Chapter 6 Light, space, and time

  Robert Grosseteste, Bishop of Lincoln and the first Chancellor of the University of Oxford in the 13th century, was one of the leading thinkers of his day, and a proponent of the works of the ancient Greeks. For him, as for many philosophers, the challenge of understanding light’s nature was critical to understanding the world. In Grosseteste’s treatise on the subject, entitled De Luce,he extols the primary importance of light: 『The first corporeal form …is in my opinion light. For light of its very nature diffuses itself in every direction in such a way that a point of light will produce instantaneously a sphere of light of any size whatsoever』.

  For Grosseteste, light defines space by its propagation instantly throughout the universe. Without light, there is no space, and therefore no forum in which events can take place. Matter, and thus the spatial extension of objects, are coupled to light, but cannot be separately defined. This intimate connection between light, space, and matter—in Grosseteste’s hands amenable to quantifiable description—informed the development of ideas regarding cosmology in the subsequent centuries.

  記住全網最快小説站𝙗𝙖𝙣𝙭𝙞𝙖𝙗𝙖.𝙘𝙤𝙢

  Space-time

  For Newton, space neither admitted nor demanded definition. He thought of space as a pre-existing entity, a sort of theatre in which events played themselves out. Large-scale motion of bodies in the heavens was integral to his idea of a set of universal laws. Einstein,by contrast, places light at the centre of space. For him, it defines space and time by virtue of setting the speed limit for signals sent from one part of the universe to another. The fact that there is a finite maximum speed turns out to make space and time inseparable. Einstein’s theory of relativity teaches us that we cannot think of one without thinking of the other. This is because our perception of space and time is based on local measurements of distances and time intervals. These measures appear differently to those moving relative to us, because of the speed limit imposed by light.

  How does this strange entwining of space and time by light arise?Let’s start with Newton’s conception of space. We can think of this as a sort of scaffold—a collection of imaginary rods of fixed length all connected together in a three-dimensional framework, as shown in Figure 28. Newton thought that this sort of structure pre-existed any event, and indeed that all events took place somewhere in this structure. Events can therefore be specified by their distance from a fixed point in the frame by counting the number of rods to reach the location of the event. Of course events occur at a certain time, too, so the scaffold must be equipped with clocks to measure the time. If a clock is placed at the junction of each of the rods, they all show the same time everywhere in space,and we can easily define a 『universal time』.

  Now we must ask several questions. First, how should we build a clock? Second, how should we ensure that they are all synchronized across space? Third, how should we build a ruler?These questions all have answers that are intimately related to the properties of light. Indeed, the answer to the last is this: one metre is the length of the path travelled by light in a vacuum during a time interval of 1/299,792,458 of a second. It is therefore linked to the answer to the first question: how accurate a clock can be built.

  28. A scafold representing space made up of a 3D lattice of measuring rods. At each intersection is a clock, all synchronized.

  Clocks

  The primary characteristic of a clock is that it 『ticks』. That is, it signals at regular intervals of time. By counting the number of ticks between events, we can assign a time interval between them. The more precise a clock is, the more regular the intervals between the ticks. In a grandfather clock, the 『ticks』 are provided by the regular swings of a long pendulum. In an electronic wristwatch, the 『ticks』are the oscillations of a piece of quartz crystal. These are much more regular than the swings of a pendulum, which can be affected by the temperature and humidity of the place where the grandfather clock is located. Therefore the quartz watch is said to keep better time than the grandfather clock.

  The most accurate clocks in the world are based on the highly regular clicks of electrons moving around in atoms. We』ve seen that atomic electrons can jump between different stable energy levels within an atom. For some atoms, the difference in energy between these levels is extremely well defined, and the rate at which electrons jump between the two can therefore be used to define a set of ticks, based on the frequency of the light used to push the electrons up and down between the two energy states.

