod
= '
µ ¶ y k1 + k2 5z ◦ ◦ = 2π · sin [40 ] + cos [40 ] 2 48 48 ³ y 2π z ´ · (y sin [40◦ ] + 5z cos [40◦ ]) ' 2π + 48 74.674 12.532 2π k1 − k2 = · (5y sin [40◦ ] + z cos [40◦ ]) 2 48 ³ y z ´ + 2π 14.935 62.0
The superposition is: f1 [r, t] + f2 [r, t] = 2favg [r, t] · fmod [r, t] where the full expressions for the average and modulation waves are: favg [r, t] = cos [kavg · r − ω avg t] · ¸ 2π 10πt ◦ ◦ = cos · (y sin [40 ] + 5z cos [40 ]) − 48 48 ¸ · 2π ◦ ◦ (y sin [40 ] + 5z cos [40 ]) − 5t) = cos 48 µ ¶¸ · ³ z ´ t y + − 2π ' cos 2π 74.674 12.532 9.6
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CHAPTER 10. WAVES AND IMAGING fmod (r, t) = cos(kmod · r − ω mod t) · ¸ 2π 2πt ◦ ◦ = cos · (5y sin [40 ] + z cos [40 ]) − 48 48 ¸ · 2π (5y sin [40◦ ] + z cos [40◦ ] − 2t) = cos 48 µ ¶¸ · ³ z ´ t y + − 2π ' cos 2π 74.674 12.532 24
Note that both the average and modulation waves are traveling waves; they are headed in different directions with different frequencies and different velocities. The temporal frequencies are ν avg = 5 2 48 Hz and ν mod = 48 Hz. If the intensity (squared-magnitude) of the sum is averaged over time at an observation plane located downstream on the z-axis, both traveling waves will average out and no stationary fringe pattern will be visible.
k1 Sum
Power
k2 Sum of two sinusoidal traveling waves with period λ1 = 8 units and λ2 = 12 units directed at ±40◦ , respectively. The same principles just discussed may be used to determine the form of interference fringes from wavefronts with other shapes. Some examples will be considered in the following sections.
10.1.3
FRINGE VISIBILITY — COHERENCE
The visibility of a sinusoidal fringe pattern is a quality that corresponds quite closely to modulation, which is a term used by electrical engineers (sparkies). Given a nonnegative sinusoidal irradiance (intensity) distribution with maximum Imax and minimum Imin (sothatImin ≥ 0), the visibility of the sinusoidal fringe pattern is: Imax − Imin V≡ Imax + Imin Note that if Imin = 0, then V = 1 regardless of the value of Imax . The visibility of the fringe pattern is largely determined by the relative irradiances of the individual wavefronts and by the coherence of the light source. To introduce the concept of coherence, consider Þrst the Young’s two-aperture experiment where the source is composed of equal-amplitude emission at two distinct wavelengths λ1 and λ2 incident on the observation screen at ±θ. Possible pairs of wavelengths could be those of the sodium doublet (λ = 589.0nm and 589.6nm), or the pair of lines emitted by a “greenie” He:Ne laser (λ = 543nm
10.1. INTERFERENCE OF WAVES
117
(green), 594nm—(yellow)). In air or vacuum, the corresponding angular frequencies obviously are 2πc ω 1 = 2πc λ1 and ω 2 = λ2 . To Þnd the irradiance pattern created by the interference of the four beams, we must compute the superposition of the amplitude of the electromagnetic Þeld, Þnd its squared-magnitude, and then determine the average over time. The sum of the four component terms is straightforward to compute by recognizing that it is the sum of the amplitude patterns from the pairs of waves with the same wavelength. We have already shown that the sum of the two terms with λ = λ1 is: · ¸ · ¸ 2πy 2πz f1 (y, z; λ1 ) + f2 (y, z; λ1 ) = 2A cos sin [θ] cos cos [θ] − ω1 t λ1 λ1 " # · ¸ y 2πz = 2A cos 2π ¡ Lλ1 ¢ cos cos [θ] − ω1 t λ1 d
