theoretical analysis of optical fiber laser amplifiers and oscillators

Theoretical analysis of optical fiber laser amplifiers andoscillators

M. J. F. Digonnet and C. J. Gaeta

Using the formalism of mode overlap, a theoretical analysis of optically pumped fiber laser amplifiers andoscillators is developed. The concept of normalized overlap coefficients is introduced to account for the ef-fects of the transverse structure of the interacting signal and pump modes on the device characteristics.Simple and accurate closed-form expressions are derived for the gain of fiber amplifiers and the thresholdand energy conversion efficiency of fiber laser oscillators in terms of the fiber and laser material parametersand the pump and signal modes. When applied to step-index Nd:YAG fiber lasers, this study predicts opti-mum fundamental mode oscillation in fibers with a V number of 5-25 with submilliwatt thresholds andnearly quantum-limited conversion efficiencies.

1. Introduction

In recent years, optical laser oscillators and amplifiersin a fiber form have received increasing attention.Active fiber devices can combine the excellent proper-ties of standard laser materials and the high-energyconfinement available in optical fibers. Their roundgeometry is also perfectly adapted to fiber system ap-plications. For these and other reasons they are an-ticipated to exhibit large energy conversion efficienciesand excellent coupling properties to single-mode fibersand to have important applications in fiber systems andnetworks. This promising potential has already beendemonstrated experimentally for a variety of activefiber materials including Nd-doped glasses,1"2 dye so-lutions,3 '4 and Nd:YAG. 5,6

In fiber laser devices, as in any active or nonlinearwaveguide, the device performance is intimately relatedto the degree of spatial overlap of the interacting waves.To study and optimize the overall efficiency of this newclass of fiber component, it is, therefore, essential todevelop a theoretical model that accounts for the effectsof the transverse mode structure of the various wavesinvolved. This problem of modal overlap in lasers has

When this work was done both authors were with Stanford Uni-versity, Hansen Laboratories of Physics, Ginzton Laboratory, Stan-ford, California 94305; M. J. F. Digonnet is now with Litton Industries,Chatsworth, California 91311.

Received 6 August 19840003-6935/85/030333-10$02.00/0.© 1985 Optical Society of America.

been previously studied by several authors in the caseof Gaussian optical beams in unguided devices7 8 andin optical waveguide laser oscillators.9 10 However, tothe best of our knowledge the case of laser interactionin optical fibers has not yet been investigated. Also, aunified approach is needed to characterize waveguidelasers and amplifiers with arbitrary index profiles andmode distributions.

In this paper we present results of an analysis of thegain, threshold, and energy conversion efficiency ofoptically end-pumped laser devices. This analysis,based on the general formalism of mode overlap,9 in-troduces the concept of normalized overlap coefficientand provides a general solution for either guided orunguided laser devices. In Sec. II the general case ofarbitrary transverse pump and signal modes is ad-dressed, and general expressions are derived. In thefollowing two sections these results are applied to thecase of free-space (unguided) and fiber (guided) devices,respectively. It is shown that the operation of fiberlaser devices can be described very simply and accu-rately with closed-form expressions of the gain,threshold, and conversion efficiency as a function of thedevice parameters. In the particular case of a step-index profile fiber, this approach leads to the design ofa best configuration, namely, a fiber V number in the5-25 range, which optimizes the device performance.All numerical examples treat the case of Nd:YAG op-tically pumped at Xp = 514.5 nm. (Similar results canbe derived in a straightforward manner for other pumpwavelengths, such as A = 808 nm, which falls on astrong absorption line of Nd:YAG.) The laser param-eters were taken from the literature, namely, a = 3.2 X10-19 cm2 (effective stimulated emission cross section)and Tf = 230 pusec (fluorescence lifetime).1

1 February 1985 / Vol. 24, No. 3 / APPLIED OPTICS 333

11. General Formalism

We shall consider a four-level laser material of lengthI which is end-pumped by an optical beam of wave-length X1p and used either as an optical amplifier or os-cillator. For amplifier applications this medium willbe assumed to be probed by a small signal (wavelengthX) cotraveling with the pump. We will then be con-cerned with the single-pass gain experienced by thesignal as it travels through the device. For laser oscil-lator applications the active material is placed in anexternal optical cavity which provides feedback andoscillation buildup. We will then turn our interest tothe device threshold of oscillation and conversion effi-ciency.

In this pumped laser system, the excitation generatesan electronic population inversion which interacts withthe photon density via spontaneous and stimulatedemission. The interaction between the pumping rateper unit volume r(x,y,z), the population inversiondensity n(x,y,z), and the signal photon density of theith mode si (x,y,z) is described by the laser rate equa-tions, which, following the notation introduced in Ref.9, may be written in a steady state as

dn(x,y,z) = r(xyz) - n(x,y,z) cn(xy,z)dt Tf nl

NX F Sj(x,y,z) = 0,

j=1

dSi = c - n(x,y,z)Si(x,y,z)dv -- Si = 0,dt ni cavity 21ni

(2)

where c is the velocity of light in vacuum, nj is the lasermedium refractive index, bi is the round-trip loss for the6th mode, and the summation in Eq. (1) is carried overthe N transverse modes present in the cavity. To writeEqs. (1) and (2), it was assumed that the gain mediumis placed in a relively high-Q cavity, so that theseequations apply only to a cavity laser amplifier.However, since the presence of the cavity clearly doesnot affect the single-pass gain of the device (the cavityis assumed to be resonant at the signal frequency butnot at the pump frequency), the single-pass gain ex-pressions that will be derived later also apply to atraveling-wave (TW) amplifier. This point can beshown mathematically by introducing in Eq. (2) theaSil/z term that holds for TW amplifiers. Spontaneousemission was also ignored in Eq. (2) as its contributionto the device output above threshold is negligible.

