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The Origin of Excess NoiseG.H.C. New aa Laser Optics and Spectroscopy Group, Department ofPhysics, Imperial College, London SW7 2BZ, EnglandPublished online: 06 Dec 2010.

To cite this article: G.H.C. New (1995) The Origin of Excess Noise, Journal of Modern Optics,42:4, 799-810, DOI: 10.1080/713824416

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JOURNAL OF MODERN OPTICS, 1995, VOL . 42, NO. 4, 799-8 10

The origin of excess noise

G. H. C . NEW

Laser Optics and Spectroscopy Group, Department of Physics,Imperial College, London SW7 2BZ, England

(Received 23 August and accepted 9 September 1994)

Abstract . The key features of excess noise are reviewed and the resultsillustrated using a confocal strip resonator example . The excess noise factor(or Petermann Kfactor ) is shown to be a strong function of the effective Fresnelnumber of the resonator ; for Fresnel numbers in the region of 50, peak valuesof K are greater than 104 . The identity between the excess noise factor andthe injected-wave excitation factor is demonstrated numerically, and bothmathematical and physical interpretations of the high values offered .

1 . IntroductionSome years ago, Petermann [1] showed that the level of spontaneous emission

noise in the output of gain-guided lasers exceeds the `canonical' amountcorresponding to one photon per mode of the radiation field. His paper at firstaroused controversy [2-8], with critics arguing that the result violated thefundamental laws of quantum mechanics . However, it was soon generally agreed thatPetermann's argument was correct, and that the root of the apparent paradox lay inthe use of the word 'mode' to describe field distributions that were not `powerorthogonal' in the normal sense . In most of the systems that one encounters in laserphysics, the modes are (at least approximately) power orthogonal and most of one'sstandard assumptions are built on this premise . Unstable resonators (see [9] for anexcellent early review) are the most obvious examples of systems that violate thepower-orthogonality condition, and their properties consequently seem unusual andeven bizarre . For example, the total power output is no longer the sum of the powersin the individual modes because cross-terms that are zero under power-orthogonalconditions are no longer so in the more general case . Optimal excitation of a modeis not achieved by mode matching in the obvious sense, but rather by the injectionof a time-reversed wave which benefits from the injected-wave excitation (IWE)factor . These systems also exhibit spontaneous emission noise in excess of the levelpredicted by the standard Schawlow-Townes formula . The noise intensity and theassociated laser linewidth are both enhanced by the Petermann Kfactor (or excessnoise factor) which is the main subject of this paper .

In the present paper, the key features of excess noise are reviewed and the resultsillustrated using the confocal strip resonator considered in an earlier publication [10]as an example . The most interesting feature of this unstable resonator is that, whereasin gain-guided lasers only fairly modest enhancements in spontaneous emission arepredicted, the Petermann K factor (or excess noise factor) can exceed 10 4 in this case .The existence of such large values invites consideration of both the mathematicaland the physical origin of excess noise, and these issues are addressed in some detail .

0950-0340/95 $1000 © 1995 Taylor & Francis Ltd .

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The identity between the IWE factor and the Kfactor is demonstrated numericallyin the strip resonator case, providing valuable insight into the origin of excess noise .

2. Basic definitionsThe general space-time structure of a self-reproducing profile propagating in the

positive z direction in an optical resonator is of the form

E+ (s, z, t) = 9 { U(s) exp [i((wt - kz)] } ,

(1)

where M stands for the real part, U(s) is a slowly varying complex amplitudeand s represents an appropriate set of one or more transverse coordinates .The self-reproducing profile propagating in the opposite direction is

E- ( s, z, t) = M {V(s) exp [i(wt + kz)]} .

(2)

Arnaud [11] and more recently Siegman [12] have pointed out that, when a modeis established in an optical resonator, V(s) in equation (2) is the adjoint ratherthan the complex conjugate of U(s) . This arises because the kernel of theKirchhoff-Huygens diffraction integral governing the backward-propagating waveis the transpose rather than the Hermitian conjugate of its forward counterpart .

