Lasers and Applications - UNSW

Lasers and Applications

Introduction to lasers

Unit 1 Introduction to lasers 2

Introduction

In this unit, we shall do the groundwork for the study of lasers. We shall speed-read through the applications of lasers and fly through the history, but slow down for a review of blackbody radiation, a topic that was vital for the development of lasers (and for quantum mechanics in general). Interestingly, blackbody radiation explains how ordinary light sources work, but it is also a necessary stepping-stone towards understanding the laser.

Learning outcomes

After studying this unit you will be able to

• Discuss the key applications of lasers • Discuss the crucial historic developments that lead to the laser • Describe and discuss the blackbody problem • Discuss the relevant physics of blackbody radiation • Calculate the blackbody spectrum given the necessary formulae


Background

A laser is a light source, much like a flashlight or torch. We put energy into it, and get a parallel beam of light out. Although the word ‘light’ usually denotes the visible part of the electro-magnetic spectrum, lasers have been made to operate from the deep ultraviolet to the far infrared, and it is in this context that we shall talk about lasers. The word ‘LASER’ is an acronym for Light Amplification by Stimulated Emission of Radiation. It is a word that was coined in 1957 by Gordon Gould, a postgraduate student, who modified another acronym, the ‘MASER’, which stands for Microwave Amplification by Stimulated Emission of Radiation. The maser was the precursor to the laser, and was (is) used to amplify microwave radiation. More about this later.

Before we get submerged in the theory of lasers, let’s take a preview of some applications of lasers. For many years after its invention, the laser was considered to be a “solution looking for a problem”. Today, lasers are used in nearly every facet of our life and we hear daily of new, exciting and ingenious ways they are put to use. In the ensuing few pages, we’ll list the key areas of applications of lasers, applications that will be further discussed in the second half of this course.

Applications of lasers

The most significant applications of lasers include telecommunications, medicine, manufacturing, the military, consumer goods and basic research.

1. Telecommunications

� Telecommunication is one of the most important applications of lasers. Light from diode lasers is modulated by the signal that contains the information, which is then transmitted through a network of optical fibres. At the other end, the modulated light is converted back to electrical signal. This technology allows the transmission of enormous amounts of information which would not be possible through conventional wires.

� Optical fibres, semiconductor diode lasers and fibre (light) amplifiers are vital to all modern communication networks.

� Interestingly, the two key elements in this technology, low loss optical fibres and the diode laser, were discovered more or less at the same time.

� After 40 years of research, optical losses in fibres have reached their theoretical minimum.


� Future developments will most likely include tuneable diode lasers compatible wavelength division multiplexing, a technique which will even further enlarge the bandwidth of fibre optic systems

� Some interesting numbers:

o In 2007, 830 million diode lasers were sold globally (for all types of applications, not only telecom)

o The worldwide market for laser-diodes in 2007 was U$4 billion

Diode laser compared to eye of a needle

(from: http://www-a.jpl.nasa.gov/news/news.cfm?release=2005-134)

2. Medicine

� Applications in medicine are some of the most interesting uses of lasers.

� Lasers are frequently used in general surgery and medical diagnostics, but are especially useful in ophthalmology. Before the laser, any procedure inside the eye required the eyeball to be cut open.

o One of the first ophthalmic applications of lasers was the repair of detached retinas. When laser light is absorbed by the retina, it is converted to heat with a rise of temperature to about 65C. At this temperature denaturing of the protein occurs which causes an adhesion between the choroid and the adjacent retina.


o The leading cause of blindness is younger people is vascular disease, particularly diabetic retinopathy which is typically caused by problems with the blood supply. Lasers can be used to destroy such abnormal blood vessels.

o In older people, the leading cause of blindness is macular degeneration, which is either due to vascular insufficiency or the opposite, the overgrowth of blood vessels in the centre of the retina. Lasers can often be used in these types of diseases.

(from http://www.houstonretina.com/laserTreatment.html)

o Another area where lasers have made a big difference is the treatment of glaucoma. Glaucoma is a disease of the optic nerve due to increased intraocular pressure. Lasers can be used to punch a hole in the iris, which facilitates flow of aqueous humour from the posterior to the anterior chamber of the eye.