It is convenient to define the distribution functionsro(x,y,z) and so,i (x,yz), normalized to unity over spaceso that

r(x,y,z) = Rro(xy,z), (3)

si(Xyz) = siisj(x1y1z), (4)

Ji(S 1,S2 , . * * SN) - 21IR

where

Ji(SS2, * . .SN) = so,i(x,y,z)ro(x,y,z) dv.cavity Co(J Tf N

1 + -E Sjsoj (x ,y,z)ni j=1

(5)

(6)

Equation (5) is the basic expression for the number ofsignal photons present in the ith mode in the presenceof other signal modes. As anticipated, this expressioninvolves the degree of spatial overlap between the pumpand signal distributions and the degree of gain satura-tion resulting from the signal modes above threshold (Sj# 0). For a device supporting N modes, there are Nsuch equations, whose solutions describe the numberof photons in each mode at a given pumping rate as wellas their dependence on the pumping rate.

Although the general solution of Eq. (6) for a highlymultimoded laser device is difficult to express in asimple form, the case of a single oscillating mode S canbe treated easily. This is in fact the single most im-portant situation since it is commonly encountered inactual devices in which mode loss and gain competitionsusually drastically limit the number of oscillatingmodes. Single (lowest-order) mode operation is alsohighly desirable in most amplifier applications. LettingS2 = S3 = .. . SN = 0 in Eqs. (5) and (6) yields

Ji(Si)2 = 52la-ifR

(7)

where

Ji(S) = J1(S1,O,10.0.,o)

S rotx,y,z)so,jkx,y,z)

cavity +or Ss(XYZ)nli

dv. (8)

For an oscillator, the pumping rate required to reachthreshold is obtained by setting S, = 0 in Eqs. (7) and(8). In the following we shall more conveniently expressthe threshold in terms of the total pump power absorbedby the active material. In an end-pumped device it isrelated to the incident pump power Pp as follows:

Pabs = Pp[1 - exp(-a)]. (9)

Here the plane-wave absorption coefficient a hasbeen replaced by the effective absorption coefficient aa

= taqp where 7p is the fractional pump energy con-tained in the fiber core, to account for the fact that onlythe core absorbs the pump power.

By definition the pumping rate is R = Pabs/hvp,where hvp is the pump photon energy. The absorbedpump power required to reach oscillation threshold is,therefore,

where R is the pumping rate, and Si is the total numberof signal photons in the ith mode. Solving Eq. (1) forn(x,y,z) and inserting the result in Eq. (2) yield thefollowing equation describing the steady-state numberof signal photons S (Ref. 9):

(10)p t hvp 61 1o--f 21 J(°)

where

J1(0) = J 1(°0, .0 °) = cavity ro(x,y,z)so,1 (x,y,z)dv. (11)

334 APPLIED OPTICS / Vol. 24, No. 3 / 1 February 1985

Equivalently, since at threshold the single-pass loss 31/2and the gain are equal, the unsaturated single-pass gainfactor of the active medium as a cavity or traveling waveamplifier is

trf Pabs (12)hv AP

Here A* is the effective pump area defined asP~~~~~~~AP = W (O) ' (13)

1J1 (0)

while Pab/A* is the effective (average) pump intensityexisting in the active medium. This quantity is afunction of the pump and signal modes through thesimple overlap integral J1(0) [Eq. (11)]. For reasonsthat will become apparent shortly it is convenient tonormalize the effective pump mode area to the fiberactive area At in the case of a fiber laser and to the pumpmode area AP averaged along the device length in thecase of an unguided laser. This introduces the notionof filling factor Fvnm defined by

A = A (fiber case), (14)Fvpnm

A = AP (unguided case), (15)Fvgnm

where the index pairs (A) and (nm) characterize thesignal and pump modes, respectively. With thisnotation the amplifier gain factor of Eq. (12) may bewritten as

= 7oFynm, (16)

where wyo is the gain that would be expected had weneglected the signal and pump mode structure, e.g., inthe fiber case

= Tf Pabs (17)Y h v A17

With this notation the effect of the transversestructure of the interacting waves on the gain of a laserdevice is described by a single additional parameter,namely, the filling factor Fvunm. As expected on thegrounds of the wave nature of the problem at hand, ananalogy can be drawn between this formalism and thatof the electronic wave function overlaps describingatomic interactions. We shall see that in practical sit-uations the F coefficents can be computed accurately,and coefficient tables can be generated for a variety oflaser configurations.

Furthermore, this formalism can be easily extendedto the case of a laser optically excited by a multimodepump. Then the filling factor entering the gain ex-pression [Eq. (16)] is replaced by the weighted factor

(F) = E PnmFv~rnm, (18)n,m

where Pnm is the fraction of pump energy carried by thepump mode (nn). This formalism is, therefore, apowerful tool which greatly simplifies analysis of thegain in optically pumped single-mode lasers and am-plifiers.

