3. The waves must have the same frequency and must also have a fi xed phase relationship. This means that over a given distance or time interval the phase difference between the waves remains constant. Such waves are called coherent.
Coherence
Interference conditions 1 and 2 are illustrated in Figure 25.1, but condition 3 and the notion of coherence warrants more discussion. When the ear detects a sound wave or the eye detects a light wave, it is actually measuring the wave intensity averaged over many cycles. For example, the frequency of blue light is about 7 1014 Hz. The eye cannot follow the variation of every cycle; instead, the eye aver- ages the light intensity over time intervals of about 0.1 s, which corresponds to approximately 7 1013 cycles, and then sends this average signal to the brain.
Suppose the waves in Figure 25.1 are two interfering light waves. In order for the waves to interfere constructively and give a large total intensity, they must stay in phase during the time the eye is averaging the intensity. Likewise, for the two waves in Figure 25.1C to interfere destructively, they must stay out of phase during this averaging time. Each scenario is possible only if the two waves have precisely the same frequency. If their frequencies differ, two waves that are initially in phase with each other (and interfere constructively) will be out of phase at a later time (and interfere destructively).
Waves are in phase. Waves are 180 out of phase.
A C
Speaker 1
Listener
Constructive interference Speaker 2
L2 l
L1
B
l
Destructive interference
l 2
Figure 25.1 A Sound waves emitted by two separate sources (two speakers) combine when they reach a listener. These waves travel distances L1 and L2 on their way to the listener, where L1 and L2
may or may not be equal. B Here the two waves arrive in phase, and their sum is larger than either of the individual waves. C If the two waves are 180° out of phase when they reach the listener, their sum is zero, and no sound (and no energy) arrives at the listener.
1Mathematically, we would say that the waves in Figure 25.1C are 180° out of phase, corresponding to a shift of one wave relative to another by l/2. Figure 25.1 considers waves that are either perfectly in phase or completely (180°) out of phase, but other relationships (specifi ed by other phase angles) are also possible.
Figure 25.2 shows two waves with slightly different frequencies. The waves are initially in phase, as their maxima and minima occur at about the same times. After many cycles, however, one of the waves “gets ahead” of the other; eventually, the wave with the higher frequency has completed an extra one-half cycle, and the two waves are out of phase. The two waves thus go in and out of phase as time passes.
When averaged over a very large number of cycles, the two waves display neither constructive nor destructive interference; on average, there is no interference at all.
So, to exhibit interference, two light waves must have exactly the same frequency.
Most visible light is emitted by atoms; you might expect that light waves pro- duced by two atoms of the same element would be “identical” and have the same frequency and could therefore exhibit interference. To exhibit interference, how- ever, two waves must satisfy an additional condition. Consider atoms in a gas that are emitting light. These atoms undergo collisions with other atoms in the gas and with the walls of the container, and each collision causes the phase of the emitted light to jump abruptly to a new value. In other words, each collision “resets” the light wave, starting its oscillation over again. These collisions are different for each atom and spoil interference effects because they cause waves that may have origi- nally been in phase to suddenly become out of phase and vice versa. Experiments show that phase jumps in typical light waves occur about every 108 s; hence, light from two different atoms of a given element are not coherent. (Certain special cases are an exception, such as when the atoms are all in the same laser. See Chapter 29.) One way to eliminate the effect of phase jumps is to derive both waves from the same source. In Section 25.2, we’ll explain how to do that.
CO N C E P T C H E C K 2 5.1 | Conditions for Interference
Consider an (attempted) experiment to observe interference of light from two different helium–neon lasers. The light from these lasers is emitted by neon atoms. Is it possible to observe interference with these two light sources?
Explain why or why not.
25.1 | COHERENCE AND CONDITIONS FOR INTERFERENCE 843 Two waves with slightly
different frequencies Waves are in phase at start.
Waves are out of phase now.
Figure 25.2 Two waves with slightly different frequencies.
Initially, they are in phase, but as time passes they become out of phase. The in-phase and out-of- phase portions average out over many cycles, so on average there is no interference. Important note:
The different colors here do not indicate different colors of light, but rather denote the amplitudes of the different waves.
E X A M P L E 2 5 . 1 Coherence Time for a Light Wave
The phase of light emitted by an atom changes (jumps) after a typical time interval t ⬇ 108 s, called the coherence time of the light. How many cycles of a light wave occur during this coherence time? Assume the light is blue, with f 7.0 1014 Hz.
RECOGNIZE T HE PRINCIPLE
A wave’s frequency is inversely related to its period T, with f 1/T, and the period is the time it takes to undergo one cycle of oscillation. Given the frequency, we can fi nd the period and from that calculate how many cycles are completed in the coherence time t.