Laser eye surgery

(from http://uuhsc.utah.edu/MoranEyeCenter/patientcare/davis.html)

� Surgeons can now use (carbon dioxide) lasers as bloodless scalpels because the optical beam cauterises the incision as it is made. Before the advent of


lasers some procedures that cause profuse bleeding were impossible to perform

� Lasers are also used to open up clogged arteries using a technique called laser angioplasty

� Cancer treatment is a small but promising field that, today, is based on photodynamic therapy (more about this later)

� In terms of the number of procedures, the most important growth areas are eyesight correction and cosmetic surgery

o In 2007, tens of millions of laser-based eyesight correction procedures were performed around the world. Because the refractive power of the eye is determined by the curvature of the cornea, by vaporizing some corneal cells with a UV laser, one can ‘tailor make’ the specific refractive power, just like wearing spectacles.

o Cosmetic surgery also relies heavily on lasers for things like removal of pigments, tattoos, unwanted hair, spider veins, ‘skin resurfacing’


o It is estimated that there are 300 million potential patients for laser-based eyesight surgery.

o Global sales of medical lasers systems in 2006 was over $3billion

3. Manufacturing

� The automotive industry relies greatly on high power CO2 lasers for cutting and welding metals. Lasers have several advantages over conventional techniques for cutting and welding materials (more about this later) and are fully compatible with robotics.

Laser robot (from http://www.ferret.com.au/t/Robot-Technology)


� Nd:YAG lasers are used in drilling holes, spot welding and marking. If necessary, laser can drill tiny holes in metals or non-metals that would not be possible by mechanical drills.

� The printing industry uses a range of lasers, mostly ion lasers, but the trend is to use high power diode lasers. Laser typesetters and laser platemakers are routinely used in the printing industry.

� The semiconductor industry is a major user of ultraviolet (excimer) lasers for photolithography. Photolithography plays an important role in the manufacture of integrated circuits with millions of transistors on a semiconductor chip.

� The entertainment industry (i.e. laser light-shows) is based on ion lasers


o In 2007, the global non-diode laser sales were U$3.7 billion

o Estimated global sale of lighting products (not just lasers) in 2007 was $50 billion

Laser welding

(from http://www.ailu.org.uk/laser_technology/news/2008-07-14/trumpf_080623.html)

4. Military

� So far, (mercifully) lasers have been found to make poor weapons. On the other hand, they are extensively used in guiding missiles to their destination, range finders and other target designators (‘smart bombs’ )


� It is suggested that in future ‘star-wars’ applications lasers will be used in space-based weapons and other airborne systems. This meant to detect enemy missile attacks and destroy ballistic missiles on the fly. Not simple: the Earth’ s atmosphere absorbs and scatters light; also not a lot of time to available: Scuds with a range of 300km are in the air for 4 min max. Missiles with longer range stay in flight for 15 minutes.

Mr Spock makes laser history

(from http://www.scienceclarified.com/scitech/Lasers/Military-Applications-of-Lasers.html)

5. Consumer goods

� DVDs, CD-ROMs, optical disks, etc all rely on diode lasers.

� Big changes have occurred with the development of laser diodes emitting in the blue end of the spectrum. Shorter wavelengths (eg. blue and U.V.) enable optical storage devices to pack more information per disc. The ‘blu-ray disc’ works at 405nm, while the standard DVD format uses 650nm. The shorter the wavelength, the more information can be stored on a disc. Blu-ray discs hold 50gigabite, which is 5-10 times more than DVDs. And research labs are already working on the next generation of laser diodes that emit in the deep UV.


� A new area that is likely to make a big impact on the consumer market is Laser TV. There are already high quality HDTVs on the market that utilize diode lasers instead of UMP lamps. Advantages include very large colour gamut (90% of the colours visible to the human eye), lower cost, lower power consumption, very long lifetime, etc

� Diode lasers are also at the heart of laser printers, barcode scanners, laser pointers, security devices, etc


o In 2007, global sales of DVD players were 127 million units. In 2000 this number was 35 million units. However, new flash memories are taking over a great section of the optical storage market.

Blu-ray disc

(from http://www.blogcdn.com/www.engadget.com/media/2008/05/5-25-08-laser-disc.jpg)

6. Basic research

� Lasers are the product of basic research, and researchers around the world are still developing new types of lasers.

