theory of superfluorescent fiber lasers

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-4, NO. 1 1 , NOVEMBER 1986 1631

Theory of Superfluorescent Fiber Lasers MICHEL J. F. DIGONNET

Abstrct-We report a theoretical ,analysis of superfluorescence in short active fiber devices for fiber system applications requiring broad- band light sources. Using a mode overlap approach, we derive simple expressions for the threshold and energy conversion efficiency of this new class of devices, and study the effect of the fiber V-number and internal loss on their overall performance. We show that near single- mode Nd : YAG single crystal fibers pumped near 810 nm, possibly with a high-power laser diode, are anticipated to exhibit thresholds on the order of a few milliwatts, conversion efficiencies in excess 35 percent, and power-independent bandwidths of several nanometers.

I. INTRODUCTION

S UPERFLUORESCENT SOURCES are needed as low temporal coherence sources in many fiber sensor ap-

plications, in particular fiber gyroscopes [ 11, and in some signal-processing fiber systems [2]. The only miniature superfluorescent sources currently available are semiconductor devices. However, commercially available superluminescent diodes exhibit a short lifetime, a poor wavelength stability, and a relatively low output power. Cou- pling to a single-mode fiber also is hindered by their poor spatial coherence [3]. An alternate possibility is the use of high gain fibers optically pumped to a sufficiently high level to generate a significant superfluorescent output via amplified spontaneous emission (ASE). To date, very limited work has been done toward the -development of such devices, primarily because of the lack of suitable laser materials in a fiber form. Large ASE outputs were observed in a RhB dye-doped thin film [4]. Single-mode glass fibers doped with an active ion appear to be good candidates for superfluorescence studies in the future as demonstrated by the high optical gains that they can provide [5]. The recent development of single crystal fibers also opens a new realm of possibilities, including the prospect of implementing high gain materials such as Nd : YAG [6], [7] or Nd : LiNb03 [8] in a fiber form. In a typical configuration, these devices would be made of a short length of fiber (0.2-2 cm) end-pumped with a source of suitable wavelength, particularly one or more laser diodes (LD) for compact practical devices.

As a preliminary step toward the development of superfluorescent fiber lasers (SFL), we describe in this pa- per a theoretical analysis of ASE in laser fibers with a view toward characterizing their anticipated behavior. The

Manuscript received January 1 , 1986; revised April 15, 1986. This work was supported in part by the Air Force under contract F33515-82-C-1749.

The author is with Litton Systems Inc., Chatsworth, CA, and the Ed- ward L. Ginzton Laboratoy, W. W. Hansen Laboratories of Physics, Stan- ford University, Stanford, CA 94305.

IEEE Log Number 86095 18.

quantities of interest include the definition and evaluation of a threshold, the estimation of the output power and conversion efficiency, and the frequency spectrum characteristics. In particular, it was important to study the fea- sibility of pumping these devices with a laser diode. The following analysis looks at both single-mode and multimode fibers, and includes the effects of mode confinement and fiber geometry on the device characteristics. In Sec- tion I1 we present the general formalism of mode overlap used to solve the rate equations in a circular guided me- dium. Formal expressions for the gain, the fluorescence output, and their relationship are derived. In Section I11 these results are applied to specific types of fibers to evaluate their potential as a SFL. We define their characteristics as a function of the fiber V-number, or fiber core size, to design an optimum device configuration.

11. GENERAL TREATMENT We consider a step-index fiber of length 1, made of an

active core of radius a and index n1 , surrounded by a pas- sive cladding of index n2. The numerical aperture is assumed to be small enough that the fiber supports linearly polarized guided modes of the LP,, type [9]. In a cylin- drical system of coordinates ( r , 4, z ) attached to the fiber at its input face, the spatial intensity distribution of the LP,, modes are labeled s,,(Y, r$), normalized to unity across the infinite transverse section of the fiber [9]. This device is end-pumped by an optical beam coupled at z = 0, with a wavelength X, corresponding to an absorption line of the laser material. The fiber ends are polished and antireflection (AR) coated to prevent the onset of resonant oscillation.