which is the sum of a stationary sinusoid and a traveling wave in the +z-direction. A snapshot of this amplitude at a Þxed time is shown: The second pair of wavefronts with λ = λ2 yield a similar result, though the period of the stationary fringes and the temporal frequency of the traveling wave differ. The expression for the sum of the two pairs is: 4 X
fn (y, z; λ) = f1 (y, z; λ1 ) + f2 (y, z; λ1 ) + f1 (y, z; λ2 ) + f2 (y, z; λ2 )
n=1
# " · ¸ ¸ 2πz 2πz y cos [θ] − ω 1 t + 2A cos 2π ¡ Lλ2 ¢ cos cos [θ] − ω 2 t = 2A cos 2π ¡ Lλ1 ¢ cos λ1 λ2 d d "
y
#
·
Note that both waves traveling waves move along the z-axis and that their wavevectors and velocities are: k1
=
k2
=
v1
=
v2
=
2πz λ1 cos [θ] 2πz λ2 cos [θ] c ω1 2πν 1 ν 1 λ1 cos [θ] = cos [θ] = ¡ 2π ¢ z = k1 z z λ cos[θ]
c ω2 2πν 2 ν 2 λ2 cos [θ] = cos [θ] = v1 = 2πz = k2 z z λ2 ·cos[θ]
Since the two traveling waves have the same phase velocity, they may be factored out to yield the sum of two different-frequency cosines multiplied by a sinusoidal traveling wave: ¸ ¸ · ¸¶ · 2πyd 2πyd 2πz fn [y, z; λ] = 2A cos cos [θ] − ω 1 t + cos cos Lλ1 Lλ2 λ1 n=1 4 X
µ
·
In identical fashion, the sum of the two stationary cosine waves may be recast as the product of cosines with the average and modulation frequencies: ·
¸ · ¸ · ¸ · ¸ 2πyd 2πyd 2πyd(λ2 + λ1 ) 2πyd(λ1 − λ2 ) cos + cos = 4A cos · cos − Lλ1 Lλ2 2Lλ1 λ2 2Lλ1 λ2 · ¸ · ¸ 2πyd(λ1 + λ2 ) 2πyd(λ1 − λ2 ) = 4A cos · cos 2Lλ1 λ2 2Lλ1 λ2 · ¸ · ¸ 2πyd 2πyd · λavg · cos · λmod = 4A cos Lλ1 λ2 Lλ1 λ2
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CHAPTER 10. WAVES AND IMAGING
The time-average of the squared magnitude of the amplitude averages the traveling wave portion of the intensity to 12 . The Þnal expression for the irradiance is the product of two sinusoidal irradiance patterns with identical maxima and zero minima: ¯ · ¸¯2 + ¸ · ¸¯2 *¯ · ¯ ¯ ¯ ¯ 2πyd 2πz 2πyd ¯ ¯ ¯ I = ¯4A cos · λavg · cos · λmod ¯ ¯cos cos [θ] − ω1 t ¯¯ Lλ1 λ2 Lλ1 λ2 λ1 · ¸ · ¸ 2πyd 2πyd 1 = 16A2 cos2 · λavg · cos2 · λmod · Lλ1 λ2 Lλ1 λ2 2 ¸ ¸¶¸ · · µ · 1 4πyd 4πyd 21 = 8A · λmod λavg · 1 + cos 1 + cos 2 Lλ1 λ2 2 Lλ1 λ2 µ · µ ¶¸¶ µ · µ ¶¸¶ 2d 2d 2 · λavg · λmod = 2A · 1 + cos 2πy · 1 + cos 2πy Lλ1 λ2 Lλ1 λ2 µ · ¸¶ µ · ¸¶ 2πy 2πy = 2A2 · 1 + cos · 1 + cos Davg Dmod after deÞning the respective periods of the two oscillations to be:
Davg Dmod
≡ ≡
Lλ1 λ2 L = 2d · λavg d
µ
1 1 + λ1 λ2
¶−1
∝ |λ1 + λ2 |−1 ∝ (λavg )−1
¯ ¯−1 Lλ1 λ2 1 ¯¯ L ¯¯ 1 = ¯ − ¯ ∝ |λ1 − λ2 |−1 ≡ (∆λ)−1 2d · λmod d λ1 λ2
−1 Note that the lengths of the spatial periods of the oscillations are proportional to λ−1 avg ∝ (λ1 + λ2 ) −1 −1 and λmod ∝ |λ1 − λ2 | = (∆λ) . In the case where the two emitted wavelengths are close together such that λ1 ' λ2 ' λavg >> λmod , the expressions for the periods of the two component oscillations may be simpliÞed:
Davg
'
Dmod
'
Lλavg 2d L (λavg )2 2d · ∆λ
After cancelling the common terms, the relative lengths of the spatial periods of the modulations are: ∆λ Davg = Dmod ·
Waves and Imaging We now return to those thrilling days of waves to consider their effects on the performance of imaging systems. We Þrst 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”.