The laser output power at a pumping rate R is given.by 9

Pout = T hCS,21n,

(19)

where T1 is the output mirror transmission, assumingthe other mirror is a high reflector (HR) (T2 = 0). SinceEq. (19) involves the number of photons S1, solution ofEqs. (7) and (8), the exact dependence of the cw outputpower Pout on the excitation power Pabs is expressed ina rather complicated mathematical form unappropriatefor physical understanding. However, the optical en-ergy density circulating in the cavity of a laser oscillatoroperated above threshold is unusually large. Comparedwith the saturation intensity the saturation term in Eq.(6) becomes very large soon after oscillation breaks in,and the overlap integral J1 (S) can be expanded to firstorder in S, with good accuracy:

1 1 CoTf S1

Ji(S) J 1(0) ni 1p(20)

where p is the fraction of pump power carried by theactive region:

ip ftive o ro(x,y,z)dv. (21)

Eliminating Si between Eqs. (19) and (20) and replacingJ,(S 1 ) and J 1(0) by their respective expressions as afunction of the absorbed pump power [Eqs. (7) and (10)]lead to the laser slope efficiency s, referenced to thepump power absorbed by the active material:

S = Pout T1 hi'8 (22)(Pabs - Pth) &i hvp

Even though this result was derived under the as-sumption that the laser was operated somewhat abovethreshold, comparison of exact and approximate solu-tions indicates that this result is also valid nearthreshold, as will be illustrated in the following sections.The second assumption that was implicitly made wasthat the oscillator conversion efficiency is reasonablylarge. This in turn assumes that the signal and pumpspatial distributions are comparable so that they exhibita relatively good spatial overlap. The range of validityof this expression, therefore, depends on the laser con-figuration, in particular, whether it is guided or un-guided. However, we shall see in the following sectionsthat this range is rather broad in most practical situa-tions and includes in particular the important region ofmaximum laser efficiency.

Within its limits of validity, Eq. (22) provides a simpleand powerful result. It states that provided the sig-nal-to-pump spatial overlap is good enough, thepump-to-signal photon conversion process is relativelyindependent of the modal distributions involved. Inother words, if the pump and signal photons occupy thesame general volume and the probability of stimulatedemission is large enough, essentially all the invertedpopulation is driven to the ground level (which againimplies a relatively efficient laser to start and, in par-ticular, a relatively low cavity loss). Under such con-ditions the laser slope efficiency [Eq. (22)] is propor-tional to

(1) p, the fractional pump power contained in theactive region of the laser (p can at best equal unity);


The average pump beam area inside the active me-dium is approximated by Ap = (r/2)W where W isan average value of the pump beam radius along theactive medium, defined as:

== J WI2(So )dz. (24)

Fig. 1. Schematic of the Gaussian pump beam path inside the lasermedium.

(2) T1h31, the ratio of output coupler loss to totalcavity loss [in the most favorable case (low cavity loss)this ratio can be very close to unity];

(3) h/hvp, the ratio of signal-to-pump photon en-ergy, which constitutes the fundamental limit of theenergy efficiency of any photon-to-photon conversionprocess.

Another important point is that Eq. (22) is indepen-dent of the laser material properties (,yrf) as well as thetype of signal mode. It is also fairly independent of thepump mode as long as this mode is sufficiently far fromcutoff (then np 1). Within limits, it holds in partic-ular for side-pumping configurations. However, thelaser output power Pout does depend on the pumpconfiguration, pump mode and material propertiesthrough the threshold Pth. Consequently, to fullycharacterize a single transverse mode laser with a givenexcitation ro(xy,z) and pumping arrangement the onlyoverlap integral that needs to be evaluated is J 1(0) [Eq.(11)].

111. Free-Space Lasers

As a first application of the above results let us con-sider the case of a free-space (unguided) laser. Thelaser configuration is specified in Fig. 1. A crystal oflength is placed between a high-reflection mirror andan output coupler, which form a high-Q cavity at thesignal frequency. The mirrors are assumed to be es-sentially transparent at the pump frequency. Thepump beam, end-fed into the laser material along thelaser cavity (z axis), is a fundamental Gaussian beamwith a waist Wp. Diffraction causes the pump beam toexpand on either side of its waist so that its radius variesalong the z axis as12

W2(Z) = W2 [1 + (X'(z ZP'))2 (23)

where Xp is the pump wavelength in vacuum, n1 is thematerial refractive index, and zP is the location of thepump waist inside the cavity (Fig. 1). Similarly, thesignal mode is assumed to have a TEMmn Gaussiandistribution imposed by the configuration of the reso-nant cavity. For all TEMmn modes the beam radiusWs (z) is given by an expression similar to Eq. (23) inwhich W ,X, and z, are the relevant parameters for thesignal. Finally, we assume pump and signal modes arecoaxial and aligned with the z axis defined by the res-onant cavity.

A similar quantity (W) can be defined for the signalmode. These average radii can be calculated in a closedform using Eq. (23). For example, for the fundamentalpump mode, assuming its waist located at the center ofthe laser material (zp = 1/2), we find that

(25a)-2 = W2(1 + f )

where Ep is the diffraction term

1P ( nWp)

As a result of the relative simplicity of the mathe-matical description of Gaussian modes, the filling fac-tors Fpnm can be evaluated in a closed approximateform. For example, for the most important case of aTEMoo signal mode, one finds 6

W2wp +W (26)

where W, and Wp are the average signal and pumpbeam radii. This was calculated from Eqs. (13) and (15)in which each radius was replaced by its respective av-erage value for the sake of simplicity. For this modeconfiguration the effective pump area is, therefore,approximately given by [Eq. (15)]

A == AP + A =(WP + W2).Foooo 2

(27)

This result is identical to that derived by other authorsin a slightly different form.9 For a given signal waistradius, the smaller the average pump radius the higherthe overlap and the higher the gain. Of course, dif-fraction limits the minimum average radius that can beachieved. For a given laser medium length, refractiveindex, and pump wavelength, the pump waist radiusWp,Opt that minimizes Wp and maximizes the gain is[from Eqs. (25)]

Wp'opt= .PMn 1

(28)

In terms of the pump Rayleigh range ZRP, this resultstates that the gain is optimum when ZRP is approxi-mately equal to the length of the active medium. Thiscondition is, of course, reminiscent of the optimum fo-cusing condition of other optical parametric processes,such as bulk optic second harmonic generation.13However, in the present case the pump and signal playsymmetric roles in the small signal gain, and the aboveconclusion also applies to the signal. The largest gainis, therefore, achieved when ZR = ZR,8 1. Thiscondition can presumably be realized in practice sincefocusing the signal and pump can be individually con-trolled.

