Threshold Condition of Masers for Higher Frequencies

Koichi Shimoda

After a brief discussion on the quality factor of a parallel-plate interferometer, threshold conditions ofoscillation for a beam maser, a solid-state maser, and a gas maser are calculated from a unified theory.Operation of a gas maser on the line broadened by both collision and the Doppler effect is discussed. Therate of producing population inversion is calculated in terms of gas pressure, quantum efficiency, satura-tion parameter, and the intensity of excitation. Finally, the optimum condition is discussed.

1. Introduction

Interest in optical and infrared masers is increasingrapidly after successful operations of solid-state masersand more recently of a gas maser.' Basic considera-tions on high-frequency masers have been discussed bySchawlow and Townes.2 The aim of this paper is todevelop a unified theory3 of various types of masers andto discuss optimum operating conditions4 in a generalway.

The ordinary operation of a maser requires that atomsor molecules in a resonator should have the distributioncorresponding to a negative temperature or a populationinversion between two states. Of course there are afew types of masers which do not require populationinversion. An example of this type, using a multiplequantum transition in the conventional microwaverange, has been studied,5 but the application of suchmethods to higher frequencies is not included in thefollowing discussion.

11. Parallel-Plate Resonator

In the absence of loss between the plates, the qualityfactor Q of a parallel-plate interferometer with a separa-tion d is calculated from the power reflection coefficientR of each plate. The stored energy in the resonator de-cays according to the expression W = Wo exp(--ot/Q),which must be equal to the reflection loss for t = d/c:

/ o coo d dR = exp - -) z 1-

where co/2 7r = c/X is the frequency of oscillation. Thenthe quality factor can be expressed by

The author is associated with the Microwave Physics Labora-tory, Institute of Physical and Chemical Research, and Depart-ment of Physics, University of Tokyo, Bunkyo-ku, Tokyo, Japan.

Received 18 September 1961.

2irdX(1 - R)

(1)

For example, for X = 10-4 cm, R = 0.99, and d = 10 cm,Qo = 6.3 X 107.

The loss of power through the side openings may beevaluated from the diffraction of reflected light. Ac-cording to the standard theory of diffraction, the angleof diffraction 0 from the edge of the plate can be approx-imately given by

d(l - cos 0) ~-, or 0 = -2 7r' E7d

(2)

The loss of power Pd by diffraction for reflectors in theform of square plates of side dimension L or circularplates of diameter L can be expressed by the increase of1/Q as

1 Pd 2 |X

Qd woW ?rL w %d

Taking L = 1 cm, d = 10 cm, X = 10-4 cm, the aboveequation gives Qd 9 X 106, which is considerablysmaller than Qo from Eq. (1).

However, the above estimate cannot be applied to theresonator of a high-frequency maser for the followingreason. If the coherent length of radiation is muchlonger than d/(1 - R), the homogeneous distribution ofthe light amplitude over the surface of the reflector doesnot exist. The distribution must have a maximum atthe center and decrease near the edges. When theamplitude near the edge is small, the diffraction given byEq. (2) is reduced since it arises from the dominant in-terference of components coming from a narrow beltwithin (d/7r)/2 from the edge of the plates.

If the field distribution on a square reflector has aform sin(n7rx/L) -sin(mwry/L), where x and y extendfrom 0 to L, the spill-off power from the edges due to

May 1962 / Vol. 1, No. 3 / APPLIED OPTICS 303

diffraction is expressed by

1 x2

Qd 47rL2 ( + )(3)

It should be noted that a mode with n = 1 and m = 0exists in a rectangular cavity but any similar mode in aparallel-plate resonator does not exist. The lowestmode has a similar field distribution to the case m = n= 1, which gives the diffraction loss as

1 X2Qd :: -v (4)

* Excited LevelNA N,/ (Metastable)

Upper Level AC-

hvNA' f Induced Emmission

Lower Level _

NA NB

Initial (Ground) Level

Fig. 1. Energy levels of atoms A and B.

Ill. Threshold Condition for the Maser

The stored energy XV of the high-frequency field inthe resonator drops by an amount

AoP - (t1ol1 = Vo 2V (5)Q 4Q

in unit time, where E2 is the space average of the fieldintensity squared, V is the effective volume of the reso-nator, and W = (E2/87r) V.