� Physicist, chemist, biologist are major users of lasers, in areas such as spectroscopy, laser fluorescence, holography, biology (eg. the human genome program), etc

� One interesting example, which has been around for a while, is laser (optical) tweezers. Laser tweezers can manipulate small objects using radiation pressure from a focussed laser beam. Objects can be biological (living cells, viruses, etc) or inorganic particles, such as semiconductor quantum dots, tiny metal spheres, or even atoms.


� Some experiments aiming to achieve controlled nuclear fusion, which one-day may produce abundant and clean energy, also rely on lasers. A technique called laser induced fusion, in which light from lasers is used to raise the temperature of a deuterium and tritium pellet to 109°C (yes, 109 degrees Celsius) in order to induce fusion, is being tried in several laboratories.

The National Ignition Facility (NIF) laser bay at Lawrence Livermore National

Laboratory in California (http://images.google.com.au/imgres?imgurl=http://www.sandia.gov/ASC/images/library/)


Brief History

The history of the development of the laser is full of twists and turns, and is a very interesting read (see for example Laser Pioneers by Jeff Hecht, ISBN 0 12-336030-7, Academic Press, 1991 or How the laser happened by C. Townes, ISBN 0-19-512268-2, Oxford University Press, 1999). However, as much as we would like, in this course we do not have time to go into the historical details. We shall only list the major turning points of the ‘laser saga’ .

The basic physics needed for the laser was known by the end of the 1920s:

� The Fabry-Perot resonator was known since the early 1800s

� 1916 Einstein postulates stimulated emission

� 1928 Landenburg and Kopfermann experimentally verified the existence of stimulated emission

But it took until 1960 to actually make a laser.

THE MASER AND ITS ROLE IN THE INVENTION OF THE LASER The maser, you remember, stands for microwave amplification by stimulated emission of radiation. After the WWII lots of scientists had experience in microwaves because of their work during the war with radars.

1. The most important work that produced an operating maser was done at

Columbia University by Charles Townes and his students. But the idea did not come from Townes.

2. Joe Weber, an American physicist was the first to discuss amplification using stimulated emission in 1952. He was interested in microwave amplification for military applications. His paper produced some interest (he got invited to RCA to give a seminar). Townes asks for reprint, but in his later papers did not refer to Weber.

3. Townes wanted to build an oscillator, a device that emits microwaves, using stimulated emission. He, with students Jim Gordon and Herb Zeigler, designed a beam of ammonia molecules that they separated into two beams: one containing the excited molecules, and one containing the unexcited molecules. This could be done by applying an inhomogeneous electric field and, since the excited/ground state molecules have different electric dipoles, it was possible to separate them in space. The excited beam was then channelled into a resonant cavity (a metal box having dimensions that of the microwave wavelength, ie few centimetres). This resonant cavity selected


EM radiation of just the right frequency to stimulate the excited molecules to emit radiation.

Maser beam: An excerpt from Charles Townes's notebook.

(from How the Laser Happened) 4. They worked on this project for 2 years. Lots of money was spent. They

were told to give up, the project was not going anywhere. April 1953: “IT WORKS!”

5. They coined the name: MASER 6. Masers are very low noise, very high spectral purity devices. The military

loved it!

Charles Townes (left) and James P. Gordon display their maser

(from http://www.scienceclarified.com/scitech/Lasers/The-Development-of-Lasers.html) 7. The stability of the frequency was one part in 109. Townes realised that this

would work as a precise frequency standard or what we now call an atomic clock.


8. At more or less the same time, two physicists in the USSR, Basov and Prokhorov, also built a maser using a different molecule, CsF.

9. In 1956, Bloembergen in the USA, and Basov and Prokhorov in the USSR, suggested a different type of maser, made from a solid material instead of a gas.

10. The advantage was that they were more compact and, since they were based on magnetic materials, they could be tuned with an external magnetic field.

11. In 1957 Kikuchi in Japan made a maser using Al2O3 doped with Cr, ‘ruby’ . The ruby maser made all the other masers obsolete. Ruby was easy to get, strong and could be widely tuned with magnetic field.

12. But, masers were BIG. Large electro-magnets, liquid nitrogen dewar, (-196C, 77K), liquid helium dewar (-269C, 4K), inside a metal box, inside the crystal. Weighed 2 tons!

13. After the initial research interest, only very a few masers were used. Mainly by astronomers.

14. Masers did have a moment of glory: Arno Penzias and Robert Wilson from Bell Labs used a maser in 1965 to discover the 3K blackbody radiation from the Big Bang. (Penzias was a student of Townes). They were awarded the Nobel Prize in 1978.

