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5 Transformation in Time: Pulse Compression and Pulse Expansion

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Propagation, Amplification, Frequency Conversion, Pulse Compression

FIG. 12.11. Experimental setup for pulse compression.

can define a group velocity, g , and group-velocity dispersion, GVD, as g D .d!=dˇ/!L [see

(8.6.26)] and GVD D .d 2 ˇ=d! 2 /!L [see Eq. (8.6.33)], respectively, where !L is the central

frequency of the beam spectrum. On the other hand, for an inhomogeneous medium such as

the prism-pair described in Sect. 8.6.4.3 or the grating-pair described below, a more useful

approach can be obtained if we let Ein / exp j.!t/ be the field of a monochromatic input

beam at frequency ! and Eout / exp jŒ!t

.!/ the field of the corresponding output beam.

For a pulsed input beam, one can then define a group delay, g , and a group-delay dispersion, GDD, of the medium as g D .d =d!/!L , [see Eq. (8.6.27)], and GDD D .d2 =d! 2 /!L ,

respectively.

12.5.1. Pulse Compression

An arrangement that is commonly used to compress ultrashort laser pulses is shown

schematically in Fig. 12.11. The pulse of a mode-locked laser, of sufficient power (in practice

a relatively modest peak-power of, e.g., Pp D 2 kW) and long time duration (e.g., p D 6 ps),

is sent through a single-mode silica optical fiber of suitable length (e.g., L D 3 m). The wavelength of the pulse (e.g.,

Š 590 nm) falls in the region of positive GVD of the fiber

( < 1.32 m, for non-dispersion-shifted fibers). After leaving the fiber, the output is collimated and passed through an optical system consisting of two identical gratings aligned

parallel to each other and whose tilt and spacing are appropriately chosen, as described below.

Under appropriate conditions, the output beam then consists of a light pulse with a much

shorter duration (e.g., p D 200 fs) than that of the input pulse and, hence, of much higher

peak power (e.g., Pp D 20 kW). Thus, the arrangement of Fig. 12.11 can readily provide a

large compression factor (e.g., 30 in the illustrated case) of the input pulse. The rather subtle

phenomena involved in this pulse compression scheme are discussed below..21, 22/

We start by considering what happens when the pulse propagates in the optical fiber.

First we recall that, due to the phenomenon of self-phase modulation, a light pulse of uniform

intensity profile, which travels a distance z in a material exhibiting the optical Kerr effect,

acquires a nonlinear phase term given by Eq. (8.6.38). In an optical fiber, however, the situation is somewhat more complicated due to the non-uniform transverse intensity profile of its

fundamental mode .EH11 /. In this case, it can be shown that the whole mode profile acquires

a phase term given by.22/

.t, z/ D !L t

!L n0

z

c

!L n2 P

z

cAeff

(12.5.1)

12.5

537

Transformation in Time: Pulse Compression and Pulse Expansion

FIG. 12.12. Time behavior (a) of the pulse intensity, and (b) of the pulse frequency, when propagating through a

single mode fiber of suitable length. The solid and dashed curves refer, respectively, to the cases of no group velocity

dispersion and positive group velocity dispersion in the fiber.

where n0 is the low-intensity refractive index, n2 is the coefficient of the nonlinear index of

the medium [see Eq. (8.6.23)], P D P.t, z/ is the power of the beam traveling in the fiber

and Aeff is a suitably defined effective area of the beam in the fiber. The instantaneous carrier

frequency of the light pulse is then obtained from Eq. (12.5.1) as

! .t, z/ D

@

D !L

@t

!L

@P

zn2

cAeff

@t

(12.5.2)

and it is seen to be linearly dependent on the negative time derivative of the corresponding

power, P. Thus, for a bell-shaped pulse as in Fig. 12.12a, the carrier frequency will vary with

time as indicated by the solid curve in Fig. 12.12b. Notice that, around the peak of the pulse,

the time behavior of the power can be described by a parabolic law and the instantaneous

carrier frequency then increases linearly with time (i.e., the pulse is said to show a positive

frequency chirp). Note however that the frequency chirp becomes negative after the pulse

inflection points, i.e., for t < tA or t > tB in Fig. 12.12b.