10.1
Interference of Waves
References: Hecht, Optics §8 Recall the identity that was derived for the addition of two oscillations of different frequencies ω 1 and ω 2 :
y1 y2 y1 + y2
= A cos [ω1 t] = A cos [ω2 t] ·µ ¶ ¸ ·µ ¶ ¸ ω1 + ω2 ω1 − ω 2 = 2A cos t cos t 2 2
i.e., the sum of two oscillations of different frequency is identical to the product of two oscillations: one is the slower varying modulation (at frequency ω mod ) and the other is the more rapidly oscillating average sinusoid (or carrier wave) with frequency ω avg . 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) = ω 1 t and Φ2 (t) = ω 2 t are different at the same 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). We also derived the analogous effect for two waves traveling along the z -axis: f1 [z, t] = A cos [k1 z − ω 1 t] f2 [z, t] = A cos [k2 z − ω 2 t] f1 + f2 = {2A cos[kmod z − ω mod t]} cos[kavg z − ωavg t] 109
110
CHAPTER 10. WAVES AND IMAGING
kmod
=
ωmod
=
vmod
=
kavg
=
ω avg
=
vavg
=
k1 − k2 2 ω 1 − ω2 2 ω mod ω1 − ω2 = kmod k1 − k2 k1 + k2 2 ω 1 + ω2 2 ω avg ω1 + ω 2 = 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 , ω 1 ,and ω 2 are all positive, and so kavg and ω avg must be also. However, the modulation wavenumber and frequency may be negative. In fact, the algebraic sign of kmod may be negative even if ω mod 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 have the same frequency ω, they must have the same wavelength λ and the same wavenumber k = 2π λ . The modulation terms kmod and ω mod 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 [k1 z − ω 1 t] f2 (z, t) = A cos [k1 z + ω 1 t] f1 + f2 = {2A cos [kmod z − ω mod t]} cos [kavg z − ωavg t]
kmod
=
ωmod
=
kavg
=
ω avg
=
k1 − k1 =0 2 ω 1 − (−ω1 ) =ω 2 1 k1 + k1 =k 2 1 ω 1 + (−ω1 ) =0 2
f1 [z, t] + f2 [z, t] = 2A cos [k1 z] cos [−ω1 t] = 2A cos [k1 z] cos [ω 1 t] where the symmetry of cos[θ] was used in the last step. Traveling waves also may be deÞned 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 propagation 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 = [kx x ˆ + ky y ˆ + kz ˆ z]
10.1. INTERFERENCE OF WAVES
111
The corresponding wave travels in the direction of the wavevector k and has wavelength λ = In other words, the length of k is the magnitude of the wavevector: |k| =
2π |k|
.