  The idea is simple enough: illuminate an atom with microwaves(recall that microwaves are just like light, but with a much lower frequency—billions of oscillations of the electric field per second rather than millions of billions as in the case of visible light). Then slowly change the frequency of the microwaves until the electrons move between the atomic energy levels most efficiently. This defines the number of cycles per second of the microwave radiation—the rate of ticks of the clock—in terms of the energy level spacing of electrons in a particular atom.

  There are many technical challenges to building such an atomic clock—cooling the atoms, preparing them in the right initial state,illuminating with microwaves in a clever way to maximize the sensitivity, detecting that the electrons are indeed in the higher state at a given time—but clocks based on caesium atoms are now the most accurate way to measure time, with a rate of ticking that deviates from complete and utter regularity by only one millionth of a second in one hundred million years.

  These clocks provide a time standard that is agreed internationally and maintained by government laboratories such as the National Institute of Standards and Technology (NIST) in the US, the National Physical Laboratory (NPL) in the UK, and PhysikalischTechnische Bundesanstalt (PTB) in Germany. They are crucial components of many technologies that underpin our daily lives. For instance, they are vital for the global positioning system (GPS) that is the basis for navigation, including the satnav commonly used in cars.

  Clock synchronization

  The next challenge is to synchronize the two clocks, so that they are calibrated. One way to do this is by sending a signal from one clock to another. You start the first clock, sending a pulse of light to the other clock telling it when your first 『tick』 occurred. The person in charge of that clock then knows what fraction of tick her clock is behind yours (since the clocks are the same construction,we can assume they tick at the same rate) and can use this information in setting the correct time.

  There’s an interesting consequence to this. Imagine you want to synchronize your clock on Earth to that on a distant planet, in a far-away galaxy. You send your pulse of light off into the direction of the planet, and then you wait. As the planet is far away, it might take a very long time for the light to get there, even given the high velocity of light. Meanwhile you are getting older and older. The person who receives your synchronization message will have received it from the young you—she will see you as you were when you sent the message.

  Likewise when we look at the night sky, and see the distant stars,we are seeing images of them by receiving light that left their surfaces a long, long time ago. And as we look to stars or galaxies that are even farther away, so we look into the deeper and deeper past, seeing the universe as it was billions of years ago. In that sense, the light that reaches us is also billions of years old—it has been travelling across space for an age since its birth in the remote past. Light is the oldest thing we can see in the universe.

  Our clocks, though, are a bit closer together. An interesting fact is that when you put one of these clocks on an aeroplane moving at 800 kph or so, you find that it ticks slower than one on the ground. That is, if you set the clocks to tell the same time, you will find that the one in the aeroplane appears from the ground to be ticking at a slower rate than the ground-based one. This is a consequence of the maximum speed at which signals can be conveyed between the two clocks—the speed of light.

  You can see this by looking at the situation in Figure 29. There,person A is on the ground and person B is in an aeroplane moving at high speed. B watches as A sends a light pulse to a mirror suspended at height H above the ground. From A’s point of view,the light pulse travels distance H. However, from B’s perspective that signal will travel a slightly longer distance than H since A appears to be moving backwards at high speed relative to B.Therefore, because the signal travels at light speed according to both A and B, and both record the same number of ticks between sending and receiving the signal, then the only explanation is that B’s clock is ticking slower than A’s clock when seen from A, and A’s clock is ticking slower than B’s when seen from B. This effect is called time dilation.

  29. Time dilation due to relative motion. Observers A and B bothmeasure the time for a pulse to arrive at the suspended mirror. Theymeasure different times because they are moving relative to oneanother.

  Einstein argued in a similar way that space should also contract.That is, a rod of B’s imaginary scaffold should look smaller to A than an identical rod in his possession. And vice versa: A’s rod looks shorter to B than her own.

  Both of these effects arise because there is a maximum speed at which signals of any kind can propagate, and this speed is the same for everyone. If that were not the case, then one could determine a preferred scaffold, or 『frame of reference』, in which the signals went at the highest speed. Einstein’s work on relativity showed that there is no preferred frame, so that Newton’s idea of a fixed, pre-defined space could not be the case. Since the maximum speed for signals of any kind turns out to be light speed, light is crucial in defining space and time.