Let n+(v, z ) and n;(v, z ) be the photon populations in fiber mode i, between frequencies v and v + dv, between positions z and z + dz, in the forward ( z = 0 to z = I ) and backward ( z = I to z = 0) directions, respectively. They satisfy the following equations of evolution [lo], [l l]:

(1b)

yi ( v , z ) is the optical gain factor at frequency v and position z , while c y j is the loss factor of mode i. The '1' and

0733-8724/86/1100-1631$01.00 O 1986 IEEE

1632 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-4, NO. 11, NOVEMBER 1986

‘ni’ terms account for spontaneous and stimulated emis- sulting from this approximation appears also in the nu- sion, respectively. From the laser rate equations the gain merator (see (2)); in the superfluorescent regime these two factor yi may be expressed as a function of a spatial over- contributions cancel out and the accuracy of the approxi- lap between the signal mode si(r, 4) and the normalized mation is greatly improved. pump intensity distribution rp(r, 4) [12]

u(v) and rf are the stimulated emission cross section and the fluorescence lifetime of the material, respectively, and hv, is the pump photon energy. The denominator ex- presses saturation in a homogeneously broadened material, and involves the N guided modes of fiber and all frequencies. The quantity CY, Pp e-(ffa+crp)Z is the fractional amount of input pump power Pp absorbed between z and z + dz, where a, is the material absorption coefficient at Ap, and ap is the loss coefficient of the fiber at the pump frequency accounting for all loss mechanisms other than the resonant absorption described by CY,.

Because of the coupling between the spatial and frequency dependences via the saturation term, (1) is difficult to solve exactly in the superfluorescent regime, and some assumptions and approximations must be made to proceed. The first assumption concerns the fluorescence spectrum. In general u(u) , and therefore the fluorescence spectrum, are complex functions of frequency, and the exact solutions for n: ( v , z ) can only be attained through cumbersome numerical analysis. For the sake of simplicity, and to retain the physical character of the problem, we shall study the case of a single laser transition, chosen to possess a Lorentzian lineshape g ( v ) , normalized, en- tered at v,, and of width Av,. The stimulated emission cross section is then [13]

It is implied here that both states of polarization (SOP) of a given mode experience the same gain regardless of the pump SOP, as will be clarified later on by a careful verification of energy conservation.

The second approximation concerns the computation of the frequency integral involved in (2). At high pump levels, saturation may be important at specific locations along the fiber, as will be described in a later section. At such locations the frequency distributions of n: ( v , z ) may be assumed to be much narrower than the unsaturated linewidth of the transition Au, as a result of the frequency selective stimulated emission process. Thus in the saturation term, u( v ) may be replaced by its value at linecenter a, = a(v,) [lo]. This approximation also can be applied in unsaturated regions, where an error in the saturation term has no effect. Note that the error in the integral re-

With these approximations, combining (1) and (2) and integrating over frequency yields

dP 0, Tf

‘ dz - hVP a p - = + - CY p ,-(aa+olp)z (Po + PiF (z))

’ j r [, N rp(r, 4) si(r, +)r dr d4

p: ( z ) + PJ7 ( z ) 1 + sj(r, 4)

j = O Is,,

T CYiP?(Z) (4)

where hv, is the (average) fluorescence photon energy, Zsat = hv,/u, rf is the saturation intensity at linecenter, and Po is the power associated with one photon in the gain band- width

Po = hv,s - VS

2 . (5 )

The new variables in (4) are the forward and backward fluorescence power PLT ( z ) , defined by

P‘(z) = hv, s n:(v, z ) du. (6) --m

In the lossless case for which closed-form expressions are seeked, the ratio of (4) for the counterpropagating waves can be integrated along z to yield

(Po + P‘ (2)) (pO + Pi ( z ) ) = constant

= Po(P, + P+(Z))

= PO(P0 + p i (0)) (7) which shows that the forward and backward output powers are identical, as expected in a lossless device. Equa- tion (7) provides a useful means of decoupling (4) in the lossless case.