q 2π . kx2 + ky2 + kz2 = λ
The temporal oscillation frequency ω is determined from the magnitude of the wavevector 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. The wave travels in the direction of the 2-D wavevector k which is in the y − z plane: k = [0, ky , kz ] The points of constant phase with with phase angle φ = C radians is the set of points in the 2-D space r = [x = 0, y, z] = (r, θ) such that the scalar product k · r = C: k·r = r·k = |k||r| cos [θ] = ky y + kz z = C Therefore, the equation of a 2-D wave traveling in the direction of k with linear wavefronts is: f [x, y, t] = A cos [ky y + kz z − 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 [kx x + ky y + kz z − ωt] = A cos [k · r − ωt]
Plane wave traveling in direction k
112
CHAPTER 10. WAVES AND IMAGING
Now, we will apply the equation derived when adding oscillations with different 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 [kavg · r − ω avg t] · cos [kmod •r − ωmod t] where the average and modulation wavevectors are: kavg
=
kmod
=
(kx )1 + (kx )2 (ky )1 + (ky )2 (kz )1 + (kz )2 k 1 + k2 = + + 2 2 2 2 (kx )1 − (kx )2 (ky )1 − (ky )2 (kz )1 − (kz )2 k 1 − k2 = + + 2 2 2 2
and the average and modulation angular temporal frequencies are: ω avg
=
ωmod
=
ω 1 + ω2 2 ω 1 − ω2 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: ω avg vavg = |kavg | ω mod vmod = |kmod | 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: ω 1 = ω2 ≡ ω The component waves travel in different directions so the components of the wavevectors differ: k1 = [(kx )1 , (ky )1 , (kz )1 ] 6= k2 = [(kx )2 , (ky )2 , (kz )2 ] Since the temporal frequencies are equal, so must be the wavelengths: λ1 = λ2 = λ → |k1 | = |k2 | ≡ |k|. The condition of equal ω ensures that the temporal average and modulation frequencies are: ω avg
=
ω mod
=
ω1 + ω 2 =ω 2 ω1 − ω 2 =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 − ωt) + A cos(k2 • r − ωt) = 2A cos(kavg • r − ω avg t) · cos(kmod • r − 0t) = 2A cos(kavg • r − ω avg t) · cos(kmod • r) Therefore, the superposition of two 2-D wavefronts with the same temporal frequency but traveling in different directions results in two multiplicative components: a traveling 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.
10.1. INTERFERENCE OF WAVES
10.1.1
113
Superposition of Two Plane Waves of the Same Frequency
Consider the superposition of two plane waves: f1 [x, y, z, t] f2 [x, y, z, t] k1 k2
= = = =
A cos [k1 · r − ωt] A cos [k2 · r − ωt] [0, ky , kz ] [0, −ky , kz ]
i.e., the wavevectors differ only in the y-component, and there only by a sign. Therefore the two wavevectors have the same “length”: |k1 |
2π λ =⇒ λ1 = λ2 ≡ λ. =
|k2 | =
Also note that:
kavg
=
kavg
=
ω avg
=
ωmod
=
k 1 + k2 2 k 1 − k2 2 ω 1 + ω2 2 ω 1 − ω2 2
kz
= |k| cos [θ] =
ky
=
2π cos [θ] λ
2π sin [θ] λ
[0, ky , kz ] + [0, −ky , kz ] 2π = [0, 0, kz ] = ˆ z cos [θ] 2 λ [0, ky , kz ] − [0, −ky , kz ] 2π = = [0, ky , 0] = y sin [θ] ˆ 2 λ =
=ω =0
The wavevectors of two interfering plane waves with the same wavelength. The superposition of the two electric Þelds is:
f[x, y, z, t] = f1 [x, y, z, t] + f2 [x, y, z, t] = 2A cos [kavg • r − ω avg t] · cos [kmod • r] h z i h y i = 2A cos 2π cos [θ] − ωt cos 2π sin [θ] λ λ
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CHAPTER 10. WAVES AND IMAGING
The second term (with no dependence on time) is a spatial wave along y, while the second term is a traveling wave in the direction deÞned by k = [0, 0, kz ]. The amplitude variation in the y direction is: i h y y ´ 2A cos 2π sin [θ] = 2A cos 2π ³ λ λ sin[θ]
which has a period of
λ sin[θ] .
