THRESHOLD

In a laser there is not only an increase in the number of cavity photons because of stimu- lated emission but also a decrease because of loss effects. These include scattering and absorption of radiation at the mirrors, as well as the “output coupling” of radiation in the form of the usable laser beam. To sustain laser oscillation the stimulated amplification must be sufficient to overcome these losses. This sets a lower limit on the gain coefficient g(n), below which laser oscillation does not occur.

One thing we can do now is to predict, given the various losses that tend to diminish the intensity of radiation within the cavity, what minimum gain is necessary to achieve laser oscillation. The condition that the gain coefficient is greater than or equal to this lower limit is called thethreshold conditionfor laser oscillation.

Ordinarily, the scattering and absorption of radiation within the gain medium of active atoms is quite small compared to the loss occurring at the mirrors of the laser.

We will therefore consider in detail only the losses associated with the mirrors.

Figure 4.3 shows a stylized version of a laser resonator, that is, an empty space bounded

z=0 z=L

(r₂, t₂, s₂) (r₁, t₁, s₁)

I₊

I_–

Figure 4.3 The two oppositely propagating beams in a laser cavity.

4.3 THRESHOLD 143

on two sides by highly reflecting mirrors. A beam of intensityIincident upon one of these mirrors is transformed into a reflected beam of intensityrI, whereris thereflection coefficientof the mirror. A beam of intensitytI, wheretis thetransmission coefficient, passes through the mirror. We might expect from the law of conservation of energy that

rþt¼1, (4:3:1)

that is, the fraction of power reflected plus the fraction transmitted should be unity.

Actually, however, some of the incident beam power may be absorbed by the mirror, tending to raise its temperature. Or some of the incident beam may be scattered away because the mirror surface is not perfectly smooth. Thus, the law of conservation of energy takes the form

rþtþs¼1, (4:3:2)

wheresrepresents the fraction of the incident beam power that is absorbed or scattered by the mirror.

Each of the mirrors of Fig. 4.3 is characterized by a set of coefficientsr,t, ands. At the mirror atz¼Lwe have

I_n⁽⁾(L)¼r2I_n^(þ)(L), (4:3:3a) and similarly

I_n^(þ)(0)¼r₁I_n⁽⁾(0) (4:3:3b) for the mirror atz¼0. Equations (4.3.3) are boundary conditions that must be satisfied by the solution of the equations describing the propagation of intensity inside the laser cavity.

What are these equations? We are now interested only in steady-state, or continuous- wave (cw), laser oscillation. Near the threshold of laser oscillation the intracavity inten- sity is very small, and therefore Eq. (3.12.10) is applicable. For light propagating in the positivezdirection, therefore, we have

dI_n^(þ)

dz ¼g(n)I_n^(þ) (4:3:4a) near the threshold, wheregmay be taken to be constant. Light propagating in the nega- tive z direction sees the same gain medium and so satisfies a similar equation (see Problem 4.1):

dI_n⁽⁾

dz ¼ g(n)I_n⁽⁾: (4:3:4b) The solutions of these equations are

I_n^(þ)(z)¼I_n^(þ)(0)e^g(n)z (4:3:5a)

and

I_n⁽⁾(z)¼I_n⁽⁾(L)exp[g(n)(Lz)]: (4:3:5b) From (4.3.5a) we see that

I_n^(þ)(L)¼I_n^(þ)(0)e^g(n)L (4:3:6) at the right mirror (z¼L), and the left-going beam has intensity

I_n⁽⁾(0)¼I_n⁽⁾(L)e^g(n)L (4:3:7) at the left mirror (z¼0). In steady state the left-going beam has a fractionr1of itself reflected at the left mirror (atz¼0), and this fraction is just the right-going beam at z¼0. A similar consideration applies at the right mirror. Thus, we have

I^(þ)_n (0)¼r₁I_n⁽⁾(0)¼r₁e^g(n)LI_n⁽⁾(L)

¼r₁e^g(n)Lr₂I_n^(þ)(L)

¼r1r2e^g(n)LI_n^(þ)(0)e^g(n)L

¼r1r2e^2g(n)L

I_n^(þ)(0): (4:3:8) Similar manipulations, applied to any of the quantitiesI_n^(þ)(L),I_n⁽⁾(L), andI_n⁽⁾(0) lead to the same result. Therefore, ifI_n^(þ)(0) is not zero, we must have, at steady state,

r₁r₂e^2gL¼1: (4:3:9) The steady-state value of gain that allows (4.3.9) to be satisfied is also the value at which laser action begins. For smaller values there is net attenuation of In in the cavity. Thus, the value ofgthat satisfies (4.3.9) is labeledgtand called thethreshold gain:

gt ¼ 1 2Lln 1

r1r2

¼ 1

2Lln (r1r2): (4:3:10) This expression can be rewritten usefully in the common case thatr1r21. Then we definer1r2¼12x, orx¼12r1r2, and use the first term in the Taylor series expansion ln(12x) 2x, valid whenx 1, to obtain

g_t¼ 1

2L(1r₁r₂) (high reflectivities), (4:3:11) which is a satisfactory approximation to (4.3.10) ifr₁r₂.0.90. The difference between (4.3.11) and (4.3.10) is connected with the assumption that the intracavity field is spatially uniform: We will see in the next chapter that spatial uniformity is a good approximation when 12r₁r₂is small, that is, when the mirrors are highly reflecting.