In the most general situation of a multimode fiber excited with an arbitrary pump distribution, all guided modes experience different gains and grow in intensity at different rates. As the SFL is pumped above threshold (which is different for different modes), the highest gain modes become strongly dominant. The output mode distribution is a function of pump power given by the solutions of (4)

DIGONNET: THEORY OF SUPERFLUORESCENT FIBER LASERS 1633

for i = 1 to N . However, these equations are coupled through their saturation terms and are difficult to solve for a fiber supporting more than a few modes.

To illustrate the magnitude of this mode selection effect, it is useful to relate the fluorescence power to the modal gain. In the lossless case, (4) can be integrated along z straightforwardly to yield

p+ (2) = po(,gi(vsJ) - 1) (8)

where gi ( v s , z) is the gain at linecenter ( v = v,)

gi(v, Z) = Y ~ ( v , 2’) d.’. s: (9)

This result simply states that the fluorescence signal per mode is equivalent to the amplification of one photon emitted at the fiber input (or output for the backward wave, for which a similar relation holds).

The mode selection effect was evaluated in an SFL in the unsaturated regime, i.e., below threshold for the highest gain mode. The fiber was chosen to have a V number V = 27ra/X,(ni - n;)”’ of 6, so that it supported 6 modes at the signal frequency, and was excited by the funda- mental pump mode. For each mode the unsaturated gain at linecenter go, i (z) , calculated from (9) and (2) (with n? = 0), is given by:

where Pa&) is the pump power resonantly absorbed by the laser material between z = 0 and z = z

and where the spatial overlap integral between the pump mode LPol and a given LP,, signal mode was written as a function of the fiber core area Af = Tu2 and of a dimensionless overlap coefficient Fi,p defined by [12]

For a given pump mode and wavelength, these coefficients are only function of the fiber V number, and become independent of I/ for large I/ numbers. We refer the reader to [ 121 for a more complete description of the properties of these coefficients.

At an absorbed pump power of 10 mW, the modes with the highest gains (LPol and LPo2) were found to carry 4 times more power at the output ports than the next high gain modes (LPjl and LP12) and 10 to 20 times more than the two remaining, lowest gain modes (LP21 and LP3J. Since this discrimination grows exponentially with pump power, the high gain modes will carry orders of magnitude more power than the other modes, and the output mode content should rapidly reduce to one or two modes, when this device is pumped several times above threshold.

Since the exact solution to the problem can not be com- puted easily in the general case of a multimode fiber excited by an arbitrary mode distribution, we focused our attention on two configurations of practical interest for which relatively simple closed form results can be derived, namely l) a multimode fiber pumped uniformly, and 2) a single-mode fiber. In the following we derive formal expressions specialized to these two cases.

A. Uniformly Pumped Multimode Fiber

We consider the case of a multimode fiber pumped with a distribution rp(r, 4) resulting from equal excitation of all the fiber guided modes. The fiber is assumed to support a number of modes large enough that the pump intensity is essentially constant in the core and virtually vanishing in the cladding. Under such pumping conditions the spatial overlap integral involved in the unsaturated gain of (10) is

where vi is the fractional energy of mode i contained in the core. In a sufficiently multimoded fiber most modes are well guided and satisfy vi 2: 1 . Therefore most modes experience about the same gain and grow along z at essentially the same rate. With this approximation we can take Pj’ ( z ) to be independent of j . The sum in the saturation term of (4) then reduces to sj ( r , q5), which can be approximated by N/Af by invoking completeness. For a uniformly pumped multimode fiber (4) therefore becomes

where P, = Zsatna2/N. In a lossless device, P ; (z) can be eliminated from (14)

with the help of (7), and the resulting equation can easily be integrated to provide the evolution of the forward fluorescence signal

(1 - 2 2) log (1 + 9) + p, pi‘ (2)

where go(z) is the unsaturated plane wave gain at linecenter, given by (10) with Fi,p = vi = 1 .

