atomic and laser physics

overview
laser = "light amplification by stimulated emission of radiation"
this is a bit of a misnomer. a laser is actually an oscillator rather than a simple amplifier. the difference is that an oscillator has positive feedback in addition to the amplifier. "light" is understood in a general sense of electromagnetic radiation with wavelength around 1 micron. thus one can have infrared, visible or ultraviolet lasers.
milestones in the history of lasers:
1917
einstein s treatment of stimulated emission.
1951
development of the maser by c.h. townes.
the maser is basically the same idea as the laser, only it works at microwave frequencies.
1958
proposal by c.h. townes and a.l. schawlow that the maser concept could be extended to optical frequencies.
1960
t.h. maiman at hughes laboratories reports the first laser: the pulsed ruby laser.
1961
the first continuous wave laser is reported (the helium neon laser).
1964
nicolay basov, charlie townes and aleksandr prokhorov get the nobel prize for “fundamental work in the field of quantum electronics, which has led to the construction of oscillators and amplifiers based on the maser-laser principle.”
1981
art schalow and nicolaas bloembergen get the nobel prize for “their contribution to the development of laser spectroscopy.”
1997
steven chu, claude cohen-tannoudji and william d. phillips get the nobel prize for the “development of methods to cool and trap atoms with laser light.”
2005
john hall and theodor h?nsch receive the nobel prize for “their contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique”.
types of laser
lasers come in many shapes and sizes. they are classified by various criteria:
• gain medium is solid, liquid or gas
• wavelength is in the infrared, visible or ultraviolet spectral region
• mode of operation is continuous or pulsed
• wavelength is fixed or tuneable.
the present state of the art includes:
• peak powers > 1012 w
• pulses shorter than 10?15 s
• cheap, efficient diode lasers available at blue (400 nm), red (620–670nm), and near-infrared wavelengths (700–1600 nm)
• other readily-available fixed-wavelengths include:
?? infrared: co2 (10.6 ?m), erbium (1.55 ?m), nd:yag (1.064 ?m), nd:glass (1.054 ?m)
?? visible: ruby (693nm), kr+ (676, 647 nm), hene (633 nm), cu (578 nm), doubled nd:yag (532 nm), ar+ (514, 488 nm), hecd (442 nm)
?? ultraviolet: ar+ (364, 351 nm), tripled nd:yag (355 nm), n2 (337 nm) hecd (325 nm), quadrupled nd:yag (266 nm), excimer (308, 248, 193, 150 nm)
3
• tuneable lasers:
?? dye (typical tuning range ~ 100 nm, dyes available from uv to near infrared)
?? ti: sapphire (700–1000 nm, doubled: 350–500nm)
?? free electron (far infrared to ultraviolet).
properties of laser light
1. monochromatic.
the emission of the laser generally corresponds to just one of the atomic transitions of the gain medium, in contrast to discharge lamps, which emit on all the transitions. the spectral line width can be much smaller than that of the atomic transition. this is because the emission is affected by the optical cavity. in certain cases, the laser can be made to operate on just one of the modes of the cavity. since the q of the cavity is generally rather large, the mode is usually much narrower than the atomic transition, and the spectral line width is orders of magnitude smaller than the atomic transition. this is particularly useful for high resolution spectroscopy and applications such as interferometry and holography that require high coherence (see below).
2. coherence.
in discussing the coherence of an optical beam, we must distinguish between spatial and temporal coherence. laser beams have a high degree of both.
spatial coherence refers to whether there are irregularities in the optical phase in a cross-sectional slice of the beam.
temporal coherence refers to the time duration over which the phase of the beam is well defined. in general, the temporal coherence time tc is given by the reciprocal of the spectral line width ??. thus the coherence length lc is given by: ??==1ctclcc (1.1)
typical values of the coherence length for a number of light sources are given in table 1.1. the figures explain why it is much easier to do interference experiments with a laser than with a discharge lamp. if the path difference exceeds lc you will not get interference fringes, because the light is incoherent. in the case of the single mode hene laser, you can set up an interferometer in which the path lengths differ by 300 m, and you will still observe fringes. the long coherence length of laser light is useful in holography and interferometry.
source
spectral line width
?v (hz)
coherence time
tc (s)
coherence length
lc
sodium discharge lamp
(d-lines at 589nm)
5 × 1011
2 × 10-12
0.6 mm
multi-mode hene laser
632.8nm line
1.5 × 109
6 × 10-10
20 cm
single-mode hene laser
632.8nm line
1 × 106
1 × 10-6
300 m
table 1.1: coherence length of several light sources.
3. directionality.
this is perhaps the most obvious aspect of a laser beam: the light comes out as a highly directional beam. this contrasts with light bulbs and discharge lamps, in which the light is emitted in all directions. the directionality is a consequence of the cavity.