  It’s worth asking how precisely we might be able to synchronize two clocks by the 『light pulse』 method discussed previously. You can see that one way to do it is by making the light pulse as brief as possible, so that the uncertainty over when it arrives is minimal.Thus it is important to know if there are limits to the brevity of light pulses. It turns out that there are, and they arise from a similar sort of wave property that limits the resolution of imaging systems, as we saw in Chapter 3.

  We might start by asking how it is that we determine the frequency of a light wave. Imagine taking one of our clocks and asking how many peaks of the wave reach us during the interval between two ticks. The greater the number of peaks, the higher the frequency. The precision with which we can determine the frequency depends on how many times we repeat this measurement, since our ability actually to tell whether we have reached the peak of a wave may not be perfect. Thus, the longer we count, the more precise our determination of frequency. This trade-off is fundamental for waves—the imprecision in frequency multiplied by the uncertainty in the time interval is a fixed product. This was first understood by Joseph Fourier, a19th-century French mathematician and scientist who played a key role in formulating the wave model of light propagation.

  Ultrashort light pulses

  Fourier’s theorem is important for clock synchronization, since it says that if we want a short pulse, we must have an indefinite frequency. Another way to say this is that a short burst of light is constituted by a broad range of colours. That’s entirely analogous to the situation Abbe identified for imaging optics: a high-resolution image, demanding a small focus, requires a wide range of ray angles to be collected. In fact, the analogy goes further. Just as Abbe showed the smallest size of a focal spot could be approximately one wavelength of the illuminating light, so Fourier showed that the shortest-duration pulse is a single cycle of the field.

  What this means in practice is that for light in the visible region of the spectrum, it is possible to produce pulses with a duration of about 2 fs. Amazingly, light sources derived from lasers can now routinely produce pulses of such startling brevity. They are based on the mode-locking technique described in Chapter 5.

  But these are not the shortest bursts of light that occur in nature,nor even the shortest that can be produced in a laboratory.That honour goes to light sources with much shorter average wavelengths. For instance, using the single-cycle argument you can easily see that if the wavelength is shortened, then the duration of an optical cycle is reduced and in principle the duration of the pulse can be reduced. This approach currently holds the record for the world’s shortest controllable light pulses.By shining very intense laser light on an atomic gas, a process known as high-harmonic generation produces light waves with a frequency multiples of several tens of the driving laser frequency.This allows pulses with durations of several tens of attoseconds(10-18 s, or a billion billionth of a second). These are unimaginably short bursts of light, with duration equal to the time it takes an electron to oscillate within an atom.

  Frequency combs

  In Chapter 5, I stated that in a mode-locked laser a single pulse traverses the optical cavity. Each time it bounces off one of the mirrors, a little bit of the light pulse is transmitted through the mirror and exits the cavity. As a consequence, outside the cavity the light appears as a 『continuous』 sequence of pulses spaced by the round-trip time of the light in the cavity, typically a billionth of a second or so. A 『snapshot』 of such a train of pulses would show these very short bursts separated from each other by a delay that is long compared to their duration, like the teeth of a comb (as shown in Figure 30). And the pulses can be made identical to one another by careful adjustment of the laser producing them, so that the electric field of each of the pulses peaks at exactly the same time with respect to the intensity envelope of the pulse.

  It turns out that this configuration means that each of the 『teeth』in the frequency comb has a very precise position at an absolute frequency. A precisely calibrated set of frequencies is a very important tool for building accurate clocks. This is because it allows a direct comparison of optical frequencies to lower (usually microwave) frequencies, which can be counted by means of electronics.

  Thus frequencies in the microwave region inhabited by the caesium atomic clock can be compared simply to much more precise electronic transitions in the optical region of the spectrum in, for example, strontium atoms or aluminium ions. Therefore the standard caesium atomic clocks used in satellite navigation,for instance, can now all be synchronized and to tick at the same rate to within one part in a billion billion (i.e. 1:10-18), due to the precision of the optical electron oscillation frequency within strontium or aluminium.