15.

Penzias and Wilson in front of their maser amplifier.

(from: http://woodahl.physics.iupui.edu/Astro100/23-05.jpg)


THE OPTICAL MASER

1. Valentin Fabrikant in the USSR studied light emission from electrically

excited gasses. From 1939, he studied stimulated emission in electrically excited gases. In 1951 he patented “A method for the amplification of electromagnetic radiation using stimulated emission”. He showed that the intensity of radiation as it passes in a medium with inverted population will increase exponentially. He also discussed how to invert the population. Because of the cold war his work was unknown outside the USSR, and even in the USSR it was not taken seriously. (After the maser/laser were built, in 1965 he was awarded the medal of science by the government).

2. In 1957, Townes and his brother-in-law, Arthur Schawlow wrote a paper describing the concept of an “optical maser”. They discussed two major problems in going from maser to optical maser: resonant cavity and optical material. Townes and Schawlow obtained patents to protect their IP.

3. Their work created considerable interest and many labs started to search for possible materials and methods of making an optical maser. Schawlow, working at Bell labs, initially worked on ruby but after a few experiments decided against it: “it will not work”.

4. Also at Columbia was Gordon Gould, a grad student working with Polykarp Kusch (Nobel laureate). Towards 1957 he finished his experimental work, and was supposed to be writing his thesis. Instead, he worked on the optical maser, a topic that he heard about from Townes. He named it LASER. Although their offices were only a few meters apart, Townes and Gould did not know of each other’ s ideas.

5. Gould had an optics background, and came up with the idea of the optical cavity, and optical pumping. He wanted to patent his device. He misunderstood the patent layer, and thought he needs a working model of the laser to patent it. He did not get a patent until 1978! (More about this later). At the time (1957) he had his lab book signed by a notary public.

6. He knew he could not work on his laser at Columbia, so he left (without writing up his thesis) and joined a small consultancy group called Technical Research Group (TRG).

7. At TRG they were interested in his laser, and applied for a US Army grant $200k. They were awarded $1M, but with strings attached.

8. Gould had an initial lead over Townes and Schawlow who were just figuring out how to make a laser cavity. However, Townes and Schawlow 6 months later were awarded the optical maser patent. In 1964 Townes was awarded, together with Basov and Prokhorov, the Nobel Prize for the maser. (Schawlow was also awarded the Nobel Prize in 1981 for laser spectroscopy.) Gould was forgotten!


RACE TO MAKE THE FIRST LASER

1. Lots of groups jumped on the laser bandwagon. Everyone wanted to be the first to actually make a laser.

2. In the late 1950s smart money was on Bell Labs to win the ‘laser race’ . Schawlow was at Bell Labs and Townes was a consultant. Money at BL was plenty, and the brightest of the bright were hired every year.

3. At Bell Labs two directions were taken: (i) ruby crystal emitting at 694.3nm and (ii) the HeNe gas mixture. Townes and Schawlow were working on the ruby and Ali Javan was working on the HeNe gas laser.

4. Initially, people relied on incorrect data regarding quantum efficiency of ruby. It was published as 1% so Schawlow gave up on ruby.

5. At TRG, they also had heaps of money but Gould was not given security clearance! People at TRG were also working on a gas laser. Without Gould’ s active involvement, it was going slowly.

6. Meanwhile, back at the ranch in California, at the Hughes Research Labs, a little known physicist, Theodor Maiman (student of Will Lamb, Nobel laureate) was working on making masers smaller. And he did. After a while he got interested in lasers. He measured ruby’ s quantum efficiency again. He found it to be 99%!

7. He made a simple and effective laser cavity by evaporating the mirrors on the ruby, and using spiral flashlamp as the pump source. The ruby laser emits in the red: 694.3nm. (Nowadays only used for laser hair removal.)

8. When he wanted to publish his results (in Phys. Rev. Letters) it was rejected. He sent it to an English journal, Nature. It was published on the 6th of August, 1960. Most US scientists heard about it through the New York Times.

T. Maiman and his ruby laser (Nature Photonics 1, 372 - 373 (2007))


9. The ruby laser was a pulsed laser. The first CW HeNe laser was built at BL

by Ali Javan (student of Towne’ s) on the 12th of December, 1960.