Equation (15) has the same general form as the equation of the power amplification in a bulk amplifier in the plane wave approximation, which was expected since in es-

1634 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-4, NO. 11 , NOVEMBER 1986

sence we assumed a plane wave pumping arrangement. It provides a simple expression for the output of a uniformly pumped multimode SFL in the lossless case. For lossy devices for which no closed-form soltx$&%a is available, (14) was solved numerically using a fiffli-order Runge- Kutta method [ 141.

B. Single-Mode Fiber We consider the case of a fiber which is single mode at

the fluorescence frequency, i.e., with a Vnumber smaller than about 2.4. At the pump wavelength it may not be single moded, but we shall assume that only the funda- mental pump mode is excited. The forward fluorescence power is given by (4), where i = 1, N = 2 (two guided modes with orthogonal polarizations), and where the mode intensity distributions are now independent of the azi- muthal angle 4

2mp(r) sl(r)r dr S, sl(r) f a l p : .

1 + 2 - (P :(z) + Pi-(z)) Isat

(16) These coupled differential equations were integrated numerically using a fifth-order Runge-Kutta method.

111. SFL CHARACTERISTICS In the following we apply the results of the previous

sections to investigate the properties of 1-percent Nd- doped YAG superfluorescent fiber lasers and their dependence on the fiber characteristics. The material parameters that were used are as = 3.2 cm2, rf = 230 ps [15], and n1 = 1.820. To illustrate GaAlAs LD-pumping applications, we assumed a pump wavelength X, = 0.810 pm, for which the material absorption coefficient cya was taken to be 4 cm-'. For simplicity we assumed the same loss coefficient for the pump and for all guided signal modes.

A . Gain and Power Distribution The evolution of the gain factor and total fluorescence

power P ' ( z ) = NP: ( z ) along the length of a single-mode device is shown in Fig. 1. At low pump-power levels (Fig. l(a)) spontaneous emission dominates and the forward and backward waves build up almost linearly. The gain factor is essentially unsaturated and decays expone from left (input side) to right as a result of pump tion. At high pump power (Fig. l(b)), stimulated emission is dominant and the fluorescence power grows exponen:, tially. At high enough pump power the fluorescence &r&s to very large levels near the input and output ends of the fiber, where the gain factor is strongly reduced by population inversion depletion (Fig. l(c)). The gain factor ex- hibits a sharp maximum near the center of the fiber where the total fluorescence power ( P + + P -) is minimum.

. ,

l.Oh , 4

POSITION ALONG FIBER (mm)

(a)

0.6 30 Pp = IO mW

\BACKWARD \ /FORWARD 1 2 0

0;o 0 I 2 3 4 5 ~~~

POSITION ALONG FIBER (rnm)

(b)

Y I

2 3 POSITION ALONG FIBER [ m m l

(c) Fig. 1 . Evolution of the gain factor and total fluorescence power along the

length of a lossless single-mode SFL. Fiber parameters are a = 3 pm, An = 0.005, 1 = 5 mm. (a) Pp = 1 mW; (b) P,, = 10 mW; and (c) P,, = 100mW.

B. Output Curves Fig; 2 illustrates the output characteristic curve (output

power versus total absorbed pump power) of both a single-mode and a multimode (uniformly pumped) SFL. In this figure the ordinate is the total pump power resonantly absorbed by the laser material Pabs = Pabs(l) given by (1 1). The characteristic curves resemble those of a resonant fiber laser, namely the output grows almost linearly above some oscillation threshold [12]. However, in a SFL the absence of optical feedback results in 1) a noticeably higher threshold, 2) a smoother onset of oscillation, and


"i a = 12.5prn

60 - 2 I

ABSORBED PUMP POWER (mW)

Fig. 2. Output versus input characteristic curves of a single-mode SFL (same parameters as Fig. 1) and of a uniformly pumped multimode SFL (a = 12.5 pm, An = 0.005) for various values of the loss coefficient a. The input power coupled into the fiber was taken to be 100 and 400 mW for the single-mode and multimode fibers, respectively.

