784 OPTICS LETTERS / Vol. 16, No. 11 / June 1, 1991

Intensity noise enhancement in the half-wave plate/polarizerattenuator

Brian H. Kolner*

Hewlett-Packard Laboratories, P.O. Box 10350, Building 26M16, Palo Alto, California 94303-0867

Received December 12, 1990

The conventional optical attenuator for linearly polarized light is usually constructed with a half-wave retardationplate and a polarizer. With one axis of the polarizer aligned so as to transmit the incident beam fully, the addition ofthe half-wave plate causes the output power to follow a cos

2 20 dependence, where 0 is the angle between the opticaxis of the wave plate and the incident polarization. When the incident light has an additional orthogonalpolarization component, the output power dependence becomes more complex and depends on the correlationbetween the two polarization fields (ExE,). More significantly, if amplitude noise in the polarization fields iscorrelated, the attenuator will couple the noise processes, which results in relative power fluctuations that increaseas the optical power is reduced. The noise produced by this coupling may even exceed the noise of eitherpolarization state alone. Measurements of the statistical behavior of the relative intensity noise of a cw mode-locked Nd:YAG laser as a function of the Wave-plate angle showed more than a tenfold increase when the outputpower was reduced to near minimum. In addition, the noise was found to be asymmetrical about the minimumpower point, 0 = 45°. It is shown that the simple addition of a polarizer ahead of the wave plate strips off theunwanted polarization field and virtually eliminates this added noise effect.

Variable optical attenuators are an important tool inalmost all optical laboratories. A common configura-tion suitable for linearly polarized light uses a half-wave retardation plate followed by a polarizer. Thisconfiguration is especially useful for high-average-power sources since both components can be dissipa-tionless.

In this Letter it is shown that if the laser source isnot perfectly linearly polarized (that is, there is somepower in the unwanted orthogonal field), then the useof the variable attenuator described above can actual-ly increase the amount of relative intensity noise as thenet power is reduced. In addition, the attenuator willcouple the noise processes inherent in the orthogonalpolarization fields, and if the noise processes are sta-tistically correlated, this coupling causes a multiplica-tive effect and the relative noise level can increasebeyond that of either polarization state alone.

The basic noise behavior of the variable attenuatorcan be understood by first assuming that the lasersource is composed of two orthogonal electric fieldcomponents, one much larger in magnitude than theother. In Fig. 1 the strong component is labeled Ex,and the weak component is labeled Ey. As the waveplate is rotated through an angle 0, the orientations ofE. and Ey rotate at the rate 20. The vector projectionof the two components on the axes of the polarizerresults in an approximately cos2 20 form for the outputpower. Note, however, that as the amplitude of thestrong x component is reduced, the amplitude of theweaker y component is enhanced. If they componenthas larger relative noise than the x, then the relativeoutput noise power of the attenuator is increased.

Most lasers are designed to operate in a single linearpolarization state. This is usually accomplished byusing intracavity polarization-dependent loss ele-

ments. For example, windows, mode lockers, dye jets,and lasers rods can be used at Brewster's angle. Also,dielectric mirror stacks used at other than normal inci-dence can be polarization sensitive. In the case wherethese techniques are not 100% effective, there will belight in the unwanted polarization state that is eitherbelow or slightly above threshold. It is well knownthat a laser is noisier near threshold than well above orbelow it,l,2 and thus it is surmised that this is thereason that the unwanted polarization produces high-er relative noise than does the wanted polarization.

To measure the relative noise as a function of wave-plate angle, a simple attenuator was assembled, andthe pulses from a mode-locked laser were analyzedwith a sampling oscilloscope. The laser was a Coher-

241Polarizer

EY 0=0°

X/2

Fig. 1. Variable optical attenuator. Rotation of the domi-nant x-axis field component for rejection by the polarizeralso rotates the weaker y-axis field for transmission.

0146-9592/91/110784-03$5.00/0 © 1991 Optical Society of America

June 1, 1991 / Vol. 16, No. 11 / OPTICS LETTERS 785

two orthogonal electric fields with complex ampli-tudes E.(t) and Ey(t) that have a magnitude and phasethat are random processes, the amplitudes can be ex-pressed as4

E.(t) = E0 x(t)exp[ikx(t)],

EY(t) = E 0Y(t)exp[iky(t)], (1)

0

0.85 1.0

10.20'

h..... 30'

145°

1.15

Normalized Pulse Amplitude

Fig. 2. Histograms of normalized mode-locked laser pulseamplitudes as a function of the attenuator wave-plate angle0.