It should be noted that the physical situation described so far has neglected the presence

of GVD in the fiber. In the absence of GVD the pulse shape does not change with propagation

i.e. the field amplitude remains a function of the variable .z g t/, where g is the group velocity (see Appendix G). The z dependence of the pulse, at any given time, is then obtained from

the corresponding time dependence by reversing the positive direction of the axis and multiplying the time scale by g (see Fig. 12.12). This means that a point such as A of Fig. 12.12a

is actually in the leading edge while a point such as B is in the trailing edge of the pulse. Note

now that, according to Fig. 12.12b, the carrier frequency of the pulse around point A will be

smaller than at C, where it is roughly equal to !L . On the other hand, the carrier frequency of

the pulse around point B will be higher than at C.

Assume now that the fiber has a positive GVD. That part of the pulse, in the vicinity of

point A, will move faster than that corresponding to point C and this will, in turn, move faster

than the region around point B. This means that the central part of the pulse, while traveling in

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Propagation, Amplification, Frequency Conversion, Pulse Compression

FIG. 12.13. Plots of calculated values of (a) self-broadening and (b) self-phase-modulation of an initial 6-ps pulse

after propagating through 30 m of single-mode fiber with positive group velocity dispersion; (c) output pulse spectrum; (d) compressed pulse produced by an optical system having a negative group velocity dispersion with linear

dispersion (after ref.,.21/ by permission).

the fiber, will be expanded. Following a similar argument one sees that the outer parts of the

pulse will be compressed rather than expanded, because there the frequency chirp is negative.

Thus, when a positive GVD is considered, the actual shape of the pulse intensity as a function

of time, at a given z position, will look like the dashed curve of Fig. 12.12a. The corresponding

behavior for the frequency change will then be as shown by the dashed curve of Fig. 12.12b.

Note from Fig. 12.12a that, owing to the pulse broadening produced by the GVD, the peak

intensity of the dashed curve is lower than that of the solid curve. Note also that, since the

parabolic part of the pulse now extends over a wider region around the peak, the linear part of

the positive frequency chirp will now occur over a larger fraction of the pulse.

Having established these general features of the interplay between SPM and GVD, we are

now able to understand how, for a long enough fiber, the time behavior of the pulse amplitude

and of the pulse frequency, at the exit of the fiber, can actually develop into those shown in

Figs. 12.13a and 12.13b. Note that the pulse has been squared and expanded to a duration of

0

p Š 23 ps while the positive frequency chirp is linear with time over most of this light pulse

duration. The corresponding pulse spectrum is shown in Fig. 12.13c and one can see that,

due to the strong SPM occurring for such a small beam in the fiber (the core diameter for the

condition depicted in Fig. 12.13 was d Š 4 m), the spectral extension of the output pulse,

0

1

L D 50 cm , is considerably larger than that of the input pulse to the fiber. The latter is,

in fact, established by the inverse of the pulse duration and, for the case considered of p Š

1

6 ps, corresponds to

L Š 0.45= p Š 2.5 cm . This means that the bandwidth of the output

pulse is predominantly established by the phase modulation of the pulse rather than by the

duration of its envelope.

Suppose now that the pulse of Fig. 12.13a and 12.13b is passed through a medium of

negative GDD. With the help of a similar argument to that used in relation to Fig. 12.12, we

can now see that the region of the pulse around point A will move more slowly than that around

C and this in turn will move more slowly than that around B. This implies that the pulse will

12.5

539

Transformation in Time: Pulse Compression and Pulse Expansion

now be compressed. Let us next suppose that the GDD of the medium, besides being negative,

is also independent of frequency. According to Eq. (8.6.27), this means that the dispersion in

group-delay, d g =d!, will also be negative and independent of frequency. Thus g decreases

linearly with frequency and, since the frequency chirp of the pulse increases linearly with time

(see Fig. 12.13b), all points of the pulse of Fig. 12.13a will tend to be compressed together at

the same time if the GDD has the appropriate value. According to Eq. (8.6.31), this optimum

value of GDD must be such that

Â 2 Ã

d

!L0 D p0

(12.5.3)

d! 2 !L

0

0

where !L0 D 2

L is the total frequency sweep of the pulse of Fig. 12.13b and p is the

duration of the expanded pulse of Fig. 12.13a. It should be noted, however, that this compression mechanisms cannot produce an indefinitely sharp pulse as, at first sight, one may be led

to believe. In fact, the system providing the negative GDD is a linear medium and this implies

that the pulse spectrum must remain unchanged on passing through such a system. This means

that the spectrum of the compressed pulse still remains as that shown in Fig. 12.13c. Even

under optimal conditions, the duration of the compressed pulse, p00 , cannot then be shorter

than approximately the inverse of the spectral bandwidth, i.e., p00 Š 1= L0 Š 0.75 ps.