3) a poorer linearity, as observed in superluminescent laser diodes [3], [ 161. This linearity can be explained with (15) or (16) which, in the limit of high pumping rates, may be approximated to yield (see Appendix A)

P+(Z) = - - P,bsp 1 hv, 2 hu,

where p is a dimensionless parameter nearly equal to unity in the uniformly pumped multimode case, and between 0 (for V = 0) and 1 ( p = 0.95 for V = 2.4) in the single- mode case. The output power is independent of the fiber numerical aperture (which is not the case at low gain). In a lossless device, and in the high gain regime, virtually all absorbed pump photons are transformed into output fluorescence photons, with the expected efficiency hv,/hvp. The first signal photon generated within the acceptance angle of the fiber triggers the relaxation of the entire in- verted population and the emission of all signal photons via stimulated emission, with an equal probability in the forward and backward directions.

In practice this situation is realized when the device is pumped more than three times above threshold (see Fig. 2). In a low-loss device (a1 << l ) , taking into account both forward and backward outputs (which could be achieved in practice by placing a high reflector on the backward port and collecting both outputs at the forward port) the total conversion efficiency is equal to the ratio of signal to pump photon energy (energy conservation). This is precisely the slope efficiency of a low-loss, high output coupling fiber laser [12]. In both types of devices, near theoretical limit conversion efficiencies are made possible by the very large probability of stimulated emission resulting from the resonating structure in one case, and from the high pumping level in the other.

CORE RADIUS ( , u r n 1

Fig. 3. Threshold versus I.'-number in a lossless and in a lossy (a = 2 cm-') SFL ( I = 5 mm). The upper horizontal scale shows the corresponding core radius for An = 0.005. The solid lines are the exact solutions ((14) and (16)); the dashed lines are the approximate solutions (18).

C. Threshold Condition

As mentioned above, the two types of devices differ essentially by their threshold. Whereas the threshold of a low-loss resonator fiber laser may be in the submilliwatt range, 161, [12] that of a SFL is at best in the milliwatt range. To obtain an estimate of the pump power required to reach the onset of superfluorescence, we somewhat ar- bitrarily defined the threshold as the absorbed pump power for which the total forward output power is equal to the fraction E = 1 percent of the absorbed pump power.

The dependence of the threshold on the I/-number is shown in Fig. 3. As expected the threshold is a critical function of the core size. It becomes increasingly high in very weakly guiding fibers and in large diameter fibers, in which mode confinement is unfavorably poor. The lowest threshold is achieved in a single-mode fiber with a V of about 1.7, for which the optical density in the waveguide is nearly maximum. In such a fiber pumped near 810 nm, the threshold can be as low as a 3.3 mW. Note the fair agreement between the single-mode and multimode predictions in the intermediate range where their validity breaks down (a = 3-10 pm), although a perfect agreement was not expected since the two types of fibers involve different pump distributions.

An approximate closed-form expression for the threshold requirement can be derived by assuming that at threshold gain saturation is small (see Appendix B). This hypothesis was found to be quite good in most cases, and as we shall see, leads to a fairly accurate approximate expression for Pth

The logarithmic term is a slow function of Pth and Af (involved in the number of modes N ) . Therefore, to a good approximation the threshold power is inversely propor- tional to 1) the gain factor usrf /hup, and 2) the overlap integral Fi,,/Af. In large core fibers Fi,, is independent of the core size and Pth varies as A f ; in small core fibers F j , p


gradually goes to zero as the core radius is reduced and the threshold increases dramatically (see Fig. 3). This behavior is analogous to that prevailing in resonant devices [ 121. The predictions of (1 8) agree to better than 5 percent with the exact solutions (dashed lines, Fig. 3) except for very small core radii (< 1 pm) where they become somewhat less accurate.