Ali Javan and colleagues with the first HeNe laser

(from http://photonics.usask.ca/photos/images/Chapter%204/(4-01)Javan&Other.jpg)

HeNe lasers emitting in the red (632.8nm) and in the green (543nm)

(from http://img.directindustry.com/images_di/photo-g/helium-neon-laser-240520.jpg) The laser ‘patent-wars’ continued for 30 years. Initially Townes and co. were given the patents, which were withdrawn in the early 1970s, and Gould was given his


patents in 1977 and 1978. As it turns out, he made lots of money from this late recognition because there were lots more companies making lasers (and therefore paying royalties) by 1978 than there were in the 1960s. However to get this far, he had to sell a large chunk of his patent rights to the law firms that were fighting for his patents. And the law firms did have to fight because the laser manufacturers refused to recognize the Gould patents. (One day there may be a movie about this)

As you see, the development of the laser was tortuous. The allocation of credit for ‘inventing’ the laser is still a touchy subject. Townes and Schawlow have been greatly honoured by the scientific community, separately receiving Nobel Prizes in 1964 and 1981. Gould collected millions of dollars in royalties from his (belated) laser patents. Theodore Maiman, who actually made the first laser, had to be satisfied with being cited in laser textbooks and encyclopaedias. V.A. Fabrikant, on the other hand, who was probably the first person to think of ‘light amplification by stimulated emission radiation’ is hardly ever mentioned even in laser textbooks.

(Laser Pioneers by Jeff Hecht, ISBN 0 12-336030-7, Academic Press, 1991)


What is so special about the laser?

As we saw, lasers are used in a wide range of applications, from supermarkets to nuclear fusion, from laser guided missiles to dental ‘drills’ . Most of these applications rely on just a few properties of lasers, properties that we’ ll talk about in detail later on. Here we’ ll only run through them to show you that there is nothing mystical about lasers:

1. High optical powers can be generated by lasers, making them useful in applications where high energy density is required.

2. Collimation of the optical beam is another useful property of lasers. It means that light propagates large distances as a ‘parallel’ beam without diverging much. Typical angular spread for laser beams is of the order of 10-

3 rad, that is, the beam only spreads about a millimetre for every meter travelled.

3. Monochromaticity or spectral purity of the laser is essential in some applications, such as spectroscopy and remote sensing.

At this stage, you may wonder why we couldn’ t get ‘laser-like-light’ from an ordinary but powerful tungsten filament lamp by combining it with appropriate filters and apertures. For example, could we get monochromatic red light (�= 632.8nm) of similar brightness, spectral purity and collimation as the common 1 mW HeNe laser emitting at �= 632.8nm by using filters and collimators in combination with a strong reading lamp, as shown in the figure below?

Reading lamp

filters aperture

laser-like beam?

Fig. 1.1 Can we get laser-like light out of an ordinary light source with filters


The answer is a definite no. It can be shown (and you will be able to show it at the end of the next unit) that to generate light with the above-mentioned parameters we would need a lamp whose filament temperature is close to a million degrees Kelvin! Thus, the combination of spectral purity, collimation and power is unique to laser light sources.

Let’ s continue to overview the properties of lasers. As you’ ll see, these next two characteristics are unusual and can only be found in lasers, but at the same time, only very few applications rely on these two peculiar qualities:

4. Coherent light is also associated with lasers. Coherence is related to the phase stability of the electro-magnetic wave. In ordinary thermal light sources the phase of the emitted waves are quite random, while in lasers the phase of the waves is locked together for a certain distance, called the coherence length. Coherence is important in applications such as interferometry and holography.

5. Pulse length: Using lasers it is possible to produce very short optical pulses (without trying very hard). By relying on the properties of light waves in a resonant cavity, it is possible to produce light pulses that are only about 10fs long (one femtosecond is 10-15 seconds). These extremely short laser pulses are used to study the dynamics of atomic systems, and in non-linear optics.

Some general comments lasers:

� While these properties make lasers extraordinary as far as light sources go, the radiation coming from the laser is ‘ordinary’ electro-magnetic radiation. So there is no magical “ laser beam” . The difference is in the way the light is generated, not in the radiation itself.

� There are no magic laser materials either. Many solids, liquids and gases have been shown to ‘lase’ , including some very common materials, such as water (vapour), nitrogen gas, alcohol, etc.