SK E TCH T HE PROBLEM No sketch is necessary.
IDENT IF Y T HE REL AT IONSHIPS
Recall some basic facts about oscillations and waves from Chapters 11 and 12. The period of this light wave is (using Eq. 11.1)
T5 1
f 5 1
7.031014 Hz51.4310215 s
which is the time required to complete one cycle of the wave. The number of cycles N contained in the coherence time t is thus
N5Dt T SOLV E
Inserting our values of t and T gives N5 Dt
T 5 131028 s
1.4310215 s5 73106 cycles What does it mean?
Although the coherence time t is very short, the frequency of visible light is very high, so there are more than one million oscillations—quite a large number—
during this coherence time. The precise value of t depends on the type of atom and its environment; the value 108 s is only a typical estimate.
2 5 . 2 | T H E M I C H E L S O N I N T E R F E R O M E T E R
An optical instrument called a Michelson interferometer is based on interference of refl ected waves (Fig. 25.3). This clever device, invented by Albert Michelson in the late 1800s, played an important role in the discovery of relativity (Chapter 27). The device contains two refl ecting mirrors mounted at right angles. At least one of these mirrors is movable; here the mirror on the far right can be moved along the horizon- tal axis. A third partially refl ecting mirror called a “beam splitter” is mounted at a 45° angle relative to the other two. The beam splitter refl ects half the light incident on it and lets the other half pass through.
Light incident from the left in Figure 25.3 strikes the beam splitter and is divided into two waves. One of these waves (denoted 1) travels to the mirror at the top where it is refl ected and then returns to the beam splitter. When it reaches the beam splitter, it is again split into two waves, and one of these waves travels downward in Figure 25.3 to the detector as wave 1. The other wave from the beam splitter (wave 2) travels to the mirror on the right, where it is also refl ected back to the beam split- ter. When wave 2 returns to the beam splitter, a portion is refl ected to the detector at the bottom of the fi gure. Waves 1 and 2 thus combine at the detector; they are the two waves that interfere, and they follow very similar paths as they travel through the interferometer. First, both waves are refl ected once by the beam splitter. Second, both waves pass through the beam splitter once. Third, both waves are refl ected once by a mirror (at the far right or top). The only difference between the two waves is that they travel different distances between their respective mirrors and the beam splitter. After the waves are created by the beam splitter, wave 1 travels a round-trip distance 2L1 and wave 2 travels a round-trip distance 2L2 before they recombine. The implications for interference are shown in Figure 25.4. The path length difference is
DL52L222L1 (25.1)
If we assume L2 L1, wave 2 travels an extra distance L and undergoes a number of extra oscillations before it reaches the detector. Each oscillation occurs over a Figure 25.3 Schematic diagram
of a Michelson interferometer.
Light is incident from the left and split into two separate waves (also called beams) by partial refl ection at the beam splitter in the center.
These waves refl ect from the mir- rors at the top and right and arrive back at the beam splitter, where they recombine and travel together to the detector at the bottom.
(Waves 1 and 2 also recombine and travel back to the left, but these beams are not used in the experiment.) The colors of the rays do not indicate the color of the light.
Movable mirror
Detector Incoming
light
2 2
Wave 1
Wave 2 Mirror
MICHELSON INTERFEROMETER
2 1 1 Beam splitter
L2 L1 1
distance equal to the wavelength l, so the number of extra oscillations made by wave 2 is
N5 DL
l (25.2)
If N is an integer, then N complete oscillations fi t into the extra path length and the two waves are in phase when they recombine, producing constructive interference.
On the other hand, if N is half integral (i.e., if N 12, 32, . . . , m 12, where m is an integer), wave 2 travels an “extra” one-half wavelength relative to wave 1 and the maxima of one wave will coincide with the minima of the other as in Figure 25.1C.
That is the condition for destructive interference. The interference conditions for a Michelson interferometer are thus
DL5ml 1constructive interference2
DL5 1m1122l 1destructive interference2 (25.3) where m is an integer. If the interference is constructive, the light intensity at the detector is large, whereas if the interference is destructive, the detector intensity is zero. Figure 25.5 shows how the light intensity at the detector varies as a function of L. The places on the intensity curve where the interference intensity is greatest are called bright “fringes,” and the places where the intensity is zero are called dark
“fringes.”2
The Michelson interferometer uses refl ection to satisfy the general requirements for interference from Section 25.1. First, refl ection at the central beam splitter pro- duces two separate waves that travel through different regions of space. Second, these waves are brought back together by the beam splitter so that they recombine at the detector. Third, because they are produced from a common incident wave, the two interfering waves are coherent.