The influence of fiber loss on the threshold is relatively minor, as illustrated in Fig. 3. Near threshold, the single pass gain is in excess of 16 dB, so that the net effect of an even sizeable loss is negligible. This situation is in sharp contrast with that in resonant fiber lasers, in which the gain at threshold is equal to the cavity round trip loss, i.e., is a strong function of the round trip fiber loss [12]. As indicated by (18), Pth increases linearly with the loss factor. An increase in the single pass fiber loss of 1 dB results in a threshcld power increase of only 6-8 percent depending on fiber V-number.

Pth is the absorbed pump power required to reach the onset of ASE. Clearly another quantity of interest is the power coupled into the fiber at threshold Pth,in, which combines the effect of both the signal and the pump loss on the power required to reach threshold. This quantity may be directly calculated from Pth with (1 1). In the ex- amples treated in this section Pth,in is between 16 percent (for a = 0) and 58 percent (for a = 2 cm-') larger than Pth, This relatively efficient utilization of the available pump power is due to the very large absorption coefficient of Nd : YAG at the selected pump wavelength.

It is worth stressing that even though these figures may appear approximate due to the somewhat arbitrary threshold definition that was used, the nature of the problem is such that the threshold figures depend rather weakly on the exact choice of the factor E . Reference to (1 8) indi- cates that a change in E by a factor of 2 would only result in a change in the threshold figures of 10-30 percent for the range of V-numbers studied here. The figures mentioned in this section are therefore believed to be good representatives of the onset of ASE in active fibers.

D. Eficiency The dependence of the slope efficiency s = dPout/dPabs

on the fiber V-number is shown in Fig. 4 for different fiber loss factors. Since the output is not truly linear, we plot- ted the slope efficiency 5 times above threshold. In a nearly single-mode fiber, s drops sharply as the core radius is reduced, i.e., as the modes become more weakly guided. The slope efficiency is maximum and essentially independent of the fiber V number for V numbers larger than about 4, as is the case in resonant fiber lasers [ 121. As the SFL is pumped many times above threshold, the slope efficiency increases asymptotically toward the limit value given by (17) and represented in Fig. 4 by the dashed curve.

The slope efficiency does not depend strongly on the fiber loss. In the saturated regime the signal loss appears both as a driving term and in the saturation term. The first term contributes to reducing the signal, while the second

CORE RADIUS ( p m ) 0 5 I O 15 20 25

0.5 SINGLE MODE

- MULTIMODE 0.4 -

> 0 z W

THEORETICAL LIMIT ia=Ol - - - - - - - - - - - - - - - - - - - - - - - - - - - - -. ,_-- -___.

0 . 3 - / * - - - Q = O

_ _ _ _ _ LL LL W

wo.2-f:::- ...... C Y = I crn-'

a 2 2 cm-' a 0 J m 0. I L

, , O ! I I ! A

0 4 8 12 16 20

V-NUMBER

Fig. 4. Slope efficiency versus V-number five times above threshold (solid curves) and infinitely far above threshold (dashed curve). The fiber length is 1 = 5 mm. The upper horizontal scale shows the corresponding core radius for An = 0.005.

0 CORE RADIUS (prn) 5 I O 15

3 0 P,= 100 mW

1 V-NUMBER

Fig. 5 . Output power versus V-number. An = 0.005, I = 5 mm, P,, = 100 mW.

one reduces the saturation effect and tends to increase the output. As a result the net effect of a loss factor a is a reduction of the output by a factor smaller than the anticipated factor of exp (- aZ) . The calculated ds/da is about 0.014 per dB/cm loss.

E. Output Power Optimization In Fig. 5 we show the predicted output power as a func-

tion of the V number in devices pumped at a constant power level (P, = 100 mW). As much as 25 mW per port can be extracted for an optimized device (V = 2.5) at this pump power level. Again this figure may be doubled by collecting both outputs at the same port. In lossy devices the penalty amounts to a relatively small reduction of the output power (2 cm-' = 8.7 dB/cm).