Since the time duration p of the pulse originally entering the optical fiber was 6 ps (see

Fig. 12.11a), the above result indicates that a sizable compression of the incoming light pulse

has been achieved.

The above heuristic discussion is based on the assumption that a chirped pulse can be

subdivided into different temporal regions with different carrier frequencies. Although this

idea is basically correct and allows a description of the phenomena in simple physical terms, a

more critical detailed examination of this approach would reveal some conceptual difficulties.

To validate this analysis, however, the analytical treatment of the problem can be performed

in a rather straightforward way although the intuitive physical picture of the phenomenon

gets somewhat obscured. For this analytical treatment, in fact, one merely takes the Fourier

transform, E! .!/, of the pulse of Figs. 12.13a and 12.13b and then multiplies, E! .!/, by the

transmission t.!/ of the medium exhibiting negative GDD. The resulting pulse, in the time

domain, is then obtained by taking the inverse Fourier transform of E.!/t.!/. Note that, for

a lossless medium, t.!/ must be represented by a pure phase term, i.e., it can be written as

t.!/ D exp. j /

(12.5.4)

where D .!/. If the medium has a constant GDD, the Taylor-series expansion of .!/

around the central carrier frequency !L gives

Â

Ã

Â Ã

1 d2

d

.! !L / C

.! !L /2

(12.5.5)

.!/ D .!L / C

d! !L

2 d! 2 !L

where .d =d!/!L is the group delay and .d2 =d! 2 /!L is the group-delay dispersion. By substituting Eq. (12.5.5) into Eq. (12.5.4) and taking the inverse Fourier transform of E.!/t.!/

Techniques of this type to produce shorter pulses by first imposing a linear frequency chirp followed by pulse

compression have been extensively used in the field of radar (chirped radars) since World War II.

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Propagation, Amplification, Frequency Conversion, Pulse Compression

FIG. 12.14. Grating-pair for pulse compression.

one then finds that, if .d2 =d! 2 /!L is negative and satisfies condition Eq. (12.5.3), the optimum pulse compression occurs. The optimally compressed pulse, calculated in this way, is

shown in Fig. 12.13d. The resulting pulse duration turns out to be p00 Š 0.6 ps, rather than the

approximate value (0.75 ps) estimated before.

We are now left with the problem of finding a suitable optical system that can provide

the required negative GDD. Notice that, since one can write GDD D d g =d!, a negative GDD

implies that the group delay must decrease with increasing !. As discussed in Sect. 8.6.4.3,

one such system consists of the two-prism couple shown in Fig. 8.26. Another such system

is the pair of parallel and identical gratings shown in Fig. 12.11..23/ To understand the main

properties of this last system we refer to Fig. 12.14, which shows a plane wave, represented

by the ray AB, incident on the grating 1 with a propagation direction at an angle Âi to the

grating normal. We now assume that the incident wave consists of two synchronous pulses,

at frequency !2 and !1 , with !2 > !1 . As a result of the grating dispersion, the two pulses

will then follow paths ABCD and ABC0 D0 , respectively, and one sees that the delay suffered

by the pulse at frequency !2 , d2 D ABCD= g , is smaller than that, d1 D ABC0 D0 = g , for the

pulse at frequency !1 . Since !2 > !1 , this means that the pulse delay dispersion is negative.

A detailed calculation then shows that the GDD can be expressed as.23/

GDD D

d2

D

d! 2

4 2c

! 3 d2 fŒ1

1

ŒsinÂi . =d/2 g3=2

Lg

(12.5.6)

where ! is the frequency of the wave, its wavelength, d the grating period, and Lg the

distance between the two gratings. Note the minus sign on the right-hand side of Eq. (12.5.6)

indicating indeed a negative GDD. Note also that the value of the dispersion can be changed

by changing Lg and/or the incidence angle Âi . It should lastly be observed that the two-grating

system shown in Fig. 12.14 has the drawback that a lateral walk-off is present in the output

beam, the amount of walk-off depending upon the difference in frequency between the beam

components (as it occurs e.g. between rays CD and C0 D0 ). For beams of finite size, this walkoff can represent a problem. This problem can however be circumvented by retroreflecting

the output beam back to itself by a plane mirror. In this case the overall dispersion resulting

12.5

Transformation in Time: Pulse Compression and Pulse Expansion

from the double pass through the diffraction-grating-pair is, of course, twice that given by

Eq. (12.5.6).