Using a Michelson Interferometer to Measure Length
The wavelengths of light emitted by various sources are known. For example, a helium–neon (He–Ne) laser emits light with a wavelength of approximately lHe–Ne 633 nm. Recall that 1 nm 1 109 m and that the wavelength of visible light is in the range of about 400 nm to 750 nm. As a rough comparison, the wavelength of vis- ible light is about one thousand times less than the thickness of a human fi ngernail.
Knowing the value of lHe–Ne, an experimenter can use a Michelson interferom- eter to measure a displacement in the following way. Using light from the laser, the mirror on the far right in Figure 25.3 is adjusted to give constructive inter- ference and its position is noted; this corresponds to one of the bright fringes in Figure 25.5. The mirror on the far right is then moved horizontally, changing the path length difference L, and the intensity at the detector moves along the curve in Figure 25.5. The intensity changes from a high value to zero and back to a high value every time the path length difference L changes by one wavelength of the light. According to Equation 25.3, moving the mirror from one bright fringe to the next bright fringe corresponds to starting at the condition for constructive interference with L mlHe–Ne and increasing the path length difference to L (m 1)lHe–Ne. If the interferometer mirror is moved so as to pass through N cycles from bright to dark and back to bright, the change in the path length difference is
1DL2change5NlHe–Ne (25.4)
When the mirror moves a distance d, the distance traveled by the light changes by 2d because the light travels back and forth between the beam splitter and the mirror
Figure 25.4 A Michelson inter- ferometer with the waves redrawn.
The two waves travel distances 2L1 and 2L2, respectively, as they travel between the beam splitter and the mirrors. The path length difference L leads to interference;
compare with Figure 25.1A.
Beam splitter creates two beams here.
Beam splitter recombines the two beams.
Incoming light
2L1
2L2 DL
Wave 2 Wave 1
2We’ll see why they are called fringes in Section 25.3, when we consider some examples of how inter- fering light waves appear when viewed by the eye.
Figure 25.5 The light intensity at the detector of a Michelson interferometer oscillates as a func- tion of the path length difference L 2L2 2L1, which is varied by moving one or both mirrors.
Dark “fringe”
(intensity at detector 0) Bright “fringe” (large intensity at detector)
Intensity
l
DL
25.2 | THE MICHELSON INTERFEROMETER 845
(Fig. 25.3). Hence, if the mirror moves through N bright fringes, the distance d traveled by the mirror satisfi es
1DL2change52d5NlHe–Ne so that
d5 N
2 lHe–Ne (25.5)
The accuracy of this measurement of d depends on the accuracy with which the wavelength lHe–Ne is known. For this reason, physicists have devoted a lot of effort to measuring the wavelength of light from certain light sources. In fact, the wave- length produced by a specially constructed helium–neon laser is known to be
lHe–Ne5632.99139822 nm (25.6)
Suppose this light source is used in a Michelson interferometer and one of the mirrors is moved a distance d such that exactly N 1,000,000 bright fringes are counted.
According to Equation 25.5, we have d NlHe–Ne/2. Since N can be counted directly (and is thus accurately known) and the wavelength in Equation 25.6 is also known with high accuracy, this approach provides a very precise way to measure length.
Moreover, people in different laboratories can use helium and neon to construct lasers that produce the wavelength given in Equation 25.6, so they can compare their separate length measurements with high accuracy. The Michelson interferometer thus makes possible a very convenient standard for the measurement of length. A similar type of interference effect is used to read information from CDs and DVDs.
CO N C E P T C H E C K 2 5. 2 | Analyzing the Michelson Interferometer A Michelson interferometer using light from a helium–neon laser (lHe–Ne 633 nm) is adjusted to give an intensity maximum (a bright fringe) at the detec- tor. One of the mirrors is then moved a very short distance so that the inten- sity changes to zero. How far was the mirror moved, (a) lHe–Ne/4, (b) lHe–Ne/2, (c) lHe–Ne, or (d) 2lHe–Ne?
E X A M P L E 2 5 . 2 Applications of a Michelson Interferometer:
Detecting Gravitational Waves
An experiment called the LIGO (for laser interferometer gravitational wave observa- tory) is designed to detect very small vibrations associated with gravitational waves that arrive at the Earth from distant galaxies. The LIGO experiment involves several Michelson interferometers; in one of these interferometers, the mirrors are placed a distance L2 4 km (kilometers) from the beam splitter. Suppose changes in the inter- ference pattern corresponding to 1% (1001) of a cycle in Figure 25.5 can be detected.