4. brightness.
the brightness of lasers arises from two factors. first of all, the fact that the light is emitted in a well-defined beam means that the power per unit area is very high, even though the total amount of
4
power can be rather low. then we must consider that all the energy is concentrated within the narrow spectrum of the active atomic transition. this means that the spectral brightness (i.e. the intensity in the beam divided by the width of the emission line) is even higher in comparison with a white light source like a light bulb. for example, the spectral brightness of a 1 mw laser beam could easily be millions of time greater than that of a 100 w light bulb.
properties 1-4 are common to all lasers. there is a fifth property which is found in only some:
5. ultrashort pulse generation.
lasers can be made to operate continuously or in pulses. the time duration of the pulses tp is linked to the spectral band width of the laser light ?? by the "uncertainty" product ?t ?? ~1: ??> 1~pt . (1.2)
this follows from taking the fourier transform of a pulse of duration tp. as an example, the bandwidth of the 632.8 nm line in the hene laser is 1.5 ghz (see table 1.1 above), so that the shortest pulses that a hene can produce would be 0.67 ns long. this is not particularly short by modern standards. dye lasers typically have gain bandwidths greater than 1013 hz, and can be used to generate pulses shorter than 100 fs (1 fs = 10-15 s). this is achieved by a technique called "mode-locking". the present world record for a light pulse is less than 1 fs. these short pulsed lasers are very useful for studying fast processes in physics, chemistry and biology, and are also being developed for high data rate transmission using optical fibre networks.
essential operating principles of a laser
light amplification + positive optical feedback
light amplification is achieved by stimulated emission. (see below.) ordinary optical materials do not amplify light. instead, they tend to absorb or scatter the light, so that the light intensity out of the medium is less than the intensity that went in. to get amplification you have to drive the material into a non-equilibrium state by pumping energy into it. the amplification of the medium is determined by the gain coefficient ?, which is defined by the following equation:
dixidxxixidxxi+=+=+)()()()(? (1.3)
where i(x) represents the intensity at a point x within the gain medium. the differential equation can be solved as follows: xixiixidxidi???e)0()(dd=?== (1.4)
thus the intensity grows exponentially within the gain medium.
positive optical feedback is achieved by inserting the amplifying medium inside a resonant cavity:
5
gainmediumhighreflectoroutputcouplerpower supplylightoutput
light in the cavity passes through the gain medium and is amplified. it then bounces off the end mirrors and passes through the gain medium again, getting amplified further. this process repeats itself until a stable equilibrium condition is achieved when the total round trip gain balances all the losses in the cavity. under these conditions the laser will oscillate. the condition for oscillation is thus:
round-trip gain = round-trip loss
the losses in the cavity fall into two categories: useful, and useless. the useful loss comes from the output coupling. one of the mirrors (called the "output coupler") has reflectivity less than unity, and allows some of the light oscillating around the cavity to be transmitted as the output of the laser. the value of the transmission is chosen to maximise the output power. if the transmission is too low, very little of the light inside the cavity can escape, and thus we get very little output power. on the other hand, if the transmission is too high, there may not be enough gain to sustain oscillation, and there would be no output power. the optimum value is somewhere between these two extremes. useless losses arise from absorption in the optical components (including the laser medium), scattering, and the imperfect reflectivity of the other mirror (the "high reflector"). taking into account the fact that the light passes twice through the gain medium during a round trip, the condition for oscillation in a laser with small gain and losses can be written:
211?lrrochr=?+?+()() scattering losses +absorption losses, (1.5)
where l is the length of the gain medium, roc is the reflectivity of the output coupler and rhr is the reflectivity of the high reflector. when the gain and losses are large (i.e. > 10%), the more general version of the oscillation condition must be used:
errelochrl221?????=? , (1.6)
where the losses due to scattering and absorption are lumped into the distributed loss coefficient ?.
in general we expect the gain to increase as we pump more energy into the laser medium. at low pump powers, the gain will be small, and there will be insufficient gain to reach the oscillation condition. the laser will not start to oscillate until there is enough gain to overcome all the losses. this implies that the laser will have a threshold in terms of the pump power.
stimulated emission
in atomic physics notes 3, we considered the spontaneous tendency for atoms in excited states to emit radiation. we now consider the optical transitions that occur when the atom is subjected to electromagnetic radiation with its frequency resonant with the energy difference of the two levels. we follow the treatment of einstein (1917).
in addition to spontaneous transitions from the upper to the lower level, there will also be:
• absorption of photons causing transitions from level 1 up to level 2
• stimulated emission in which atoms in level 2 droping to level 1 induced by the incident radiation.
6