Wavefront curvature is contained within the complex amplitudes U(s) and V(s) .Note that, if U represents a diverging wavefront, U* represents a convergingwavefront irrespective of the direction of propagation ; so, if one changes U to U*,the sense of the curvature is reversed (figure 1 (a)) whereas, if one sets V = U*(the case of the phase conjugation), the sense of the curvature remains the same

(c)U

VU'

V=U"`

V=U

Figure 1 . The effect of complex conjugation . Irrespective of the direction of propagation,a diverging wave is converted into a converging wave .

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(figure 1 (b)) . In this latter case, the backward wave is of course the time reversal ofits forward counterpart since changing t to - t in equation (1) yields

E + (s, z, - t) = R { U(s) exp [i( - wt - kz)] } _- R { U*(s) exp [i(wt + kz)] } .

(3)

Note that the situation illustrated in figure 1 (b) broadly obtains for a mode in a stableresonator, even if the reflected wave is not normally the strict phase conjugate of theincident beam. When V = U (without complex conjugation), on the other hand, theforward- and backward-travelling waves are both diverging (see figure 1 (c)), whichis the situation in the gain-guided systems that were the subject of Petermann'soriginal work on excess noise .

These seemingly trivial points are important when it comes to evaluating theexcess noise factor . As shown below (equation (13)), when the counter-propagatingwaves of an optical resonator mode shaped as in figure 1 (c), the excess noise factorwill be significantly (possibly much) higher than unity, whereas figure 1(b) is thesignature of a situation where little or no noise enhancement occurs .

3 . Orthogonality relationsThe orthogonality relations satisfied by laser cavity modes are reviewed in this

section. The results are not new but are set out here for clarity and convenience . Mostof the material can be found in [9,11-15] or in references cited therein .

As explained by Arnaud [11] and elaborated in detail by Siegman [12], the mostgeneral property of optical resonator modes is that of biorthogonality as expressedin the relationship

f Ur(S)Vn (s) ds = ABnm, (4)

where the integral is to be performed over the entire cross-section of the beam, 6nmis the Kronecker delta and A is a normalization constant. Note that, in a stableresonator where V- U*, equation (4) reduces to the familiar power orthogonalityrelationship

Um U * ds = ASmn (5)

whereas, in a gain-guided laser where V - U, the same equation impliesorthogonality (without complex conjugation) namely

f UmUn ds = Abmn .

The most general expression for the total power is [14]

W = 1: Wnn + E Wnm ,n

n#mwhere

(6)

(7)

Wnm = g f Un(s)Um(s) ds

(8)

and g is a scaling factor . Clearly, the cross-terms vanish in the power-orthogonal casewhere the total power is simply the sum of the powers in the individual modes, butthis familiar rule does not apply in the general situation .

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4. Excess noise factorIn non-power-orthogonal situations, the level of noise within a self-reproducing

Fox-Li type of mode may be higher, and possibly very much higher, than its valuein the standard power-orthogonal case . The general form of the enhancement factor(the Petermann Kfactor) for the `improper' mode n is

KK = (f . I U„(s)I 2 ds f I V„(s)1 2 ds J//

U„ (s) V„(s) ds

This form of K„ has the advantage of highlighting the essential mathematical originof large excess noise factors which is the strong cancellation in the correlation integralof U with V that occurs when the shapes of the phase fronts of the two waves aresignificantly different (as in figure 1 (c)) . It has already been emphasized that, whenthe shapes are the same (figure 1 (b)), V - U* in which case it is at once clear fromany of the above definitions that the excess noise factor is effectively unity .

In summary, we have identified the important principle that, in any laser wherethe oppositely travelling wavefronts are closely matched, K will be close to unitywhereas, when the match is poor, significant excess noise will occur .