The system of Fig. 12.11 has been used to produce compression of both picosecond

and femtosecond laser pulses over a wide range of conditions..24/ For example, pulses of

6 ps duration (and 2 kW peak power), from a synchronously-pumped mode-locked dye

laser, have been compressed, using a 3-m-long fiber, to about 200 fs .Pp D 20 kW/. These

pulses have again been compressed, by a second system as in Fig. 12.11, using a 55-cmlong fiber, to optical pulses of 90 fs duration. One of the most interesting results achieved

involves the compression of 50 fs pulses, from a colliding-pulse mode-locked dye laser down

to 6 fs, using a 10-mm-long fiber..25/ To achieve this record value of pulse duration, for such

a configuration, second-order group-delay dispersion ŒGDD D .d 2 =d! 2 /!L , and third-order

group-delay dispersion ŒTOD D .d 3 =d! 3 /!L were compensated using both two consecutive

grating pairs (each pair as in Fig. 12.14) and a four-prism sequence as in Fig. 8.26. In fact, the

TOD of the two compression system could be arranged to be of opposite sign, so as to cancel

each other.

A limitation of the optical-fiber compression scheme of Fig. 12.11 arises from the small

diameter .d Š 5 m/ of the core of the fiber. Accordingly, the pulse energy that can be

launched into the fiber is necessarily limited to a low value . 10 nJ/. A recently introduced

guiding configuration to produce wide-bandwidth SPM spectra, uses a hollow-silica fiber

filled with noble gases (Kr, Ar) at high pressures (1–3 atm)..26/ With an inner diameter for the

hollow-fiber of 150–300 m, a much higher-energy input pulse . 2 mJ/ could be launched

into the fiber. Using a fiber length of 1 m, wide SPM spectra . 200 nm/ have been obtained

starting with input pulses of femtosecond duration .20–150 fs/. With the help of a specially

designed two-prism sequence, in a double-pass configuration, and also using two reflections

from a specially designed chirped mirror,.27/ 20 fs pulses from the amplified beam of a modelocked Ti:sapphire laser were compressed to 4.5 fs..28/ These pulses containing 1.5 cycle

of the carrier-frequency, are the shortest pulses generated to date and have a relative large

amount of energy . 100 J/.

12.5.2. Pulse Expansion

It was already pointed out in Sect. 12.3 that, for chirped-pulse amplification, one needs

first to subject the pulse to a large expansion in time. In principle this expansion can be

achieved by a single-mode fiber of suitable length (see Fig. 12.11 and Fig. 12.12a). However, the linear chirp, produced in this way (see Fig. 12.13b), cannot be exactly compensated

by a grating-pair compressor Fig. 12.14, due to the higher order dispersion exhibited by this

compressor. For pulses of short duration (subpicosecond), this system would thus only provide

a partial compression of the expanded pulse to its original shape. A much better solution.29/

involves using an expander which also consists of a grating-pair, but in an anti-parallel configuration and with a 1:1 inverting telescope between the two gratings, as shown in Fig. 12.15..30/

To achieve the desired positive GDD, the two gratings must be located outside the telescope

but within a focal length of the lens, i.e., one must have .s1 , s2 / < f , where f is the focal length

of each of the two lenses. In this case, under the ideal paraxial wave-propagation conditions

541

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12

Propagation, Amplification, Frequency Conversion, Pulse Compression

FIG. 12.15. Pulse expander consisting of two diffraction gratings, in an antiparallel configuration, with a 1:1

inverting telescope between them.

and negligible dispersion of the lens material, the GDD can be shown to be given by.30/

GDD D

d2

4 2c

.2f

D 3 2

2

d!