The accuracy and validity of the approximation madeto derive the closed-form of the laser slope efficiency

(25b)


15

I-z

-

z

LLI

cr

e0

-JUi

12

9

6

3

00 20 40 60 80

PUMP WAIST RADIUS (MICRONS)100

Fig. 2. Dependence of the slope efficiency of a free-space Nd:YAGlaser at Xp = 0.5145 m (T = 1%, 61 = 5%, W = 25um): (a) exactsolution; (b) approximate solution obtained by replacing W, and Wpby their respective average value; (c) approximate solution given by

Eq. (22).

[Eq. (20)] were estimated in the present case of a free-space laser by comparing it to the exact overlap integral[Eq. (8)]. This is illustrated in Fig. 2, which representsthe slope efficiency of a Nd:YAG crystal laser vs thepump waist radius. As expected the approximation ismore accurate for values of a - W p/W' near unity.(Pump and signal then occupy approximately the samevolume.) When the pump is focused either too tightlyor too loosely, Wp >> W, (a << 1), the signal misses agood part of the population inversion, and the efficiencydrops. Maximum conversion efficiency may be reachedonly when the inverted population distribution and thesignal mode present some degree of similarity. For thisparticular example, the agreement with the approxi-mate form of the slope efficiency is excellent for Wpranging over half of an order of magnitude (8-40 Arm).

Similar expressions can be obtained for higher-ordersignal modes. For example, for the TEM 0 1 mode it canbe shown that the filling factor is6

-w4

F1 = (29)Foo=(W2 + W2)2

It is also interesting to analyze the laser behavior farabove the TEMoo threshold where higher-order modesare likely to break into oscillation. As an example thethreshold of TEM 10 in the presence of TEMoo can bestraightforwardly calculated from Eq. (6) and shown tobe

Pth(TEMlo) = Pth(TEMoo)

0- - a0o\1 10 /~

situation would clearly be modified in favor of theTEM10 mode (or other higher-order modes) if the pumpbeam was offset from the cavity axis or if a higher-orderpump mode was used. As intuitively anticipated, thecontrol of the pump distribution is primordial to sin-gle-mode laser operation. Controlled pump distribu-tions, as opposed to the highly multimoded situationoccurring in most flashlamp-pumped commercial solidstate lasers, should, therefore, extend the single-modeoutput range of end-pumped lasers to much higherpower levels. This is typically done in commercial la-sers by introducing an aperture which further decreases6 00/'5nm. However, in general, it also increases thecavity loss 6oo and reduces the overall laser efficiency.

IV. Fiber Lasers

A. Fiber Laser Configuration

In this section we consider the case of a fiber laserwith a step-index profile and a circular geometry. Theoptical cavity is assumed to be made of two flat mirrorsplaced against the fiber ends, so that all optical signalsremain guided throughout the cavity. The core (indexnf) is the only active part of the fiber surrounded by a(passive) cladding of index n2(n2 < n) infinite in theradial dimension. The fiber V number is defined by

2iraV = A -

We assume that the fiber numerical aperture (N.A.) =n2 2 is small (N.A. << 1) so that the fiber guided

modes are almost linearly polarized, and the so-calledLPnm mode classification can be used advantageously.' 4

We refer the reader to the Appendix for a descriptionof the sets of modes used in our calculations.

B. Fm Coefficients

In the following we only consider modes varying asCOS2 (po) [see Eqs. (A2) and (A5)]. Because of thecomplexity of the mathematical functions involved, theradial integrations cannot be expressed in a closed formand were analyzed by computer.

In Fig. 3 we show the dependence of Fvinm on thefiber V number for a few low-order pump and signal

2.5

(30)

where boo and bio are the cavity round-trip losses forTEMoo and TEM 1o, respectively, and a 0o =W2(TEMoo)/WP as before. It appears that to reach theonset of TEM10 oscillation, aoo must be smaller than60o/co1, which is in general smaller than unity. Thisreflects the fact that when the pump area is much largerthan the area of either signal mode (a0o << 1), the smallsignal gains yoo(TEMoo) and y1 (TEM1o) becomecomparable. Then, as the pump power is increased, thephoton density due to TEMoo does not build up toorapidly and does not saturate the gain Gylo too fast. This

E

0

2.0

1.5

1.0

0.5

0.00 10 20

NORMALIZED FREQUENCY V

Fig. 3. Variation of a few Foinm coefficients with the fiber V number(at the signal frequency) computed with X = 1.064 Aim, X, = 0.5145

ALm, nj = 1.820, and n2 = 1.815.