5 . Injected wave excitationFor optimum excitation of a particular laser mode, the obvious technique would

seem to be to match the profile of an injected signal to that of the mode to be excited .This prescription is based, however, on the assumption of power orthogonality .The correct procedure in the general (non-power-orthogonal) case is rather to excitethe mode in a time-reversed sense, that is with a forward wave U(s) = V*(s) or witha backward wave V(s) = U*(s) . The factor by which the power of the mode, whenexcited in this way, exceeds that using the more obvious methods is the IWE factor .Haus and Kawakami [8] and Siegman [14] have shown that the IWE factor ismathematically identical with the excess noise factor . In the next section, we shallsee how time-reversed excitation works in the particular case of a confocal stripresonator and this provides an important clue to the origin of excess noise .

E U„(s)V„(s) ds2

(9)

Note that the value of K,, is independent of the normalization of U„ and V. . Siegman[12,14, 15] (who uses the symbols u and 0 instead of our U and V) chooses to set

In the present paper, however, we choose instead to normalize both U„ and IV„ inthe same way, so that both terms in the numerator of equation (9) are unity .The definition of the excess noise factor now reduces to

2

f , IU„(s) 12 ds= 1 (10)

and to adjust the normalization of V„ so that

f

U„ (s) V,(s) ds = 1 . (11)

In this framework the definition of K„ naturally becomes

K„= f

IV„(s)1 2 ds . (12)

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X

Figure 2 . Schematic diagram of the unstable confocal strip resonator .

6 . The confocal strip resonatorAs an illustration of the results set out above, we return to the example of the

confocal unstable strip resonator studied previously by Lauder and New [10] .As shown in figure 2, this consists of a concave mirror and a convex (feedback) mirrora distance L apart with respective focal lengths f and L + f and sharing a commonfocus. The resonator has a magnification M = I +L/f, and the effective Fresnelnumber FE of the system is given by

FE =(M - 1)a2 ,

( 14)L2

where a is the half-width of the convex mirror and A is the optical wavelength .As before, we examine the case where Llf = 0 . 9 (so that M= 1 . 9) and values of FEare in the range 49-50 . Mode profiles are computed using Southwell's [16,17],inverse source method and these are then refined by one or more round-trip `casts'of the wavefront using the standard Kirchhoff-Huygens integral in conjuction witha fast Fourier transform routine [10] . Typically, 20 virtual apertures and 16 384elements in the transverse mode profiles were employed .

The casting procedure has two functions : firstly to eliminate edge effects that aredifficult to avoid in the Southwell procedure, and secondly to provide generalimprovements in accuracy. Caution must be exercised, however, because only thelowest-order mode (the mode with the largest eigenvalue and consequentlythe lowest round-trip loss) is ultimately stable to repeated casting . Any error in theprofile of another mode will inevitably contain a component of the lowest-ordermode; in view of its lower round-trip loss, this will ultimately dominate if theKirchhoff-Huygens procedure is repeated for long enough

Since this is a strip resonator, s is replaced by the single transverse coordinatex . And as all the formulae in sections 3 and 4 above apply at any z coordinate [12],it is convenient to define the forward-(rightward-)propagating mode U„ (x) on theplane XX in figure 2, just as the wavefront strikes the convex (or feedback) mirror .In view of the magnification factor M, U„ (x) extends from x = - Ma to x = + Maand a little beyond ; since M is nearly 2, roughly 50% of the energy in the mode istherefore lost on every round trip, the precise percentage depending of course on theshape of the mode profile .

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The backward-travelling (adjoint) wave Vn(x) defined on YY, immediately afterreflection of UU(x) from the convex mirror, is given by

z

Vn(x)- U„(x) exp [ - 27tiFE (ax)' jxj < a,

(15)

0,

Ix j > a .