! d cos2 Â

s1

s2 /

(12.5.7)

where ! is the frequency of the wave, d is the grating period, and Â is the angle shown

in Fig. 12.15. Equation (12.5.7) indeed shows that one has, in this case, a positive value of

GDD. To understand this result we refer again to Fig. 12.15, where the plane wave incident on

the first grating, represented by the ray AB, is assumed to consist of two synchronous pulses,

at frequency !2 and !1 , with !2 > !1 . As a result of the grating dispersion, the two pulses

will then follow paths ABCD and ABC0 D0 , respectively, and one sees that the delay suffered

by the pulse at frequency !2 , d2 D ABCD= g , is now larger than that, d1 D ABC0 D0 = g , for

the pulse at frequency !1 . Since !2 > !1 , this means that the pulse delay dispersion is now

positive. It should be observed that the two-grating telescopic-system shown in Fig. 12.15

has the drawback that a lateral walk-off is present in output the beam, the amount of walk-off

depending upon the difference in frequency between the beam components (e.g., between rays

CD and C0 D0 ). For beams of finite size, this walk-off can represent a problem. This problem

can however be circumvented by retroreflecting the output beam back to itself by a plane

mirror. In this case the overall dispersion resulting from the double pass through the system

of Fig. 12.15 is, of course, twice that given by Eq. (12.5.7)

To compare the positive GDD of this pulse expander with the negative GDD of the

grating-pair of Fig. 12.14, we first remember that, according to the grating equation, one

has sin Âi . =d/ D sin Â 0 , where Âi is the angle of incidence at the grating and Â 0 is the corresponding diffraction angle. One can now substitute this grating equation into Eq. (12.5.6)

and compare the resulting expression with Eq. (12.5.7). One then readily sees that, if Â 0 D Â,

the two expressions become identical, apart from having opposite sign, provided that

.Lg =cos Â/ D 2f

s1

s2

(12.5.8)

It should be stressed that this equivalence only holds under the ideal conditions considered

above. In this case, and when Eq. (12.5.8) applies, the expander of Fig. 12.15 is said to be conjugate to the compressor of Fig. 12.14. Physically, the conjugate nature of this expander comes

about from the fact that the telescope produces an image of the first grating which is located

Problems

beyond the second grating and parallel to it. The expander of Fig. 12.15 is thus equivalent to

a two-parallel-grating system with negative separation and, under condition Eq. (12.5.8), this

system has exactly opposite dispersion, for all orders, to that of the compressor of Fig. 12.14.

In practice, due to lens aberrations and dispersion, the expander of Fig. 12.15 works well for

pulse durations larger than 100 fs and for expansion ratios less than a few thousands. For

shorter pulses and larger expansion ratios, the 1:1 telescope of the expander is usually realized

via a suitably designed cylindrical-.31/ or spherical-mirror configuration..32/ In particular, the

use of the cylindrical-mirror configuration has resulted in an expander with expansion ratio

greater than 104 and with second-, third-, and fourth-order dispersion being matched, for a

suitable choice of material dispersion in the amplification chain, to that of the compressor..31/

PROBLEMS

12.1. The Nd:YAG laser beam of example 12.1. is first propagated in free-space for a distance of

1 m, starting from its waist, and then focused by a positive lens with f D 10 cm focal length.

Calculate the waist position after the lens and the spot-size parameter at this waist. [Hint: To

calculate this waist position, the lens can be considered to consist of two positive lenses, f1 , and

f2 .f1 1 C f2 1 D f 1 /, the first lens compensating the curvature of the incoming wavefront, thus

producing a plane wave front, while the second focuses the beam: : : .]

12.2. The output of a Q-switched Nd:YAG laser .E D 100 mJ, p D 20 ns/ is to be amplified by a 6.3mm-diameter Nd:YAG amplifier having a small signal gain of G0 D 100. Assume that: (1) The

lifetime of the lower level of the transition is much shorter than p . (2) The beam transverse

intensity profile is uniform. (3) The effective peak cross-section for stimulated emission is Š

2.8 10 19 cm2 . Calculate the energy of the amplified pulse, the corresponding amplification,

and the fraction of the stored energy in the amplifier that is extracted by the incident pulse.

12.3. A large Nd: glass amplifier, to be used for amplifying 1-ns laser pulses for fusion experiments,

consists of a disk-amplifier with disk clear-aperture of D D 9 cm and overall length of the disks of

15 cm. Assume: (1) A measured small signal gain, G0 , for this amplifier of 4. (2) An effective,

stimulated-emission peak cross-section for Nd:glass of D 4 10 20 cm2 (see Table 9.3). (3)

That the lifetime of the lower level of the transition is much shorter than the laser pulse. Calculate

the total energy available in the amplifier and the required energy of the input-pulse to generate

an output energy of Eout D 450 J.

12.4. Following the analysis made in deriving Eqs. (12.3.1) and (12.3.9) (assume ˛ D 0) as well as

the rate-equation calculation for a quasi-three level laser [see Eqs. (7.2.21)–(7.2.24)], prove

Eq. (12.3.15).