The power delivered by atoms or molecules in a weakfield of E coscoot is calculated by a well-known method.6

The result for the case when the oscillation frequencyvP is close to the resonant frequency of atoms is givenhv3

(E,)U) 2 <Sin 26 6A,= nlhpo ( -~ 52>Ž (6)

where n is the number of excited molecules supplied inunit time, and t is the time of interaction with the radia-tion (between successive collisions in a gaseous case).The average in Eq. (6) is taken over the distribution ofthe resonant frequency of each atom v = co/2 ir, where

= (w - wo)t/2, and over the distribution of t.The component of the dipole matrix element p along

the direction of the field is given by g 2= 1/3 Al. Then

the threshold condition of maser oscillation is obtainedfrom A\P, > AP. The threshold rate of supply of atomsin the upper level is

3hV <sm212> -I

flti, = (7)

A. Beam-Type Maser

A beam of atoms or molecules is passed nearly parallelto the surface of a parallel-plate resonator in order toreduce the Doppler broadening of the line. When anatom in the beam has a transverse velocity vt normalto the plate, the Doppler effect gives a frequency shift ofvtvo/c and a can be expressed as

7rvoVtt v, LC v Xa

where L is the length of the resonator and v is the longi-tudinal velocity. Hence the line is narrowed in propor-

tion to the directivity of the beam.Beam-type masers can be used when the probability

of spontaneous emission /rs is negligible or TS >> L/v sothat t = L/v. When 114 < 7r, an approximation sin25/32

1 can be used. When 1 >> 7r, one obtains

< j2 >a 2v(vt)

where (t) is the average of the transverse velocity.Then the threshold flow of atoms in the upper state maybe given by'

(8)3hV v v

ntl=27r2jA2Q L 2L X ).

B. Solid-State Maser

The Doppler broadening is negligible in a solid-statemaser because the frequency of thermal vibration ismuch higher than the relaxation frequency so that theDoppler shift averages zero. In the case of an idealsolid, one may assume a = 0 and the half half-width ofthe line is Av = 1/(2 7rr). Then one obtains

1 t2' (t2)1" = 2 r2 .

The threshold rate of excitation is therefore given by

3hVntl, 8 7r

2.

2Q T2 (9)

and the necessary number of excess atoms in the upperstate is

3hVAPNol = ftiT = 42r/A2Q (10)

However, in an actual crystal, atoms spread over arange of frequencies, due to the internal stress and thelocal field in the crystal. On the plausible assumptionthat the distribution of the frequency of each atom isgiven by a Gaussian distribution, the calculation can beearried out in the same way as for the gas maser whichis described below.

The transverse relaxation time T2 in a solid and theDoppler broadening in a gas are responsible for dephas-

304 APPLIED OPTICS / Vol. 1, No. 3 / May 1962

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ing the high-frequency polarization of atoms or mole-cules. The longitudinal relaxation time T in a solidand the time interval of strong collisions in a gas de-populate the upper level.

C. Gas Maser

The spread of the frequencies of the atoms in a solidcan be treated as the Doppler broadening of the line ofatoms in a gas. In either case the relaxation of thepopulation of levels can be expressed by exp(-t/r).The distribution of the velocity component v along thedirection perpendicular to the surface of the wave isproportional to exp(-v2 /a2 ), where c = (2kT/M)'/2is the most probable velocity.

The threshold condition for masers in which the lineis broadened both by the Doppler effect and by colli-sions is discussed in the following: The collision is as-sumed to be strong. The weak collision which does notthermalize the levels but varies the phase of the radia-tion is neglected here for simplicity of discussion.

The factor in Eq. (6) averaged over-all atoms in thegas can be written as

Se i '2a = -2r ri si (7rvovt/c) e-v2

/a,2 e-t ldvdt.

< 2 av 7rt 0J 7rvV/C)'

(11)

After integration with respect to t, Eq. (11) may be re-written as4

Ksin >v = >-2j. P e±+' dx

where

C

= 2 -aa voT

is proportional to the gas pressure.Then, the threshold rate of supplying atoms in the up-

per stage is obtained by substituting Eq. (12) into Eq.(7) as

ll = 223hV (apo)2 pexp( p2)"t j2'2Q \) _J 1(p) (13)

where

F(p) = 2 ef_2 dx

is the error integral. Asymptotic solutions for cases ofhigh pressure and of low pressure are easily found.4 Itis evident, however, that the optimum operation of amaser takes place under al intermediate pressure.

IV. Rate of Excitation

There are a variety of methods of producing popula-tion inversion. Typical methods are state-separationof atoms or molecules in the beam, optical pumping,collision excitation of the second kind by excited atoms,chemical reaction, etc. Depopulation of the lower stateplays an important role in some cases (e.g., the beam-type maser), while in most other cases the selective ex-citation of atoms to a particular upper level is essential.In such cases the rate of excitation can be expressed inthe common form

= 4T

2p e' e e-dx, (12) neX = + I

ifp, 1+ p-' (14)

>~e ' e'

2p no

nth

p nn

nex

1.0

May 1962 / Vol. 1, No. 3 / APPLIED OPTICS 305

0.5

04

0.3

0.2

0.1

0

Fig. 2. Graphical representation of p2no/nth and p2no/nex versus p2.