It is the exponential term in this equation that, through the strong cancellation thatit produces in the integral in the denominator of equation (13), is the mathematicalorigin of large excess noise factors .

The profile of the lowest-order even mode (defined for convenience as n = 0) isdisplayed in figure 3 for FE = 49 .4, together with that of the corresponding adjointwave (shown as the broken curve) ; note that the areas under the normalized profilesare 0 . 5 because only one side of the strip is included in the figure . The shapes of thetwo intensity profiles are of course identical for x < a, but the adjoint wave is zerobeyond the edge of the feedback mirror (x > a) ; of course it also includes the phasecurvature of equation (16) which is not evident in the figure . Equation (13) can nowbe used to work out the value of KO which is 290 in this case . The excess noise factorbecomes larger as FE is increased, reaching a peak of nearly 11500 close toFE = 49 .89, as the solid curve in figure 4 shows ; the broken curve in the figure is theresult obtained using unrefined mode profiles and clearly illustrates the benefits ofrefinement by the Kirchhoff-Huygens procedure .

The strong peaking of the excess noise factor seen in figure 4 is associated withrelatively minor changes in Un(x) . Figure 5 shows the mode profile at F E = 49 .89,with the profile of FE = 49 .4 included as a broken curve . At the higher value of FE,the mode exhibits a marked depression on axis, but at first sight one would hardlyexpect the difference between the two curves to cause a change in KK(x) by a factorof almost 30 .

The sensitivity of the excess noise factor to small changes in Un(x) can beunderstood by studying the integral

Cn(x) =J

U„(x')V„(x') dx' .

(16)

Note that the integral in the denominator of equation (13) is Cn(°), which is thesame as Cn(a) since VV(x) = 0 for x > a ; I C, (a)I 2 is therefore the inverse of K,, .From figure 6 which displays ICo(x)I for FE = 49 .4 and 49 . 89, it is easy to understandhow the sensitive dependence of the small final value ICo(a)j arises from cancellationeffects within the integral .

The biorthogonal property of the strip resonator modes can be demonstratedby considering the lowest (n = 0) and next-lowest (n = 2) modes at FE = 49 .4 .Their respective intensity profiles are compared in figure 7, where it is seen that then = 2 mode goes almost to zero on axis. Graphs of Co(x) and CO(x) are presented infigure 8 (the value of IC2(a)l implies an excess noise factor of around 4300), togetherwith the function IB02(x)l where

BOZ(x) = J

Uo(x')V2(x') dx' .

(17)

According to equation (4), the biorthogonality of the two modes is demonstrated if

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wU)0z

UIwUXw

12000

0F-UQ 8000 -

4000-

Origin of excess noise

80 5

x~a

Figure 3 . Intensity profile of the lowest order (n = 0) mode for F E = 49 . 4 ; the profile of theadjoint wave is shown as a broken curve .

049 .4

49 .6

49 .8

50.0

50.2EFFECTIVE FRESNEL NUMBER

Figure 4 . The excess noise factor as a function of effective Fresnel number (-): (---),curve obtained using unrefined mode profiles ; (*) data obtained by a separatecomputation of the IWE factor .

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xU

6.0E-2 -

4 .0E-2 -

2.0E-2 -

0 .0E+0

0.00 1

x/aFigure 5 . Intensity profile of the lowest-order (n = 0) mode for FE = 49 . 89 ; th,

mode in figure 3 is shown as a broken curve for comparison .

r

lA

fN N4

0x/a

Figure 6 . IC(x)l for FE = 49 .4 and FE = 49 . 89 as indicated .

49.40

49 .89

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0 i

x/aFigure 7 . Intensity profile of the next-lowest-order (n=2) mode for FE = 49 . 4 ; the profile

of the lowest order mode is shown as a broken curve .

I

x/a

Figure 8 . IC(x)I for (a) the n = 0 and (b) the n=2 mode for FE = 49 . 4. (c) The modulus ofB02 (x) in equation (17), which should go to zero at s = a in the case of perfectbiorthogonality .