� The size of the lasers vary from very small, such as some laser diodes whose largest dimension can be much smaller than a millimetre, to lasers used in nuclear fusion research that are the size of a three story building.


Optical radiation and optical modes

Blackbody

To understand laser emission we first have to know how light is generated by conventional light sources. In fact, the theory that explains how an ordinary tungsten filament lamp works is also the appropriate starting point for the laser. In describing the theory of lasers, we’ ll have to deal with three new concepts: density of optical modes, stimulated emission and population inversion. But to get to these concepts, we first have to describe an important but non-existing object, one that played a crucial role in the development of modern physics, including the laser. This (imaginary) object is the blackbody.

A blackbody is a perfectly ‘black’ object. It is an object that absorbs all radiation that falls on it. In addition, such a body would be independent of the materials used in making it. We could study the radiation emanating from such an object without having to worry about its material properties. As we’ ll soon see, such an object would be an ideal light source, the spectrum of which could be predicted by theory.

How can we make such a perfect black body? The design is very simple. Think about a block of material with a large void inside of it. We drill a small hole to connect the outside world with the void. Light entering the hole will suffer many reflections inside the cavity but at the end of the day (nearly all light) will be absorbed. That is, if the hole is small enough, only a negligibly small fraction of the radiation will be able to escape. The hole effectively acts like the surface of a perfectly black body. All light incident on it is absorbed.

The power absorbed by the walls of the void is converted to heat, which raises the temperature of the cavity. Therefore, in order for this object to be in thermal equilibrium, the energy it absorbs must either be carried away (eg by thermal conduction or convection) or be reemitted in the form of radiation. In a situation when the heat cannot be carried away because the object is thermally isolated, the blackbody will reradiate all the absorbed energy. That is, in thermal equilibrium, in the absence of thermal conduction and convection, absorption and emission of energy are exactly balanced. The radiation coming from a blackbody is called thermal or blackbody radiation. Thermal radiation explains the radiation from the sun, the light bulb, even the way modern digital thermometers measure the temperature of the human body.

Describing the shape and magnitude of the blackbody emission curve was a difficult problem at the turn of last century. Many leading physicist tried to explain the observed spectrum but were unsuccessful. One such model was developed by Rayleigh (and later corrected by Jeans) who suggested that, in line with standard thermodynamic reasoning, each degree of freedom in the blackbody be assigned an energy value of kT, where k is Boltzmann’ s constant and T represents the absolute


temperature. Multiplying the number of degrees of freedom with kT should give us the energy stored by the blackbody, and since absorption and emission are identical, we could calculate the radiation coming from such a cavity.

So the question of the radiation from a blackbody boils down to calculating the number of degrees of freedom. We’ ll try to work it out in a minute, but first let’ s look at the definition of a ‘degree of freedom’ in electro-magnetic (EM) theory. In EM theory, a degree of freedom represents a unique distribution of electric and magnetic fields within the given structure. Such a distribution is also called a mode. Using EM theory, it is not difficult to calculate the distribution of modes for simple structures. And that is all we need here. In doing so, we can also calculate the number of modes (or degrees of freedom) in the given structure, which is really what we are after in order to calculate the shape and magnitude of the blackbody emission curve.

Density of modes

To keep things simple, let’ s start with a one-dimensional void or cavity. Such a cavity consists of two mirrors separated by a distance L. Theory tells us that the EM waves that can exist in such a cavity are such that their amplitude is zero at the mirrors. That is, in this cavity the allowed modes are waves that have an integer number of half wavelengths that fit perfectly between the two mirrors. The wavelengths of such waves are given by

�m = 2L / m

where �m is the wavelength of the mth mode, L is the cavity length (distance between the mirrors) and m is an integer. It is customary to use spatial frequency, k = 2� / �, instead of wavelength when describing the modes. That is, in this one-dimensional cavity, the spatial frequencies of the allowed modes are given by

km = n� / L

These are called the longitudinal modes of this cavity.

We shall now calculate the number of modes (degrees of freedom) in a one-dimensional cavity. Obviously there are infinite number of modes, as ‘m’ can be any integer. It makes more sense to calculate the number of modes between 0 and a specific value of k, say kmax. Since the spacing between modes is �/L, the number of modes in this intervals is:

N = kmax / (�/L)

which, in terms of the optical frequency, � is

N = 2�nL/c

where we converted the wavelength � to frequency using � = cn /�, where n is the refractive index and c is the speed of light.