Using these figures as a guideline, let us evaluate the performance of a specific device, made of a clad Nd : YAG fiber with a numerical aperture of 0.135, a core diameter of 10 pm (V = 4), a loss of 4.3 dB/cm (a = 1 cm-') and a length of 0.5 cm. Using a high-power phased-array semiconductor laser emitting a total power of 400 mW as a pump source [17], we can expect to couple at least 100


mW into this fiber by proper imaging and first order cor- rection of the pump beam astigmatism [18]. The maximum absorbed power would then be about 73 mW. We expect the onset of superfluorescence to occur at an inci- dent pump power of 11.5 mW, and the maximum forward output to be about 15 mW. Assuming a 10-percent elec- trical to optical conversion efficiency for the pump laser, this SFL would have an efficiency on the order of 0 . 4 percent.

F. Linewidth Because frequency components near the center of the

gain curve experience a higher gain than components away from linecenter, the fluorescence spectrum narrows as the pump power is increased. In the general situation of a possibly saturated device, the fluorescence spectral distribution at z may be characterized by a function f ( v , z) defined as the ratio of the number of photons at frequency v to the number of photons at linecenter v, [lo]

where we have used the general expression of n+ as a function of the gain obtained by direct integration of (la) along z in .the lossless case. We define the fluorescence linewidth Av(z) as the separation between the two frequencies where the number of fluorescence photons is half the linecenter value, i.e., where f ( v , z ) = 1/2. For a Lorentzian lineshape, the linewidth at the output can easily be shown to be

1 /2

Avi = A- , - 1\ . (20)

In the two types of SFL considered here, all modes experience the same gain and the same linewidth narrowing

To express Au as a function of the output power we A v ~ = Au.

eliminate g i ( v s , I ) between (7) and (20) to obtain

where x = P +(Z)/NP0 = P+ (Z)/Po. At low output levels, x << 1 and Av = Avs, as ex-

pected. At high-output powers (x >> l), Av behaves like (log and varies very slowly with output power (Fig. 6). In a near single-mode SFL, the linewidth narrows very rapidly above threshold to a quasi-asymptotic value on the order of Avs/4. For a fixed output power, the output per mode P+ ( 1 ) and the gain gi ( v , I ) decrease with increasing core radius, which explains the smaller linewidth narrowing predicted in multimode SFL's (see Fig. 6). Over most of the operating range of an SFL, the linewidth is anticipated to be constant and equal to a sizeable fraction of the unsaturated linewidth, as expected in superfluorescent laser diodes [ 161.

I .01 I

L 01 I , I I 0 2 4 6 8 I O

OUTPUT POWER (mW1

Fig. 6. Signal linewidth as a function of output power for a single-mode (a = 3 pm) and multimode SFL's (a = 10 and 25 pm). An = 0.005, 1 = 5 mm.

This model assumed at the outset a single Lorentzian for the material lineshape. In reality, the fluorescence spectrum is composed of several lines (for Nd : YAG, near 0.9, 1.06, 1.32, and 1.85 pm) with different linewidths and gain cross sections, and gain competition will result in a frequency selection between lines. Above threshold, the strongest lines are expected to dominate, and their in- dividual quantitative behavior to be fairly well described by the above model.

IV. CONCLUSIONS We have developed a model for the gain and amplified

spontaneous emission in active fibers for superfluorescent source applications. This analysis accounts for the degree of spatial mode overlap to describe the device behavior in terms of its output and gain characteristics in both single mode and multimode fibers. Simple closed-form expressions for the threshold, the slope efficiency and the band- width of this class of devices were derived. In Nd : YAG devices end pumped near 810 nm, it predicts thresholds of a few milliwatts and a high-conversion efficiency, possibly as high as 38 percent per output face, in low-loss, near single-mode fibers. The threshold is minimum in a fiber with a I/ number of 1.7, while the slope efficiency is maximum for I/ > 4 . This type of device appears to be only slightly affected by even moderately high fiber losses. This analysis suggests the possibility of pumping a near single-mode fiber with a single high-power laser diode for compact SFL applications! Such a device would emit an unpolarized signal with a nearly power independent band- width (on the order of several nanometers for Nd: YAG) with a power of several milliwatts per port. Because of their projected high output power and excellent coupling to standard single-mode fibers, superfluorescent fiber lasers are anticipated to have important applications in single-mode fiber systems requiring short coherence sources.