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B02(a) = 0, a property that is obeyed to a reasonable approximation . The residualerror almost certainly arises from inaccuracy in U2(x) which cannot be refined to thesame degree as Uo(x) for the reasons explained earlier .

7. Interpretation and discussionThe first reference to the possibility of an excess noise factor appears to be in

Siegman's [13] 1979 paper . On the basis of a simple geometrical argument, he notedthat in an unstable resonator there were more atoms that could contribute tospontaneous emission than to gain . While undoubtedly correct, this argument coversonly one aspect of the problem ; it is difficult to see how it could explain the extremelylarge values of K occurring in the example studied above . On the other hand, thevery size of K invites some physical explanation . What simple picture can be offeredto explain the existence of noise levels more than ten thousand times larger than ina normal stable resonator?

The clue is provided by the identity between the Kfactor and the IWE factormentioned in section 5 above. This suggests that, if one could understand thephysical origin of large IWE factors, one would understand large Kfactors as well .To discover the physics of IWE, the Kirchhoff-Huygens procedure was used to tracethe evolution of the field in the strip resonator under optimal injection by atime-reversed replica of a Fox-Li mode . The n = 0 mode was first established in theresonator for a particular value of F E , and the direction of propagation then reversed,the wavefront Uo(x) on XX being sent back leftwards towards the concave mirror(see figure 2) . It is important to recognize that the subsequent evolution is not thesame as a time-reversal of the normal self-reproducing mode propagation, becauseat each subsequent reflection from the convex mirror the portion of the mode thatin forward propagation is lost to the outside world is absent .

10

0x/a

Figure 9. Transient evolution of the lowest-order mode for FE = 49 . 89 following timereversal of the wave-front . The intensity profile is shown two, four, six, eight and tentransits after the event .

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The geometry of the resonator ensures that, for the first few transits followingtime reversal, the field energy is tightly confined close to the resonator axis .This phenomenon is clearly illustrated in figure 9 which traces the transientevolution of the lowest-order mode at FE = 49 . 89, showing, in particular, theintensity profiles two, four, six, eight and ten transits after time reversal . Axialconfinement has the crucial consequence that the energy loss per transit istemporarily much smaller than that experienced by the self-reproducing mode . Afterall, for the n = 0 mode, roughly half the energy is normally lost on each transit byspillage round the convex mirror ; so, if one could ensure that in an idealized casethere was no loss at all for (say) 14 transits after time reversal, the energy advantageof time-reversed excitation would be about 2 14 , a value of the required magnitude .This is the physical origin of high IWE factors . In the actual strip resonator, therelative advantage experienced by the reversed wave approaches a factor of two inthe first few transits but falls away gradually until the mode is fully re-establishedafter about 40 transits. To demonstrate the identity of the IWE and Kfactors in thestrip resonator, the cumulative energy advantage was computed during the returnto the self-reproducing state . The results obtained in this way are represented byasterisks in figure 4 ; these are not the points through which the solid curve is drawnbut are derived from an entirely independent set of numerical data . Of course, thefact that the asterisks lie on the curve is inevitable given the identity of the IWE andK factors, but the exercise does nevertheless constitute a nice demonstration of howhigh IWE factors arise physically .

The natural physical explanation of excess noise is that it arises from spontaneousemission directed towards the concave mirror that, by temporary confinement alongthe axis, acquires an energetic advantage over stimulated emission supporting thelossy self-reproducing Fox-Li mode . This is in line with the two relatedinterpretations of the K factor offered by Goldberg et al . [18] . These workers showfrom a quantum-optical analysis that excess noise can be attributed to a combinationof (internal) spontaneous emission and (external) vacuum fluctuations leaking intoa lossy cavity from outside ; the precise balance between these contributions dependson the arbitrary choice of operator ordering, but both components are subject topreferential amplification in a lossy cavity, leading to a laser linewidth enhanced bythe K factor .