(C)

I (.. (b)I(a

I I I I I I

(31)

Table I. Asymptotic Values (V-- co) of the Modal Overlaps Fpnm for aFew Low-Order Signal and Pump Modes

Signal -pump 01 02 11 12

01 2.098 1.800 1.435 1.67702 1.800 2.743 0.9422 1.57611 1.435 0.9422 2.327 1.68112 1.677 1.576 1.681 2.943

modes in a Nd:YAG fiber laser. For each curve thesmallest allowed V number is the highest cutoff valueVc of the mode pair LPp,/LPnm, below which at leastone of the modes does not propagate. As V is decreasedtoward Vc, the corresponding F coefficient decreasesto vanish at V = Vc. This result stems from the factthat as one of the modes reaches cutoff, it spreads out-ward into the fiber cladding, and less energy is involvedin the active core region. For V numbers that are largecompared with V, the F coefficients rapidly convergeto asymptotic values which are only a function of thetwo interacting modes: when both modes are far abovecutoff their power density distributions become es-sentially independent of the fiber V number and so doestheir overlap. Since this convergence is rather rapid,in a practical device involving a large V-number fiberall low-order filling factors may be replaced by theirfar-from-cutoff limit with a good approximation. Thefilling factors are then universal numbers independentof the pump and signal frequencies and of the fiber ge-ometry (provided it has a step-index profile) and onlyfunctions of the mode numbers.

Table I gathers the values of the filling factors in thelarge V limit, computed for a few low-order pumpmodes. In the far-from-cutoff regime the guided modesare independent of wavelength, so that the overlap be-tween these modes is invariant under a commutationof modes, and

Fvinm = Fnmv(V >> Vc) (32)

as reflected by the symmetry of Table I about its diag-onal. This property can be usefully invoked to reducecomputation time and the volume of tabulated data.For a fundamental signal mode (LP01) pumped with afundamental pump mode, the F coefficient is F0101 =2.098; i.e., the effective pump mode area A* entering theexpression of the laser gain [Eq. (16)] is approximatelyhalf of the area of the fiber core. As intuitively antici-pated, when pumped with higher-order modes, thecorresponding overlap with the LP01 signal mode isreduced. On the other hand, F coefficients can besomewhat larger (3-6) for higher-order mode config-urations of the type LPnm/LPnm as a result of theirlarger signal and pump power local densities. 6 How-ever, such configurations involve modes of limitedpractical interest which generally exhibit higher prop-agation loss in fibers.

The general evolution of Fvnm as the mode numbersare increased is shown in Fig. 4 for the Foinm seriespertaining to the important case of a fundamental signalmode. From these curves a number of interesting ob-servations can be made. First, the largest F coefficient

occurs with the fundamental pump mode with Folo1 =2.098; in this configuration the interacting mode dis-tributions are as well matched as can be. (In this largeV limit they are actually identical.) Second, for a fixedvalue of n (n = 0) the Foinm coefficients increase withincreasing m. This behavior arises from the fact thatthe distribution of a LPnm mode has a maximum nearthe fiber axis (r = 0) and that this maximum increaseswith increasing m (see Fig. 8). Since the signal modeLP01 has a broad maximum near r = 0, the overlap witha LPnm mode increases with increasing m. Third, theFoinm series has a finite asymptotic value for large m,and this value is independent of n (for n < m). Forhigh-order fiber modes all modal lobes (see Fig. 8) carrynearly the same energy,6 and their overlap with a givenmode (here LP01 ) is independent of m or n, as observed.Finally, this asymptotic value, equal to 1.7568 for theFoinm series, is reached more rapidly for small valuesof n. For all practical purposes, for n < 10 the FoPnmcoefficients take this value for m > 10-20. Similarconclusions apply to any F,,nm series with fixed valuesof v and A.

A last case of interest is that of a fiber laser in whichthe pump is distributed among a large number of guidedmodes. This situation arises in particular when thefiber laser is excited by a Lambertian source (such as aLED) or a strongly diverging source (such as a LD). Inthis highly multimoded configuration the pump modedensity is nearly uniform across the fiber core andvanishingly small in the cladding. For large enough Vnumbers (say, V > 10-20) the pump distribution can beapproximated by a square profile ru (r) ( standing foruniform) independent of the azimuthal direction 0. Itsoverlap with any signal mode LPVM, characterized by thecoefficient F,,,u, is then simply [Eq. (11)]

(33)

where ms is the fraction of signal energy contained in thefiber core. For signal modes reasonably far from cutoff,7S 1, and the F coefficient is approximately modeindependent and equal to unity. As expected, all modeswith a reasonably good confinement, i.e., most guidedmodes in a multimode waveguide, experience the samegain.

2.5

2.0

SCz

1.5

1.0

0.5

00 4 8 12 16 20

m - NUMBER

Fig. 4. Evolution of the overlap coefficients Foinm for increasingvalues of the modal number m.


I I I I I I I I

n = _ \

-2

8

I

F."., = s'

This last observation emphasizes two important ad-vantages of pump-to-signal mode matching in a fiberlaser device. First, mode matching can significantlyimprove device performance. For example, the gain ofa fundamental signal mode amplifier will be approxi-mately twice as large with a fundamental than with ahighly multimoded pump distribution (Folo = 2.098vs Fol, 1.0). Second, in an oscillator mode selectioncan be achieved quite easily by making the gain of thedesired signal mode higher than the gain of any othermode. For example, the LP01 mode can have athreshold 17% lower than the next mode (LP1 1, see Fig.4) in a fiber laser pumped with LP01. With a highlymultimoded pump, this type of mode selection wouldclearly not be available as all signal modes experiencethe same gain. Differential mode loss is then thedominant competition mechanism, but in a practicallow-loss device it might not be sufficient to preventmultimode oscillation at higher pump power levels.Although mode selection on the basis of differential gainis by no means unique to fiber laser devices, the smoothmode control and end-pumping configuration availablein'a fiber make it particularly attractive and simple toimplement in a fiber device.