The intensity of the superposition is:
· ¸ h z i 2πy sin [θ] |f [x, y, z, t] |2 = 4A2 cos2 2π cos [θ] − ωt cos2 λ λ The cosine terms can be rewritten using: 1 cos2 [θ] = (1 + cos [2θ]) 2 As before, the Þrst term varies rapidly due to the angular frequency term ω ' 1014 Hz. Therefore, just the average value is detected: · ¸ ® 2πy sin [θ] 1 |f [x, y, z, t] |2 = 4A2 cos2 · λ 2 · µ · ¸¶¸ 1 4πy sin [θ] = 2A2 1 + cos 2 λ y ´ = A2 1 + cos 2π ³ λ sin[θ]
Note that this derivation may be applied to Þnd the irradiance of one of the individual component waves: ® I1 = |f1 [x, y, z, t] |2 ® I2 = |f2 [x, y, z, t] |2 ® ® ® I0 = |f1 [x, y, z, t] |2 = |A cos [k1 · r − ωt] |2 = A2 cos2 [k1 · r − ωt] 1 = A2 · 2
So the irradiance of the sum of the two waves can be rewritten in terms of the irradiance of a single wave: · ¸ ® 2πy sin [θ] = 4I0 cos2 |f [x, y, z, t] |2 λ µ ¶ cos [2πy · 2 · sin [θ]] = 2I0 1 + λ y ´ = 2I0 1 + cos 2π ³ λ 2 sin[θ]
λ The irradiance exhibits a sinusoidal modulation of period Y = 2 sin[θ] and its irradiance oscillates between 0 and 2I0 · (2) = 4I0 , so that the average irradiance is 2I0 . The period varies directly with λ and inversely with sin(θ); for small θ, the period of the sinusoid is large, for θ = 0, there is no modulation of the irradiance. The alternating bright and dark regions of this time-stationary sinusoidal intensity pattern often are called interference fringes. The shape, separation, and orientation of the interference fringes are determined by the incident wavefronts, and thus provide information about them. The argument of the cosine function is the optical phase difference of the two waves.
10.1. INTERFERENCE OF WAVES
115
At locations where the optical phase difference is an even multiple of π, the cosine evaluates to unity and a maximum of the interference pattern results. This is an example of constructive interference. If the optical phase difference is an odd multiple of π, the cosine evaluates to -1 and the irradiance is zero; this is destructive interference. Again, the traveling wave in the images of the amplitude and intensity of the superposed images moves in the z-direction (to the right), thus blurring out the oscillations in the z-direction. The oscillations in the y-direction are preserved as the interference pattern, which is plotted as a function of y below. Note that the spatial frequency of the intensity fringes is twice as large as that of the amplitude fringes.
10.1.2
SUPERPOSITION of TWO PLANE WAVES with DIFFERENT FREQUENCIES
For further illustration, consider the case the two waves travel in the same directions, so that k1 6= k2 , but with different temporal frequencies ω1 6= ω 2 . This means that |k1 | 6= |k2 |. The average and modulation wavevectors are found as before, but the modulation wave now travels because both kmod 6= 0 and ω mod 6= 0. Consider the example of two component waves: f1 [r,t] directed at an angle θ1 = +40◦ ' 23 radian with λ1 = 8 units and ω 1 = 18 radians/second, and f2 [r, t] directed 1 at θ2 = −40◦ ' − 23 radian with λ2 = 12 units and ω 2 = 12 radians/second. The corresponding average and modulation frequencies are: ωavg
=
ω mod
=
µ ¶ 2π 1 1 ω1 + ω2 = + = 2 2 8 12 µ ¶ 2π 1 1 ω1 − ω2 = − = 2 2 8 12
5π 2π = radians/second 9.6 24 2π radians/second 48
and the average and modulation wavevectors are: kavg
= =
kmod
= '
µ ¶ y k1 + k2 5z ◦ ◦ = 2π · sin [40 ] + cos [40 ] 2 48 48 ³ y 2π z ´ · (y sin [40◦ ] + 5z cos [40◦ ]) ' 2π + 48 74.674 12.532 2π k1 − k2 = · (5y sin [40◦ ] + z cos [40◦ ]) 2 48 ³ y z ´ + 2π 14.935 62.0
The superposition is: f1 [r, t] + f2 [r, t] = 2favg [r, t] · fmod [r, t] where the full expressions for the average and modulation waves are: favg [r, t] = cos [kavg · r − ω avg t] · ¸ 2π 10πt ◦ ◦ = cos · (y sin [40 ] + 5z cos [40 ]) − 48 48 ¸ · 2π ◦ ◦ (y sin [40 ] + 5z cos [40 ]) − 5t) = cos 48 µ ¶¸ · ³ z ´ t y + − 2π ' cos 2π 74.674 12.532 9.6
116
CHAPTER 10. WAVES AND IMAGING fmod (r, t) = cos(kmod · r − ω mod t) · ¸ 2π 2πt ◦ ◦ = cos · (5y sin [40 ] + z cos [40 ]) − 48 48 ¸ · 2π (5y sin [40◦ ] + z cos [40◦ ] − 2t) = cos 48 µ ¶¸ · ³ z ´ t y + − 2π ' cos 2π 74.674 12.532 24
Note that both the average and modulation waves are traveling waves; they are headed in different directions with different frequencies and different velocities. The temporal frequencies are ν avg = 5 2 48 Hz and ν mod = 48 Hz. If the intensity (squared-magnitude) of the sum is averaged over time at an observation plane located downstream on the z-axis, both traveling waves will average out and no stationary fringe pattern will be visible.