APPENDIX A OUTPUT POWER IN THE HIGH GAIN LIMIT

In the limit of high gain ( gi (vs , z ) >> 1), the output per mode P: (I) becomes much larger than the saturation


power P, = ZsatAf/N, and the solution for P+ ( I ) can be written in a very simple closed form. In the multimode configuration with a = 0 , (15) may be approximated by neglecting the logarithmic term in the left-hand side. After summation over all modes one obtains

N

P+( l ) = c - go(Z). PS (All

By replacing Ps by its definition and g o ( l ) by (10) (with F. ‘ . P = vi), (Al) becomes

i = l 2

The sum p = Er= v , /N is nearly equal to, and always smaller than, unity. Equation (A2) provides a direct verification of energy conservation.

The corresponding expression for the single-mode case is derived from (16) (with a = ‘O), where the the overlap integral, noted R(P+), is first expanded by replacing the mode profile sl(r) in the saturation term by its third-order power series expansion in r

sl(r) = K ~ J ~ u - .( :> (A31

where u is the transverse propagation constant and K~ the intensity normalization constant of the LPol signal mode. In the high gain limit, the overlap integral may then be written as

1 F R ( P T ) = - (A41

1 + 2 3 (P:(z) + P;(z)) Af Isat

where F is a factor independent of P f(z) equal to

F&bl are dimensionless overlap coefficients given by

(for 2m + 1 = 1, FL\L, = Folol defined by (12)). In the lossless case the solution of (16), in which the

integral is now replaced by (A4) and P;(z) is eliminated with the help of (7), can be written as

(A71

where we have made the high gain approximation PF(2) >> Ps = IsatAf/2. The dimensionless parameter p is

F p = -,

K 1 Af

Computer calculations indicate that it satisfies 0 < p < 1 for all Vnumbers. Equations (A2) and (A7) provide not only simple expressions for the theoretical limit output of a SFL, but also a verification of the internal consistency of our model, in particular between the definition of the net gain cross section (3) and the number of SOP involved in the fluorescence process.

APPENDIX B THRESHOLD CONDITION

By definition we chose the threshold Pth to be the absorbed pump power for which the total forward ASE output power is the fraction E = 1 percent of the absorbed pump power:

p + ( l ) = EPabs. (B1)

Computer simulations indicate that at threshold gain saturation is small, typically less than 10-20 percent. As- suming an unsaturated gain, the general equation for the forward fluorescence power (4) may be solved in the case of a lossy fiber. P “(I) is then given approximately by [ 111

P + ( l ) E po(ego,i(c - 1) e-e‘ (B2)

where go,i (Z) is the unsaturated gain given by (10). In- serting (B2) into (Bl) yields the condition

NPo(expgo,‘(’) - 1) e-“’ = EP,bs. 033)

Solving for P a b s = P t h we find

In all cases of practical interest the term EP[h/PoN e-e‘ is larger than about 35, and the logarithmic argument may be simplified by neglecting the unity term. The threshold condition then reduces to

ACKNOWLEDGMENT The author is thankful to E. Desurvire and M. Fejer for

helpful discussions.

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* Michel J. F. Digonnet was born in Paris, France, in 1955. He received the Engineering Degree in physics and chemistry from ‘Ecole Superieure de Physique et de Chimie industrielle de la Ville de Paris’, Paris, France, and the ‘Diplome d’Etude Approfondies’ in optics from the University of Paris, Orsay, France, in 1978. He received the M.S. degree in 1980 and the Ph.D. degree in 1983 both in applied physics from Stanford University, Stanford, CA. His graduate research included the development of single- mode fiber couplers and single crystal fiber devices.

Since then he has been with Litton Guidance and Control, Chatsworth, CA, doing his research at Stanford University as a visiting scholar. His current research interests include single crystal fibers for amplifier and source applications, miniature solid state lasers, and the development of integrated optics devices for optical fiber sensors.