C. Gain and Threshold

From the definition of A, [Eq. (14)] the fiber laserthreshold [Eq. (10)] can be -

Pth = ha, ,. Af (34)O a-f 2 F,.nm

Another pertinent quantity is the incident pump powerthreshold, i.e., the pump power coupled into the fiberat z = 0 at threshold, which can be expressed from Eq.(34) as

Pth,inc = hp', ,,. Af 135t ,rf 2 Fnm [1 - exP(-a1)](

where a', was defined earlier [see Eq. (9)].Of interest to the device design is the dependence of

the threshold and gain on the fiber core radius. Sincethe laser gain grows like the pump intensity and,therefore, like the reciprocal of the fiber active area, itis clearly interesting to reduce the fiber diameter.However, two other major factors also contribute to thisdependence and limit this effect: (1) the Fv,,nm coeffi-cients, which depend on the-fiber V number, and (2) theabsorption of the pump by the fiber. [For small enoughcore sizes the mode is very weakly guided, and its energyis not absorbed by the core, as accounted by the 77pfactor in Eq. (35).] Another somewhat weaker depen-dence arises through; the loss factor ,, which also de-pends on the fiber V number,15 but because of itscomplexity this effect is not considered here. We shallalso assume that the launching efficiency of the pumpinto the fiber is not affected by the core size, which is thecase for most laser pump sources but not necessarily forsemiconductor sources.

Taking all three contributions into account, Pth,inc iSexpected to vary with the core size as follows. For largecore radii, the modal overlap and effective absorption

CrnDr CAflI (MlARNI~I

0E 4 D 10 20 304

:I 8 =5 %/U)WU3

Fig

2

0a.Q-

a-

W_/

90 0 8 16 24

V - NUMBER

Fig. 5. Dependence of the threshold of a Nd:YAG fiber laser on thecore radius for the LPo1/LP01 mode configuration. Parameters are

the same as in Fig. 3, with Xa = 0.6 cm-' and I = 1 cm.

coefficient are independent of a, and the only depen-dence comes from the pump intensity. Then thethreshold grows quadratically with a. For vanishinglysmall core radii, the combination of all three effectsmust give an infinite threshold since in the absence ofthe active medium the gain vanishes. Consequently amaximum gain (amplifiers) and minimum threshold(oscillators) are expected for some optimum core radiusa opt.

This behavior is illustrated in Fig. 5 for a LP0 1/LP 0 1configuration. Here we consider the incident, insteadof the absorbed, pump power threshold. [In the smallcore region of interest the latter becomes vanishinglysmall (p - 0) regardless of the device laser propertiesand fails to characterize adequately the device behav-ior.] With a well-chosen core radius (a apt) theoptical gain per unit pump power is so large that evenin a 5% round-trip cavity the laser threshold can be aslow as 10-50 MtW (for Nd:YAG). For practical fibernumerical apertures (N.A. 0.08-0.25), the optimumcore radius apt which minimizes the fiber laserthreshold is in the 1 .3-3.7-Mxn range, which correspondsto a fiber V number at the signal wavelength of Vopt 1.7. As expected Vopt is comparable with the valueVna. which maximizes the (average) pump intensity inthe fiber (VMnax 1.1 in a step-index fiber). 16 Vopt isnot a universal number as it depends somewhat onpump wavelength (Vopt 1.9 at Xp 810 nm).However, it lies typically in the 1.5-2.5-range (single-mode range) and provides a convenient rule of thumbto minimize the threshold of a fiber laser or optimize thegain of a fiber amplifier.

D. Slope Efficiency

The dependence of Pout on the total pump powerabsorbed by the fiber active medium Pabs is plotted inFig. 6 for a few values of the cavity loss ,, (LPo/LPolconfiguration). Solid curves are exact solutions ob-tained from computer evaluations of Eqs. (7) and (8),while dashed curves represent the approximate form ofEq. (22). As expected, the exact dependence of Pout onPabs is very nearly linear. (Detailed analysis shows that


5

4

Ei

w0a-

I-

a.

0ixw(n

ix

mAL

3

2

00 10

ABSORBED PUMP POWER (mW)

20

Fig. 6. Exact (solid curve) and approximate (dash curves) theoreticaldependence of the output power of a fiber laser oscillator on the ab-sorbed pump power for increasing round-trip cavity loss. Signal and

pump modes are LPol; fiber V number is 40.

2I-

- 8Z6

C-

a)LUw

U)

0

CORE RADIUS (MICRONS)0 12.5 25.0 37.5

0 10 20V - NUMBER

50.0

30 40

Fig. 7. Dependence of the slope efficiency on the fiber V numberfor a LP0 1 signal mode and various pump modes. The parametersare the same as in Fig. 3. The dashed curve represents the far-from-cutoff approximation [Eq. (22)] in the case of the fundamental

pump mode.

the slope efficiency actually increases slightly as thelaser is pumped harder, typically by less than 20%from just above to far above threshold.) Comparisonof the exact and approximate solutions (Fig. 6) indicatesthat the approximation used is quite accurate and valideverywhere above threshold. The discrepancy is small,between 5 and 10% in the examples of Fig. 6, and smallerfor higher efficiencies as anticipated.

The dependence of the slope efficiency of a fiber laser[referenced to absorbed pump power, as in Eq. (22)] onthe fiber V number is shown in Fig. 7 in the importantcase of a fundamental signal mode (LP01) for differentpump modes. The slope efficiency increases rapidlyfrom zero at V - Vc (pump cutoff, no pump energy in-volved in the fiber core) to an asymptotic value at V >>Vc (where the mode profiles no longer depend on V).