k1 Sum
Power
k2 Sum of two sinusoidal traveling waves with period λ1 = 8 units and λ2 = 12 units directed at ±40◦ , respectively. The same principles just discussed may be used to determine the form of interference fringes from wavefronts with other shapes. Some examples will be considered in the following sections.
10.1.3
FRINGE VISIBILITY — COHERENCE
The visibility of a sinusoidal fringe pattern is a quality that corresponds quite closely to modulation, which is a term used by electrical engineers (sparkies). Given a nonnegative sinusoidal irradiance (intensity) distribution with maximum Imax and minimum Imin (sothatImin ≥ 0), the visibility of the sinusoidal fringe pattern is: Imax − Imin V≡ Imax + Imin Note that if Imin = 0, then V = 1 regardless of the value of Imax . The visibility of the fringe pattern is largely determined by the relative irradiances of the individual wavefronts and by the coherence of the light source. To introduce the concept of coherence, consider Þrst the Young’s two-aperture experiment where the source is composed of equal-amplitude emission at two distinct wavelengths λ1 and λ2 incident on the observation screen at ±θ. Possible pairs of wavelengths could be those of the sodium doublet (λ = 589.0nm and 589.6nm), or the pair of lines emitted by a “greenie” He:Ne laser (λ = 543nm
10.1. INTERFERENCE OF WAVES
117
(green), 594nm—(yellow)). In air or vacuum, the corresponding angular frequencies obviously are 2πc ω 1 = 2πc λ1 and ω 2 = λ2 . To Þnd the irradiance pattern created by the interference of the four beams, we must compute the superposition of the amplitude of the electromagnetic Þeld, Þnd its squared-magnitude, and then determine the average over time. The sum of the four component terms is straightforward to compute by recognizing that it is the sum of the amplitude patterns from the pairs of waves with the same wavelength. We have already shown that the sum of the two terms with λ = λ1 is: · ¸ · ¸ 2πy 2πz f1 (y, z; λ1 ) + f2 (y, z; λ1 ) = 2A cos sin [θ] cos cos [θ] − ω1 t λ1 λ1 " # · ¸ y 2πz = 2A cos 2π ¡ Lλ1 ¢ cos cos [θ] − ω1 t λ1 d
which is the sum of a stationary sinusoid and a traveling wave in the +z-direction. A snapshot of this amplitude at a Þxed time is shown: The second pair of wavefronts with λ = λ2 yield a similar result, though the period of the stationary fringes and the temporal frequency of the traveling wave differ. The expression for the sum of the two pairs is: 4 X
fn (y, z; λ) = f1 (y, z; λ1 ) + f2 (y, z; λ1 ) + f1 (y, z; λ2 ) + f2 (y, z; λ2 )