ent Antares cw mode-locked Nd:YAG laser producing70-ps pulses at an 80-MHz rate with 23 W of averagepower. Two air-glass interfaces served as 4% beamsplitters to reduce the overall power of the beam,which was then passed through a multiple-order 1.06-,m wave plate and a calcite polarizer that made up thevariable attenuator. All reflecting surfaces ahead ofthe wave plate and polarizer were used at nearly nor-mal incidence to avoid any polarization effects. Thereceiver consisted of a compensated variable reflec-tance attenuator (NRC 925B), a high-speed photodi-ode,3 and a Hewlett-Packard 54120 sampling oscillo-scope. The sampling oscilloscope contains a histo-gram feature that allows one to make statisticalstudies of both amplitude and timing fluctuations.The time window in which voltage counts were takenwas set to approximately zero picoseconds at the peakof the mode-locked pulse. Histograms containing 105counts each were taken at 20 angular intervals of thewave plate in the ranges 00 < 0 < 340 and 56° < 0 < 900and in 10 intervals in the range 360 < 0 < 560. Figure2 shows a set of histograms taken at the indicatedintervals to show the spreading of the Gaussian-likeprobability density function as the wave plate is ad-justed from maximum power (0 = 0°) to minimumpower (0 = 45°). Note the slight narrowing as 0 ap-proaches 450.

To maintain consistency, all histograms were re-corded with the same peak photocurrent from thehigh-speed photodiode. Too much power into thephotodiode causes compression and an apparent re-duction in the amplitude noise. Too little powercauses an excess noise contribution from the input ofthe sampling oscilloscope (thermal plus quantizationplus noise figure). The operating point was selectedto be just below the point at which a slight (1%) broad-ening of the photodiode impulse response was ob-served (ipk = 3 mA). After the mean and the standarddeviation were determined, the standard deviationwas normalized to the mean peak voltage and recordedas a percentage.

If the output of the laser is modeled as consisting of

where E 0 A(t) and E 0y(t) are real nonzero mean randomprocesses and ok(t) and oy(t) are uniformly distribut-ed on the interval [-7r, 7]. The full space-time depen-dence of the electric field is then recovered by multi-plying by expl(cot - kz)]. The half-wave retardationplate with its optic axis oriented at 0 with respect tothe x axis rotates the electric-field vectors by 20. Tocalculate the power transmitted through the polarizer,we find the vector projection of the rotated fields onthe polarizer's axis. This combination of the originalorthogonal fields leads to the coupling of the noiseprocesses through the cross-product terms in thePoynting vector. With this in mind, it can be shownthat the time-averaged intensity out of the attenuatortakes the form

(P) = 2° {EOX2(t)cos 2(20) + EOY2 (t)sin2 (20)2

+ E,,(t)E 0y(t)cosk[0x(t) - Y(t)]sin(40)l, (2)

where e0is the vacuum permittivity and c is the speedof light. The interval of the time average is long com-pared with an optical cycle and short compared withthe fluctuations of the random processes. Thus(Eox2(t)) = Eo0

2(t) and so on for the other terms. It isevident that there are contributions from the power ineach separate field as well as a term proportional to thecross correlation or coherence between the random-ness of the fields.

In general, to proceed further would require knowl-edge of the probability density functions of the ran-dom processes. However, with a few simple assump-tions, the general behavior of the power fluctuationscan be modeled and can thereby illustrate the depen-dence on the wave-plate angle 0. Since the electric-field amplitudes E 0 A(t) and E 0y(t) are nonzero meanrandom processes, then the square of the fields are alsononzero mean random processes. Each second-orderelectric-field term in Eq. (2) can then be written as thesum of a constant power plus a zero mean randomprocess. This will simplify the calculations of themean and standard deviations below. Equation (2) isnow

(P) = [PX + APl(t)]CoS2(20) + [Py + AP (t)]sin2 (20)

+ [PXY + APxY(t)]sin(40), (3)

where Px, PYs, and Pxy are constant powers (i.e., meanpowers for long averaging times) and the other termsrepresent the fluctuations in the powers of the x and yfields and the fluctuations in the power produced bythe correlation between the two fields.

Now, the mean power (P) is expressed by simplyomitting the time-varying terms in Eq. (3). The vari-ance is then calculated in the usual way,

Counts

12,000

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786 OPTICS LETTERS / Vol. 16, No. 11 / June 1, 1991

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0.8

0.6 >!9CD

0.40fCD

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0.00 15 30 45 60 75 90

Wave-plate Angle (degrees)

Fig. 3. Average power and normalized standard deviationof the half-wave plate/polarizer attenuator as a function ofthe wave-plate angle. Circles: average power data with asolid curve theoretical fit ((P)). Triangles: standard devi-ation of power fluctuations as a percentage of the averagepower with a solid curve theoretical fit (V/(P)). Squares:same as the triangles with inclusion of a prepolarizer to stripoff unwanted polarization components; the solid line isshown for reference.