The asymptotic value of the slope efficiency given bythe approximate form of Eq. (22) agrees with good ac-curacy. Note that the general behavior of the slopeefficiency referenced to incident pump power is essen-tially the same, except for a lower asymptotic value sinceonly a fraction of the incident power is actually ab-sorbed.

As mentioned earlier, we find that in fairly efficientlasers, the slope efficienI y does not depend strongly onthe modes involved for reasonably similar modes.Under such conditions the relative energy confinement,combined with the lai ge probability of stimulatedemission, provides efficient stimulated relaxation ofmost of the inverted population regardless of the exactphoton distributions.

E. Fiber Oscillator Design

An important consequ ence of the above results is thatthe major advantage o a fiber device over its bulkcounterpart is its higher gain (amplifiers) and lowerthreshold (oscillators). However, no significant im-provement in the slope efficiency of a laser oscillator isanticipated by going frcm a bulk to a fiber configura-tion. Since efficient laser materials already exhibit arelatively low threshold when operated in a bulk form,they will not be significantly more efficient in fiber form.A fiber configuration is truly advantageous for materialswith relatively poor laser properties which normallydisplay high oscillation thresholds in a bulk form. Ofcourse, this is not to say that highly efficient laser ma-terials do not gain from a fiber geometry, which offersseveral additional advantages such as improved thermaland pointing stability and potential miniaturization.

Combining the results of the two previous sectionsallows one to make the best choice of core radius in thedesign of a low-threshold high-slope-efficiency fiberoscillator. A compromise must be made between lowthreshold (ideally a V number of 1.5-2.5) and high slopeefficiency (V > 3-10, depending on the pump mode).In the fundamental case of LP01 modes, and for theparameters used in our example, the optimum value isin the 8 -1 0 -Am range or a V number of 6-8.

For practical technological reasons, it may be diffi-cult, at least for some materials, to fabricate fibers muchsmaller than -40 ,um in diamter with good optical andtransmission properties. However, use of a fibersomewhat larger than optimum size may not signifi-cantly degrade the performance of a laser oscillator.For example, reference to Fig. 5 shows that by doublingthe core radius from an optimum 10 to 20,um, the fiberlaser only requires an additional 0.5 mW to reachthreshold, while the slope efficiency remains nearlyunchanged. In an -30% efficient laser, this 0.5 mWonly results in a 150-MW reduction of the laser outputpower. Consequently, a 4 0-50-Mm core diam fiber lasermade of an efficient material will exhibit essentially thesame performance as a device with an optimized coresize. Furthermore, since in general fiber propagationlosses are also reduced by increasing the fiber core size,in a practical situation the optimum core size may besomewhat larger than that predicted by the present


I

/ l, /8 =1.2%/

>/ 8 2.0 %/

- XP=514 .5 nm /

T/ = 1% /

/ /// ///~~~~

/// , , 4.0%/IIII

model. As a rule of thumb, the performance of a fiberlaser oscillator will be nearly optimum for fiber Vnumbers in the 5-25 range, depending on experimentalconditions.

V. Conclusion

The above analysis provides a convenient thoroughdescription of the intrinsically complex issue of modeinteraction in multimoded laser devices. Use of thenormalized overlap coefficient formalism was shown tobe particularly rewarding for its simplicity and widerange of applications. When applied to guided fiberamplifiers, it allows one to write the gain factor in thesame form as in a free-space amplifier. The gain is theninversely proportional to the fiber core area and pro-portional to a filling factor F. These F coefficients areonly functions of the pump and signal modes involvedin the interaction region for far-from-cutoff modes andcan be easily tabulated. For fundamental signal modeoperation, the F coefficient and the gain are maximumwhen the pump is also in the fundamental mode, whilea uniform pump yields a gain -50% smaller.

This analysis also shows that in fiber lasers the slopeefficiency can be written in a simple accurate form. Forfar-from-cutoff pump modes it is independent of thefiber V number and of the pump mode. In practicalsituations the overall laser efficiency is predicted to beoptimum for a V number in the 5-25 range. These re-sults, derived for a step-index active-core fiber, can beextended in a straightforward manner to more complexindex and gain profiles as well as to other types ofwaveguide geometry. In particular, the free-space laserresults (Sec. III) are directly applicable to the case offibers supporting Gaussianlike modes such as parabolicprofile fibers.

As expected, the main advantage of a fiber configu-ration is the possibility of achieving very high gains andlow-oscillation thresholds. Also, as in other types ofwaveguide laser, differential mode gain offers the pos-sibility of achieving single-mode oscillation. Devel-opment of a new class of laser and laser material, so farrejected for their poorer laser properties and prohibi-tively high oscillation threshold in a bulk form, may,therefore, be advantageous in fiber form. The overlaptheory developed here will then be particularly usefulin the understanding and design of these new devices.

This work was supported by the Air Force undercontract F33615-82-C-1749.

Appendix: LPnm Fiber Modes

Since the fiber numerical aperture is assumed small,the fiber modes are taken to be of the LP type, de-scribed, for example, by Marcuse. 15 In (x,y,z) or-thogonal basis whose z axis points along the fiber axis,the optical field of a LP mode is either x or y polarized.Each polarization mode has nonvanishing field com-ponents [(Ex,Hy) or (Ey,Hx)] described analytically bya set of two expressions, valid in the core and cladding,respectively. For the purpose of evaluating modal

overlap integrals only the mode power density is re-quired given for the LP, mode by

s,(r,o) = 1Ex(vs,)Hy(vsy).2

(Al)

It can be easily shown that the normalized mode powerdensity, which is identical to the photon distributionfunctione used in the text, is the same for the x- andy-polarized LP,A mode and can be expressed analyti-cally as

C {,,,J(r) cos2 vk

s,,,(r0 =,J sin2vOJMrc = 2(Ka) -2 cos2v

c K2( K (yr) -'" 2,a) l sin2vo/

for r a, (A2a)

for r a, (A2b)

where a is the fiber core radius 'y, K and the transverseand longitudinal propagation constants, J, the Besselfunction of order v, K, the modified Bessel function, andC, Y a normalization constant such that the power Pcarried by each mode is a constant. Note that eachLPV,, mode (except v = 0) can have either a cosine or sineazimuthal angle dependence.