4.3 THRESHOLD 145

Note that if we are given the mirror reflectivitiesr₁andr₂ and their separationL, and therefore the threshold gain, we can determine the population inversion necessary to achieve laser action from (3.12.6) and the atomic absorption (or stimulated emission) cross section.

Our derivation of (4.3.9) assumes that the gain medium fills the entire distanceL between the mirrors. This assumption is valid for many solid-state lasers in which the ends of the gain medium are polished and coated with reflecting material. In gas and liquid lasers, however, the gain medium is usually contained in a cell of lengthl, L that is not joined to the mirrors (Fig. 4.4). In this case the threshold condition is

gt¼ 1

2lln (r1r2) 1

2l(1r1r2) (high reflectivities): (4:3:12) The threshold condition (4.3.12) [or (4.3.10)] assumes that “loss” occurs only at the mirrors. This loss is associated with transmission through the mirrors, absorption by the mirrors, and scattering off the mirrors into nonlasing modes. Absorption and scattering are minimized as much as possible by using mirrors of high optical quality.

Transmission, of course, is necessary if there is to be any output from the laser.

Other losses might arise from scattering and absorption within the gain medium (from nearly resonant but nonlasing transitions). Such losses are usually small, but they are not difficult to account for in the threshold condition. Ifais the effective loss per unit length associated with these additional losses, then the threshold condition (4.3.12) is modified as follows:

gt ¼ 1

2lln(r1r2)þa: (4:3:13) For our purposes these “distributed losses” (i.e., losses not associated with the mirrors) may usually be ignored.

It is instructive at this point to consider an example. A typical 632.8-nm He – Ne laser might have a gain cell of lengthl¼50 cm and mirrors with reflectivitiesr1¼0.998 and r2¼0.980. Thus

g_t ¼ 1

2(50)ln(0:998)(0:980) cm¹¼2:210⁴cm¹ (4:3:14) is the threshold gain.

z=0

Gain cell l

z=L

Figure 4.4 A laser in which the gain medium does not fill the entire distanceLbetween the mirrors.

Using this value forgtand Eq. (3.12.6), we may calculate thethreshold population inversionnecessary to achieve lasing:

DN_t¼ N₂g₂ g₁N₁

t

¼ 8pg_t

l²AS(n)¼ g_t

s(n): (4:3:15) TheAcoefficient for the 632.8-nm transition in Ne is

A1:410⁶s¹: (4:3:16)

For T 400K and the Ne atomic weight M 20 g, we obtain from Table 3.1 the Doppler widthdnD1500 MHz. Thus,

S(n)6:310¹⁰ s (4:3:17) and

DN_t (8p)(2:210⁴cm¹)

(632810⁸cm)²(1:410⁶s¹)(6:310¹⁰s)

¼1:610⁹atoms=cm³: (4:3:18)

This is a lot of atoms, but it is nevertheless quite a small number compared to the total number of Ne atoms. For a (typical) Ne partial pressure of 0.2 Torr, the total

TABLE 4.1 Quantities and Formulas Related to Gain and Threshold The Gain Coefficient

g(n)¼ l²A

8pn² N2g2

g1

N1

S(n)

¼s(n) N2g2

g1

N1

l¼c

n¼wavelength of radiation

A¼EinsteinAcoefficient for spontaneous emission on the 2!1 transition n¼refractive index at wavelengthl

N2,N1¼number of atoms per unit volume in levels 2 and 1 g2,g1¼degeneracies of levels 2 and 1

S(n)¼lineshape function (Table 3:1)

Threshold Gain

gt¼1

2l ln(r1r2)þa 1

2l(1r1r2)þa l¼length of gain medium

r1,r2¼mirror reflectivities

a¼distributed loss per unit length

4.3 THRESHOLD 147

number of Ne atoms per cubic centimeter is [Eq. (3.8.20)] about 4.810¹⁵. Thus, the ratio of the threshold population inversion to the total density of atoms of the lasing species is only

DNt

N ¼ 1:610⁹ 4:810¹⁵¼1

310⁶: (4:3:19)

Sometimes the quantitye^glis called the gain, and expressed in decibels, that is, G_dB¼10 log₁₀(e^gl)¼10 log₁₀(10^0:434gl)¼4:34gl: (4:3:20) The threshold gain in our example is, thus,

(G_dB)_t¼(4:34)(2:210⁴cm¹) (50 cm)¼0:048 dB: (4:3:21) In the laser research literature gain is usually expressed in reciprocal centimeters, although the decibel is the preferred unit in fiber optics.

In Table 4.1 we collect the formulas and terms we have used in discussing gain and threshold.

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