I = ((p) _ (p))2

= AP 2 (t) cos4 (20) + APy2 (t) sin4 (20)

+ 2AP.(t)APY(t)cos2(20)sin2(20)

+ 2 AP (t)APyW(t) cos2(20)sin(40)

+ 2APY(t)AP~Y(t) sin2 (20)sin(40)

+ APxY2 (t)sin2 (40). (4)

The coefficients on the first two terms in this ex-pression represent the variance in the fluctuations ofthe x-axis power and the y-axis power, respectively.They can be measured independently by setting thewave plate toO = 00 to obtain ¢x2 = APx2(t) and then to0 = 450 for a~y2 = APY2 (t). The coefficient on the nextterm is the cross correlation between the power fluctu-ations on the two axes, and the coefficient on the lastterm is related to the covariance between the two or-thogonal electric-field components. The remainingtwo terms involving sin(40) are more difficult to as-cribe a physical meaning to but are obviously relatedto the product of each axis' power fluctuations and theprocess describing the correlation power APxy(t).

Figure 3 shows a plot of the square root of Eq. (4)normalized to the mean power (P). Also shown aredata measured using the built-in histogram recordingfeatures of the Hewlett-Packard 54120 sampling oscil-loscope (the triangles). The curve described by Eq. (4)was fitted to the data by carefully adjusting the lastfour coefficients. In addition, the average power outof the attenuator is plotted: both the measured (thecircles) and the calculated (the solid curve) valuesfrom Eq. (3). Several surprising features are apparentin Fig. 3. The first and most significant is that, ingeneral, the relative amplitude noise increases as theoutput power decreases. This was readily explainedby the model put forth above wherein the weak and

noisy orthogonal component is rotated to pass throughthe polarizer as the stronger, quieter field is attenuat-ed. Second, there is a dip in the relative noise at theminimum-power point 0 = 450. This can be under-stood by noting that at 0 = 450 only the fluctuations onthe y axis are measured. As the wave plate is rotatedfrom this point, terms that contain the product of thex-axis fluctuations with the y-axis fluctuations start toadd to the relative noise but eventually die out as the xaxis dominates. Finally, there is an asymmetry aboutthe 0 = 450 point. This arises because on one side ofthis point the correlated amplitude fluctuations add inphase while on the other side they add out of phase.The reader can verify this by drawing a sketch of twoorthogonal but unequal electric-field vectors with asmall time-varying length extension. For an orienta-tion close to 0 = 450, where the vector projections addin phase, rotation to the other side of 0 = 450 results inthe projections' adding out of phase.

To verify that it was indeed the unwanted polariza-tion field responsible for the noise effects, a polarizerwas added ahead of the attenuator to strip off most ofthe orthogonal component. The data at the bottom ofFig. 3 (the squares) show the substantial improvementobtained. The noise was reduced by more than anorder of magnitude in the worst-case region 400 < 0 <500. The straight line is included for reference.

In summary, it has been shown that in the presenceof an imperfectly polarized laser source, the standardhalf-wave plate/polarizer attenuator can dramaticallyincrease the relative intensity noise of the attenuatedbeam. A simple model was advanced based on thecoupling between the two orthogonal field compo-nents in the attenuator and the nonzero mean randomprocesses that describe the noise in the two fields.The simple addition of a polarizer ahead of the attenu-ator was seen to virtually eliminate this noise effect.This should prove useful for applications in nonlinearoptics such as fiber grating pulse compression, wherethe intensity of fairly high-power beams must be set toa precise level. Finally, it may be that this effect isubiquitous and present in many systems that use os-tensibly linearly polarized lasers, birefringent trans-mission media, and polarization-selective devices. Itwill be interesting to see if it is observed in othersystems.

The author thanks P. R. Robrish and L. S. Cutler forstimulating discussions on this topic and N. C. Luh-mann for reading and commenting on the manuscript.

* Present address, Department of Electrical Engi-neering, University of California at Los Angeles, 405Hilgard Avenue, Los Angeles, California 90024-1594.

References

1. C. Freed and H. A. Haus, Appl. Phys. Lett. 6, 85 (1965).2. J. A. Armstrong and A. W. Smith, in Progress in Optics,

E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI.3. S. Y. Wang, K. W. Carey, and B. H. Kolner, IEEE Trans.

Electron Devices ED-34, 938 (1987).4. J. W. Goodman, Statistical Optics (Wiley, New York,

1985).