-a

A-C

-a

0 a

2.0 x 10-4

LP05

VA A/-3 0 a

1.2 x 10-3

0,30

0RADIAL POSITION

a

Fig. 8. Intensity radial profiles of a few LP modes (far from cutoff).


In the theoretical analysis of fiber laser devices, thepower density is normalized to the cavity volume ratherthan its cross section as is more customarily done. Anyz dependence of the mode energy distributions must,therefore, be included to compute c ,,,. For simplicitywe assume a small single-pass gain, so that the signalintensity is practically constant along the fiber length.This assumption could be avoided by introducing theexact z dependence of the signal, but it can be shownthat it only slightly changes the result while consider-ably complicating calculations. With this approxi-mation the signal normalization coefficient is

222ys 1C",.= 1re,,V2 1J.-A(Ksa)Jv+l(csa)I kAJ)

where s refers to signal quantities, Vs is the fiber Vnumber at the signal wavelength [Eq. (31)], and e, is acoefficient resulting from the azimuthal integrationequal to

12 for v = 0,e,0 = ifov,0.(A4)

In an end-pumped fiber laser device the pump poweris expected to be relatively strongly absorbed as ittravels down the fiber, so that the pump power densityhas a non-negligible z dependence. Calling aa the ab-sorption coefficient of the laser material at Xp, the pumppower density is given by

cnmJ2(Kpr) eXP(-aaZ) 2n r <a (A5a)Cn n P , I~~~~~s n f i a

rnm(r4,oz) = J2(Kpa) 2 rcos2no Ab

Kn( Ya) ,n(yPr) exp(-az) lsin2nor > a.

The normalization constant is now

Cnm = 2-y2*i - al (A6)l7ren 11 Jn_1 (K,,a)J.+,,(Ka)1 [1 - exp(-aal)]

As an example we show in Fig. 8 the power density ofa few fiber modes in a 50-Mum diam core fiber of Nd:YAG(n1 = 1.820, n2 = 1.815) at a signal wavelength of 1.064

ttm. Here the V number is relatively large (40), thefiber can support a large number of modes (V 2/2 =800),14 and the low-order mode distributions are es-sentially independent of signal wavelength.

References1. C. J. Koester and E. Snitzer, "Amplification in a Fiber Laser,"

Appl. Opt. 3, 1182 (1964).2. J. Stone and C. A. Burrus, "Neodynium-Doped Fiber Lasers:

Room Temperature cw Operation with an Injection Laser Pump,"Appl. Opt. 13, 1256 (1974).

3. H. Injeyan et al., "Amplification of Light Propagating Througha Fiber by Evanescent Wave Coupling," in Technical Digest,Conference on Lasers and Electrooptics (Optical Society ofAmerica, Washington, D.C., 1981), paper THJ5.

4. N. Periasamy and Z. Bor, "Distributed Feedback Laser Actionin an Optical Fiber," Opt. Commun. 39, 298 (1981).

5. J. Stone and C. A. Burrus, "Self-contained LED-Pumped Sin-gle-Crystal Nd:YAG Fiber Laser," Fiber Integrated Opt. 2, 1(1979).

6. M. Digonnet and H. J. Shaw, "Diode-Pumped Fiber Laser," FinalTechnical Report AFWAL TR-83-1110 (July 1983).

7. J. P. Budin et al., "On the Design of Neodynium Miniature La-sers," IEEE J. Quantum Electron. QE-14, 831 (1978).

8. S. R. Chinn et al., "Low-Threshold, Transversely ExcitedNdPsO14 Laser," IEEE J. Quantum Electron. QE-1l, 747(1975).

9. K. Kubodera and K. Otsuka, "Single-Transverse-Mode LiNd-P40 1 2 Slab Waveguide Laser," J. Appl. Phys. 50, 653 (1979).

10. K. Kubodera and K. Otsuka, "Laser Performance of a Glass-CladLiNdP 4 01 2 Rectangular Waveguide," J. Appl. Phys. 50, 6707(1979).

11. M. Birnbaum et al., "Laser Emission Cross Section of Nd:YAGat 1064 nm," J. Appl. Phys. 52, 1212 (1981).

12. H. W. Kogelnik and T. Li, "Laser Beams and Resonators," Appl.Opt. 5, 1550 (1966).

13. G. D. Boyd and D. A. Kleinman, "Parametric Interaction ofGaussian Light Beams," J. Appl. Phys. 39, 3597 (1968).

14. D. Gloge, "Weakly Guiding Fibers," Appl. Opt. 10, 2252(1971).

15. D. Marcuse, "Quantum Electronics, Principles and Applications,"in Theory of Dielectric Optical Waveguides, (Academic, NewYork, 1974).

16. D. Marcuse, "Gaussian Approximation of the FundamentalModes of Graded-Index Fibers," J. Opt. Soc. Am. 68, 103(1978).

.


theoretical analysis of optical fiber laser amplifiers and oscillators

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