12.5. With reference to problem 12.2., assume now that the input pulse duration is much shorter than

the lifetime, 1 , of the lower laser level . 1 Š 100 ps/. Using data obtained in example 2.10.

and knowing that the fractional population of the lower laser sub-level of the 4 I 11=2 state is

f13 Š 0.187, calculate the energy of the amplified pulse and the corresponding amplification.

Compare the results with those obtained in problem 12.2.

12.6. A large CO2 TEA amplifier (with a gas mixture CO2 :N2 :He in the proportion 3:1.4:l) has

dimensions of 10 10 100 cm. The small signal gain coefficient for the P.22/ transition

has been measured to be g D 4 10 2 cm 1 . The duration of the input light pulse is 200 ns,

543

544

12

Propagation, Amplification, Frequency Conversion, Pulse Compression

which can therefore be assumed much longer than the thermalization time of the rotational levels of both the upper and lower vibrational states. The laser pulse is however much shorter than

the decay time of the lower laser level. The true peak cross-section for the P.22/ transition is

Š 1.54 10 18 cm2 , while, for T D 300 K, the fractional population of both initial and final

rotational states can be calculated to be f D 0.07. Calculate the output energy and the gain available from this amplifier for an input energy of 17 J. Also calculate the energy per unit volume

available in the amplifier.

12.7. Prove Eq. (12.3.12).

12.8. The frequency of a Nd:YAG laser beam . D 1.06 m/ is to be doubled in a KDP crystal. Knowing that, for KDP, no . D 1.06 m/ D 1.507, no . D 532 nm/ D 1.5283, and

ne . D 532 nm/ D 1.48222, calculate the phase-matching angle, Âm .

12.9. Prove Eq. (12.4.30).

12.10. Calculate the threshold pump intensity for parametric oscillation at 1 Š 2 D 1 m in a

5-cm-long LiNbO3 crystal pumped at 3 D 0.5 m Œn1 D n2 D 2.16, n3 D 2.24, d Š 6

10 12 m=V, 1 D 2 D 2 10 2 . If the pumping beam is focused in the crystal to a spot of

100 m diameter, calculate the resulting threshold pump power.

12.11. Calculate the second-harmonic conversion efficiency via type I second-harmonic generation in

a perfectly phase-matched 2.5-cm-long KDP crystal for an incident beam at D 1.06 m having

an intensity of 100 MW=cm2 [for KDP n Š 1.5, deff D d36 sin Âm D 0.28 10 12 m=V, where

Âm Š 50ı is the phase-matching angle].

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Electron. Lett., 23, 1026–1028 (1987).

14. Emmanuel Desurvire, Erbium-Doped Fiber Amplifiers (John Wiley and Sons, New York, 1994).

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545

A

Semiclassical Treatment

of the Interaction of Radiation

with Matter

The calculation that follows will make use of the so-called semiclassical treatment of the

interaction of radiation with matter. In this treatment the atomic system is assumed to be quantized and it is therefore described quantum mechanically while the e.m. radiation is treated

classically, i.e., by using Maxwell’s equations.

We will first examine the phenomenon of absorption. We therefore consider the usual

two-level system where we assume that, at time t D 0, the atom is in its ground state 1 and

that a monochromatic e.m. wave at frequency ! made to interact with it. Classically, the atom

has an additional energy H 0 when interacting with the e.m. wave. For instance this may be

due to the interaction of the electric dipole moment of the atom μe with the electric field E

of the e.m. wave .H 0 D μe E/, referred to as as an electric dipole interaction. This is not

the only type of interaction through which the transition can occur, however. For instance,

the transition may result from the interaction of the magnetic dipole moment of the atom μm

with the magnetic field B of the e.m. wave (μm B, magnetic dipole interaction). To describe

the time evolution of this two-level system, we must now resort to quantum mechanics. Thus,

just as the classical treatment involves an interaction energy H 0 , so the quantum mechanical

approach introduces an interaction Hamiltonian H 0 . This Hamiltonian can be obtained from

the classical expression for H 0 according to the well-known rules of quantum mechanics. The

precise expression for H 0 need not concern us at this point, however. We only need to note

that H 0 is a sinusoidal function of time with frequency equal to that of the incident wave.

Accordingly we put

H 0 D H00 sin !t

(A.1)

The total Hamiltonian H for the atom can then be written as

H D H0 C H 0

(A.2)

547