What is the change in the mirror–beam splitter distance that can be detected by the LIGO experiment? Assume the light used in LIGO has a wavelength l 500 nm.
RECOGNIZE T HE PRINCIPLE
For a general interference experiment, the path length difference L must change by one wavelength l to go from a bright fringe to a dark fringe and then back to a bright fringe. A 1% change of a fringe thus corresponds to a l/100 change in L. When a mirror of a Michelson interferometer moves a distance d, the path length difference changes by 2d. We can fi rst fi nd the value of d that gives a 1% fringe change and from that get the change of the mirror–beam splitter distance.
SK E TCH T HE PROBLEM
Figures 25.3 through 25.5 show a Michelson interferometer and the interference conditions.
IDENT IF Y T HE REL AT IONSHIPS
The change in the path length difference equals 2d, so each complete intensity cycle from bright to dark to bright in Figure 25.5 corresponds to moving the mirror a dis- tance l/2. If we can detect 1001 of this change, the corresponding displacement of the mirror is
d5 l/2 100 5 l
200 (1)
SOLV E
Inserting the given value of the wavelength into Equation (1), we get d5 l
2005 150031029 m2
200 5 2.531029 m What does it mean?
The value of d found here is 2.5 nm—approximately 10 times the diameter of an atom—which is quite impressive. This result also shows that when adjusting the mirrors of a Michelson interferometer to actually observe interference, an experi- menter must be able to move the mirrors with a precision of a few nanometers.
Such precision can be accomplished with carefully designed screw adjustments (a low-tech approach, but there are other ways, too).
2 5 . 3 | T H I N - F I L M I N T E R F E R E N C E
Soapy water is normally transparent and colorless, but the photo of a thin soap fi lm in Figure 25.6 shows very bright colors resulting from interference of refl ected waves from the fi lm’s two surfaces. The colorful “bands” in Figure 25.6 are called
“fringes” and correspond to the locations of constructive and destructive interfer- ence for light waves with different wavelengths (colors) in the incident light.
To understand where these colors come from, we start with the case shown in Figure 25.7A, in which a thin soap fi lm rests on a fl at glass surface. For simplic- ity, we also assume the incident light is monochromatic, that is, that it has a single wavelength. (Later we’ll consider the behavior with white light, which is light con- taining many different colors.) The upper surface of the soap fi lm in Figure 25.7A is similar to the beam splitter in the center of the Michelson interferometer, refl ecting part of the incoming light (ray 2) and allowing the rest of the incident light to be transmitted into the soap layer after refraction at the air–soap interface. This trans- mitted light is partially refl ected at the bottom surface, producing the wave that travels back into the air as ray 3. The two outgoing rays denoted as 2 and 3 meet the conditions for interference: they travel through different regions (one travels the extra distance through the soap fi lm), they recombine when they leave the fi lm as parallel rays, and they are coherent because they originated from the same light source and initial wave, ray 1.
To determine whether these interfering waves are in phase or out of phase with each other, we must consider what happens in the extra distance traveled by ray 3. For simplicity, let’s assume the incident and refl ected rays are all approximately normal to the fi lm (Fig. 25.7B), which corresponds to looking straight down at the soap fi lm. For a fi lm of thickness d, the extra distance traveled by ray 3 is just 2d.
We must next account for the index of refraction of the soap fi lm. Recall that light propagates in a vacuum at a speed c 3.00 108 m/s. The frequency f and wave- length llvac are related by
lvacf5ct (25.7)
Insight 25.1 HIGH-PRECISION INTERFEROMETRY
The LIGO interferometers can actu- ally do much better than we have assumed in Example 25.2. In fact, the real LIGO interferometers can detect mirror movements more than one million times smaller than our result for d!
The goal of the LIGO experiment is to detect the very tiny mirror move- ments physicists think will be caused by gravitational waves that originate outside our galaxy. It is believed that supernovas (exploding stars) pro- duce rapid motion of large amounts of mass, resulting in gravitational waves. When these waves reach the Earth, they should deform the Earth, leading to movement of the LIGO mirrors that can be detected through changes in the interference pattern.
By using a long distance between the beam splitter and the mirrors, the LIGO interferometers are sensitive to very small percentage changes in that distance.
25.3 | THIN-FILM INTERFERENCE 847 Figure 25.6 When a soap fi lm is viewed with white light (as from the Sun), one observes colored interference fringes (colored bands). These colors are due to constructive interference; different colors (different wavelengths) give constructive interference for differ- ent values of the fi lm thickness and for different viewing angles.
Interference fringes
© Terry Oakley/Alamy