  30. Train of identical nearly single-cycle optical pulses. The spectrum of the pulse train looks like the teeth of a comb, hence it is called a frequency comb.

  『Optical clockwork』 of this kind allows the comparison of disparate frequencies with such remarkable precision that it provides a means to test the tenets of relativity, and thus to understand better the role of light in defining space and time. Frequency, and thus time, is the physical quantity that can be measured with the highest precision of any quantity, by far.

  Optical telecommunications

  Frequency combs are also important in telecommunications links based on light. In Chapter 3, I described how optical waves could be guided along a fibre or in a glass 『chip』. This phenomenon underpins the long-distance telecommunications infrastructure that connects people across different continents and powers the Internet. The reason it is so effective is that light-based communications have much more capacity for carrying information than do electrical wires, or even microwave cellular networks. This makes possible massive data transmission, such as that needed to deliver video on demand over the Internet.

  Many telecommunications companies offer 『fibre optic broadband』deals. A key feature of these packages is the high speed—up to 100megabytes per second (MBps)—at which data may be received and transmitted. A byte is a number of bits, each of which is a 1 or a 0. Information is sent over fibres as a sequence of 『bits』, which are decoded by your computer or mobile phone into intelligible video, audio, or text messages. In optical communications, the bits are represented by the intensity of the light beam—typically low intensity is a 0 and higher intensity a 1. The more of these that arrive per second, the faster the communication rate. The MBps speed of the package specifies how rapidly we can transmit and receive information over that company’s link.

  Why is optics so good for communications? There are two reasons.First, light beams don’t easily influence each other, so that a single fibre can support many light pulses (usually of different colours)simultaneously without the messages getting scrambled up. The reason for this is that the glass of which the fibre is made does not absorb light (or only absorbs it in tiny amounts), and so does not heat up and disrupt other pulse trains.

  Further, a light beam propagating in glass has to be very intense in order for it to influence another light beam. For instance, when you cross the beams from two laser pointers you don’t see either beam distort or deviate from its original path even though the two beams pass right through each other. (If your laser pointers had enormous power, you might just see such an effect, but only because the room is full of air. In a vacuum they would still not influence each other.) This means that the 『crosstalk』 between light beams is very weak in most materials, so that many beams can be present at once without causing a degradation of the signal. This is very different from electrons moving down a copper wire, which is the usual way in which local 『wired』 communications links function. Electrons tend to heat up the wire, dissipating their energy. This makes the signals harder to receive, and thus the number of different signal channels has to be kept small enough to avoid this problem.

  Second, light waves oscillate at very high frequencies, and this allows very short pulses to be generated, as described earlier. This means that the pulses can be spaced very close together in time,making the transmission of more bits of information per second possible. Indeed rates of 40 Gbps (a Gb, or Gigabit, is a billion bits)are possible in current-generation commercial long-haul systems.Electrical signals in copper wires are limited in the duration and spacing of pulses coding the information by the heating effects noted previously, which tend to get worse at higher frequencies.Copper wires run out of steam, as it were, at much lower bit rates.

  Fibre-based optical networks can also support a very wide range of colours of light. Glass transmits a broad range of wavelengths,with particularly low scatter and absorption loss in a spectral window from 1.3–1.55 μm. The rate at which photons are lost in fibre at these wavelengths is about 5 per cent per kilometre. These losses can be made up by amplifying the light while in the fibre so that transmission over very long distances (such as across the Atlantic Ocean) is possible without any conversion of the light to electrical signals or vice versa.

  The telecommunications spectral window is divided into many individual frequency 『slots』—much like the frequency comb shown in Figure 30. Each spectral component is a separate communications channel. There can be 150 or so slots in the window. Within each channel, a 40 Gbps optical signal can be operated. Therefore the total bit rate for communications is 150times 40 Gbps, or 6 Tbps (1 Tb, 1 terabit, is 1,000 billion bits).

  The ever-increasing demand for communications bandwidth due to increased use of the Internet and the services it provides have spurred optical engineers to new heights of creativity in harnessing the potential of light.