Is there a fundamental upper limit to the magnitude of the K factor? It can easilybe shown that the underlying trend is for K to increase with increasing FE . Moresignificantly, a simple geometrical argument based on losses incurred under IWEsuggests that, all other things being equal, the IWE factor in two transversedimensions should be the square of the value in one dimension. However, apreliminary study of a cylindrical resonator suggests that the massive enhancementsthat squaring would imply are not in fact realized ; it appears that the mode profilesconspire to offset the effect of the extra dimension . For example, the modes of aconfocal unstable resonator in cylindrical geometry are peaked on the axis, and theround-trip spillage is therefore far less than it would be if the profiles resembledtop-hat functions . More work is needed to establish this point in detail .

As a historical footnote, it should be recalled that, in the same year thatPetermann [1] published his original paper and Siegman [13] examined theorthogonality properties of unstable resonator modes, Corkum and Baldis [19]reported the particular sensitivity of unstable resonators of external feedback . Allfour can now be seen to have been examining the same basic phenomenon .

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Origin of excess noise

8. SummaryThe key properties of excess spontaneous emission have been reviewed using a

confocal strip resonator as an example . Excess noise factors greater than 104 occurin this case . The relationship with the phenomenon of enhanced IWE has beendemonstrated numerically and both mathematical and physical insight into theseinteresting phenomena offered .

AcknowledgmentsNumerous valuable discussions with Tony Siegman and Paul Mussche over the

past 8 years are gratefully acknowledged . Thanks are also due to Paul Corkum whodrew the author's attention to his 1979 experiments on the sensitivity of unstableresonators to extracavity feedback . Michael Rippin performed the preliminary workin two transverse dimensions mentioned at the end of section 7 .

Financial support for the project was provided in part by the Engineering &Physical Sciences Research Council and by the European Community .

References[1] PETERMANN, K ., 1979, IEEE J. quant . Electron ., QE-15, 566 .[2] STREIFER, W., SCIFRES, D . R., and BURNHAM, R. D., 1981, Electron . Lett ., 17, 984 .[3] PATZAK, E ., 1982, Electron . Lett ., 18, 278 .[4] MARCUSE, D., 1982, Electron . Lett ., 18, 920 .[5] YARIV, A ., and MARGALIT, S ., 1982, IEEE J. quant . Electron ., QE-18, 1831 .[6] ARNAUD, J ., 1983, Electron . Lett ., 19, 688, 798 .[7] AGRAWAL, G ., 1984, J. opt . Soc. Am . B, 1, 406 .[8] HAUS, H . A., and KAWAKAMI, S ., 1985, IEEE J. quant . Electron ., QE-21, 63 .[9] SIEGMAN, A . E ., 1974, Appl. Optics, 13, 353 .

[10] LAUDER, M. A., and NEW, G . H . C ., 1988, Optics Commun., 67, 343 .[11] ARNAUD, J ., 1976, Beam and Fibre Optics (New York: Academic Press) pp . 122-3, 175 .[12] SIEGMAN, A. E ., 1986, Lasers (Oxford University Press) .[13] SIEGMAN, A . E ., 1979, Optics Commun ., 31, 369 .[14] SIEGMAN, A . E ., 1989, Phys. Rev . A, 39, 1253 .[15] SIEGMAN, A . E ., 1989, Phys. Rev . A, 39, 1264 .[16] SOUTHWELL, W . H., 1981, Optics Lett ., 6, 487 .[17] SOUTHWELL, W . H., 1986, J. opt . Soc . Am . A, 3, 1885 .[18] GOLDBERG, P., MILONNI, P. W., and SUNDARAM, B ., 1991, Phys. Rev . A, 44, 1969 .[19] CORKUM, P . B., and BALDis, H . A ., 1979, Appl. Optics, 18, 1346 .

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