In the future we’ ll be more concerned with the density of modes, not the total number of modes. The density of modes, �(�), is defined as the number of modes per unit frequency per unit length of the cavity is given by:

�(�) = (1/L) (dN/d�) = (1/L) 2n/c = 2n/c modes /Hz m

So now we know how to calculate the density of modes in a one-dimensional cavity, so we could calculate the spectrum coming from a one-dimensional blackbody. But we are more interested in three-dimensional things, so let’ s look at the properties of a three dimensional cavity. For a three-dimensional (3D) cavity, the density of modes can be calculated using similar arguments to those above although the geometry of the problem is not as simple as in the one-dimension case. You’ ll find the derivation of this in your textbook, as well as the density of modes, which is given by:

�(�) = 8�n3�2 / c3 modes /Hz/m3

Blackbody emission spectrum

Now that we know how to calculate the number of modes (degrees of freedom) in 3D, we can get back to Rayleigh and Jeans’ model. You may remember, they predicted that the energy density from a blackbody at a given frequency is given by the mode density (degrees of freedom) times kT:

u(�) = 8�n3�2 / c3 x kT

where u(�) is the energy density (Joules per m3). But this result is wrong! It describes a spectrum, which is quite unlike the experimentally observed spectrum. It only matches the experimental results at long wavelengths (low frequencies) but diverges towards infinity at short wavelengths (high frequencies). That is, according to this model, all heated objects should be emitting infinite amount of energy at short wavelength, which of course makes no sense. So it’ s back to the drawing board. That is what Max Planck (probably) said in 1900 when he revisited at this problem.

Planck accepted the mode density arguments of Rayleigh and Jeans but rejected the assignment of kT energy per mode. Instead, Planck suggested his now famous hypothesis: the mode, at frequency �, could only have energy in integer multiples of h� where ‘h’ is a constant. (This was a radical revision of the physics of the day. Remember, we are in the early 1900s.) For a mode oscillating at a frequency �, he suggested, the energy could only be E = 0, h�, 2h�, 3h�, .. (We now know that h� represents the energy of a photon.) The average energy in such a mode is therefore given by

�∞

=0

)( dEEEpEave


where p(E) is the probability that the mode has energy E. Planck assumed that the energy in the modes followed Boltzmann statistics, which states that in a non-degenerate system at temperature T, the probability p(E) of a particle having energy E is given by the expression:

�∞

−

−

=

0

/

/

)(dEe

eEp

kTE

kTE

Since the value of E is an integer times h�, the above integral becomes a series (see details in the textbook) that can be shown to equal:

1/ −= kThave e

hE ν

ν

At low frequencies, that is for low energy photons (h� << kT), this formula reduces to Eave � kT. That is, at low frequencies the average energy per mode is about kT, just like classical physics predicts. This explains why the Rayleigh-Jeans model worked at low frequencies. At high frequencies, where h� >> kT, the average photon population is less than one per mode. Or in other words, when h� >> kT, the probability of finding modes oscillating at high frequencies is very small. (See the example in the textbook on page 178). So at high frequencies, the Eave � kT relationship is not true any more because there are hardly any modes excited at these high frequencies. Multiplying the mode density of a three-dimensional cavity by Planck’ s expression for the average energy per mode yields the correct formula for the energy density of the blackbody:

18

)( /3

2

−= kThe

hc

u ννπνν

The Planck’ s theory and the experimental results are in excellent agreement (see textbook for details).

The energy density, according to this formula, is plotted in Fig 1.2, along with the Rayleigh-Jeans law, for a blackbody at 57000C. As you can see, the R-J law only agrees with the Planck curve at low frequencies (long wavelengths) but is a total failure at high frequencies.


Just to reiterate, it was Planck’ s assumption of quantised energy that did the trick. This, of course, was the first step towards quantum mechanics, and the rest, as they say, is history. (Planck was awarded the Nobel Prize for his work in 1918.) Now read the chapter in the textbook that deals with the blackbody radiation and look at the various derivations in detail. You should understand the derivations, but you do not have to memorise them.

Fig. 1.2 Energy density as a function of wavelength of a blackbody radiator according to Planck and Rayleigh-Jeans.

Lasers and Applications - UNSW

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