n=1
# " · ¸ ¸ 2πz 2πz y cos [θ] − ω 1 t + 2A cos 2π ¡ Lλ2 ¢ cos cos [θ] − ω 2 t = 2A cos 2π ¡ Lλ1 ¢ cos λ1 λ2 d d "
y
#
·
Note that both waves traveling waves move along the z-axis and that their wavevectors and velocities are: k1
=
k2
=
v1
=
v2
=
2πz λ1 cos [θ] 2πz λ2 cos [θ] c ω1 2πν 1 ν 1 λ1 cos [θ] = cos [θ] = ¡ 2π ¢ z = k1 z z λ cos[θ]
c ω2 2πν 2 ν 2 λ2 cos [θ] = cos [θ] = v1 = 2πz = k2 z z λ2 ·cos[θ]
Since the two traveling waves have the same phase velocity, they may be factored out to yield the sum of two different-frequency cosines multiplied by a sinusoidal traveling wave: ¸ ¸ · ¸¶ · 2πyd 2πyd 2πz fn [y, z; λ] = 2A cos cos [θ] − ω 1 t + cos cos Lλ1 Lλ2 λ1 n=1 4 X
µ
·
In identical fashion, the sum of the two stationary cosine waves may be recast as the product of cosines with the average and modulation frequencies: ·
¸ · ¸ · ¸ · ¸ 2πyd 2πyd 2πyd(λ2 + λ1 ) 2πyd(λ1 − λ2 ) cos + cos = 4A cos · cos − Lλ1 Lλ2 2Lλ1 λ2 2Lλ1 λ2 · ¸ · ¸ 2πyd(λ1 + λ2 ) 2πyd(λ1 − λ2 ) = 4A cos · cos 2Lλ1 λ2 2Lλ1 λ2 · ¸ · ¸ 2πyd 2πyd · λavg · cos · λmod = 4A cos Lλ1 λ2 Lλ1 λ2
118
CHAPTER 10. WAVES AND IMAGING
The time-average of the squared magnitude of the amplitude averages the traveling wave portion of the intensity to 12 . The Þnal expression for the irradiance is the product of two sinusoidal irradiance patterns with identical maxima and zero minima: ¯ · ¸¯2 + ¸ · ¸¯2 *¯ · ¯ ¯ ¯ ¯ 2πyd 2πz 2πyd ¯ ¯ ¯ I = ¯4A cos · λavg · cos · λmod ¯ ¯cos cos [θ] − ω1 t ¯¯ Lλ1 λ2 Lλ1 λ2 λ1 · ¸ · ¸ 2πyd 2πyd 1 = 16A2 cos2 · λavg · cos2 · λmod · Lλ1 λ2 Lλ1 λ2 2 ¸ ¸¶¸ · · µ · 1 4πyd 4πyd 21 = 8A · λmod λavg · 1 + cos 1 + cos 2 Lλ1 λ2 2 Lλ1 λ2 µ · µ ¶¸¶ µ · µ ¶¸¶ 2d 2d 2 · λavg · λmod = 2A · 1 + cos 2πy · 1 + cos 2πy Lλ1 λ2 Lλ1 λ2 µ · ¸¶ µ · ¸¶ 2πy 2πy = 2A2 · 1 + cos · 1 + cos Davg Dmod after deÞning the respective periods of the two oscillations to be:
Davg Dmod
≡ ≡
Lλ1 λ2 L = 2d · λavg d
µ
1 1 + λ1 λ2
¶−1
∝ |λ1 + λ2 |−1 ∝ (λavg )−1
¯ ¯−1 Lλ1 λ2 1 ¯¯ L ¯¯ 1 = ¯ − ¯ ∝ |λ1 − λ2 |−1 ≡ (∆λ)−1 2d · λmod d λ1 λ2
−1 Note that the lengths of the spatial periods of the oscillations are proportional to λ−1 avg ∝ (λ1 + λ2 ) −1 −1 and λmod ∝ |λ1 − λ2 | = (∆λ) . In the case where the two emitted wavelengths are close together such that λ1 ' λ2 ' λavg >> λmod , the expressions for the periods of the two component oscillations may be simpliÞed:
Davg
'
Dmod
'
Lλavg 2d L (λavg )2 2d · ∆λ
After cancelling the common terms, the relative lengths of the spatial periods of the modulations are: ∆λ Davg = Dmod ·