where I is the number of quanta of the input power forexcitation, y is the quantum efficiency, and i3 is a satura-

tion parameter.It is known that the rate of excitation by absorption

of a coherent electromagnetic radiation is given in theform of Eq. (14). It is shown here that the same ex-pression is applicable to the excitation by collision ofthe second kind or by chemical reaction. Consider thatthe working atom A is excited by the other energeticparticle B, and the increase of the excited atom A iswritten by

- = NANB*WO - NA*NBWd NA- (15)dt TA

where w, and Wd are the probabilities of excitation andde-excitation respectively, and Ta is the relaxation timeof the atom A. For effective performance of the maserthe case of W» >> Wd is desired, so that the term with Wd

in Eq. (15) may be neglected. If it is not negligible, byassuming that NB is proportional to the gas pressure,one may write

1NBWd - WA

ad 1 =:1 1 and ±=- 7 .T TuA T8A

Then the rate of exciting the atom A to its upper state isgiven by

NANB*w= -* (16)

The change of population of the initial level (usuallythe ground level) of the atom A is given by

dNA NA(o) -NA

dt_ =NANB*W. + A (17)idl T

where NA(0) is the population of the atom A in the ab-sence of excitation. If the mean life of B* is written asTB, the number of quanta of the net input power for ex-citation is I = NB*/TB. Then the solution of Eq. (17)for the stationary state in the presence of excitation issubstituted into Eq. (16) to obtain the rate of excita-tion

NA(O)wCI TBn,, = NAWCIT = A+ WITB

1 + WeITgr(18)

In some cases it is reasonable to assume that TB isproportional to T and NA(0) is proportional to T-1.

Then Eq. (18) is reduced to Eq. (14). However, theabove assumption does not hold when the relaxationtime r of atom A is primarily determined by the spon-taneous emission which is of course independent of thecollision.

V. Optimum Pressure for Maser

Now the condition for the maser-type amplification oroscillation is obtained from Eqs. (13) and (14) or (18).It may be expressed as the supplying of a sufficient

amount of excess population of the upper state over thelower state to which the atom makes a transitionthrough the process of induced emission.

For simplicity of discussion, it may be assumed thatthe excitation to the lower state occurs in proportion tothat to the upper state. The thermal excitation is con-sidered to be small in the high-frequency maser. t Thenthe effective rate of excitation is written as (1 -r)ne.and the condition is expressed by

(1 - r)ne_ > nth. (19)

This condition can be rewritten from Eqs. (13) and(14) in the form

(20)(1-r) (6 +-) < p e.I

where no is a constant given by

3hV (, o'\ 2oV=2 -2 Q V C ) -

The right-hand term of (20) is p2no/nth, and is showngraphically in Fig. 2 as a function of p

2. The left-

hand term, p2nO/n 0 ,, is shown by a straight line. Theslope of the line is (1 - r) no/-yI and the intercept of theordinate is equal to (1 - r)no f/-y. Maser oscillationoccurs in the region where the straight line lies below thecurve. The minimum input power of excitation is ofcourse given by a tangent which is shown by a brokenline in Fig. 2.

Discussions of the performance for particular cases arehere, but it can be concluded from Fig. 2 that when(1 - r) * no f/y is small, the maser can be operated at asmall input power and at a low pressure. As the value of(1 - r)no 0/-y increases, larger gas pressure is requiredand the minimum input power increases. From Eqs.(18) and (21) the magnitude of the intercept is given by

noI3 3hV(1 - r) 17 =8,r'/2g2QCA

where CA = NA(O)TA. Since this is independent of We,

the optimum pressure does not depend on We. The slopeis

=Y WCATBI

which depends on We. Hence the required excitingpower varies in proportion to We-'.

References1. A. Javan, W. R. Bennett, Jr., and D. R. Herriott, Phys. Rev.

Letters 6, 106 (1961).

t If thermal excitation to the lower state is important, the ef-fective rate of excitation nex necessary for the maser is given by

noun - N'A > nthT,where N'A is the population of the lower state.

306 APPLIED OPTICS'/ Vol. 1, No. 3 / May 1962

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2. A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 5. T. Yajima, J. Phys. Soc. Japan 16 (1594); ibid. No. 9(1958). (1961).

3. K. Shimoda, Sci. Papers Inst. Phys. Chem. Research (Tokyo) 6. K. Shimoda, T. C. Wang, and C. H. Townes, Phys. Rev. 102,55, 1 (1961).. 102,

4. K. Shimoda, Sci. Papers Inst. Phys. Chem. Research (Tokyo) 1308 (1956).55, No. 3, (1961), to be published. 7. R. Karplus and J. Schwinger, Phys. Rev. 73, 1020 (1948).

~~M~< Z

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