The Uncertainty Principle in Classical Optics · to Heisenberg’s uncertainty relation x p x > h. Figure 2 shows the intensity and phase ... imposed by the uncertainty principle

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expand as they propagate along Zand, although their centers driftapart, there is the distinct possibilitythat they will never be completelyseparated. Roughly speaking, we ex-pect the beams to remain more orless collimated between z = 0 and z =D2/�, the Rayleigh range2 for a beamof diameter D and wavelength �. Ifat the Rayleigh range the distancebetween the beam centers is greaterthan D, the beams should be separa-ble; otherwise their drifting apart willgo hand in hand with their expan-

sion, and the beams remain entangled asthey propagate beyond the Rayleigh range.The necessary condition for separability isthus (D2/�)�� > D, or equivalently,

D�kx > 2�. (1)

The lower bound 2� on the product of Dand �kx appearing in Ineq. (1) is not ex-act, but depends on the definition of beamdiameter D and the adopted criterion forseparability, which are typically imprecise.For all practical purposes, the number ap-pearing on the right-hand side of Ineq. (1)should be on the order of unity, say,greater than 1 but less than 10.

Invoking the quantum nature of light,if the aperture diameter D is interpreted asa measure of the uncertainty �x about thephoton position along X, while �kx is re-lated (through the relation p = -hk) to thelinear momentum uncertainty �px alongthe same axis, then Ineq. (1) is equivalentto Heisenberg’s uncertainty relation �x �px > h.

Figure 2 shows the intensity and phaseprofiles of two plane waves as well as thoseof their superposition at the aperture de-picted in Fig. 1 (diameter D = 500�). Thephase distributions in Figs. 2(b) and 2(d)indicate that one of the beams is slightlytilted towards the upper right corner ofthe XY-plane, while the other is tilted byan equal amount towards the lower leftcorner. The angular separation betweenthese beams is �� = 0.23° = 0.004 radians.The combined beam’s intensity distribu-tion in Fig. 2(e) reveals the angular separa-tion of the two superimposed beamsthrough a tell-tale fringe pattern.

The Uncertainty Principle in Classical OpticsMASUD MANSURIPUR

In the classical electromagnetic the-ory the wave-vector k = (2�/�)�

underlies the Fourier space of propa-gating (or radiative) fields. The k-vector combines into a single entitythe wavelength � and the unit vector� that signifies the beam’s propaga-tion direction. The Fourier trans-form relation between the three-di-mensional space of everyday experi-ence and the space of the wave-vec-tors (the so-called k-space) gives riseto relationships between the two do-mains analogous to Heisenberg’s un-certainty relations.

Considering that in quantum theorythe electromagnetic k-vector is propor-tional to the photon’s momentum1

(p = -hk , where -h = h /2�, h being thePlanck constant), one should not be sur-prised to find relationships between di-mensions of a beam in the XYZ-space andits momentum spread in the k-space. Suchrelationships impose fundamental limitson the ability of measurement systems todetermine the various properties of elec-tromagnetic fields.

In this article we address two problemsthat have widespread applications in opti-cal metrology, spectroscopy, telecommu-nications, etc., and discuss the constraintsimposed by the uncertainty principle onthese problems. The first topic of discus-sion is the separation of two overlappingbeams of identical wavelength havingslightly different propagation directions.This will be followed by an analysis of thelimits of separating co-propagating beamshaving slightly different wavelengths.

Angular separationand the limit of resolvabilityFigure 1 shows an aperture of diameter D,which transmits two plane waves of thesame wavelength � propagating in slightlydifferent directions. Denoting the angularseparation between the beams by ��, wefind that the projections of the two k-vec-tors along the X-axis differ by �kx ≈(2�/�)��. In geometrical optics, rayspropagate along straight lines and, there-fore, the two beams must separate fromeach other after a certain propagation dis-tance. In wave optics, however, the beams

44 Optics & Photonics News � January 2002

Figure 1. Two beams of the same wavelength �,propagating in slightly different directions, passthrough an aperture of diameter D.The angle be-tween the two k-vectors is ��, giving rise to �kx ≈(2�/�)��.The beams separate from each other atthe observation plane located a distance z fromthe aperture, provided the uncertainty relation D�kx ≥ 2� is satisfied.

Figure 2. Plots of intensity (left) and phase (right)at the entrance aperture of the system of Fig. 1.Two uniform beams, one propagating with a slighttilt toward the upper right, another with a slighttilt toward the lower left, enter a D = 500� aper-ture.The angular separation of the beams is �� =0.23º.The individual beams are shown in the top(a, b) and the middle (c, d) rows; their superposi-tion appears at the bottom (e, f ).

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January 2002 � Optics & Photonics News 45

use a high-NA lens, thus enhancingthe polarization effects. The short-est focal length f is obtained whenthe NA of the lens is close to unity,that is, f ≈ 1_

2D. Figure 5 shows com-puted plots of intensity distributionat the focal plane of an NA = 0.99,f = 250� lens, when the incidentbeam is the two-beam superposi-tion depicted in Fig. 2.3 The threecolumns of Fig. 5 represent threedifferent polarization states. In (a)both incident beams are linearly po-larized along X, which explains theelongation of the spots in this par-ticular direction. In (b) the twobeams are linearly polarized at 45°to the X- and Y-axes, i.e., the direc-tion along which the spots are sepa-rated from each other. The plot in(c) corresponds to the case whenboth beams are circularly polarized.Frames (d)-(f) are the logarithmicversions of those in (a)-(c), showingtheir detailed structure by empha-sizing the weaker regions. Since theassumed values of D = 500� and ��= 0.004 rad satisfy the uncertaintyrelation in Ineq. (1), the focusedspots are seen to be resolved irre-spective of their polarization state.

Angular discrimination by means of a Fabry-Pérot etalonAnother device that can, in principle, ac-complish the separation of two beams viaangular discrimination is a Fabry-Pérotetalon,4,5 such as that shown in Fig. 6. Thisparticular etalon is tuned to transmit aplane wave of � = 633 nm at the incidenceangle of � = 45°. Figure 7 shows theetalon’s computed reflection and transmis-sion coefficients, rs = |rs|exp (i�rs) and ts =|ts| exp(i�ts), versus � for an s-polarizedplane-wave of � = 633 nm. It turns outthat the shapes of the transfer functions|rs (�)| and |ts(�)| are not quite suitable for complete separation of two finite-diameter beams of differing propagationdirections.

Computed plots of intensity distribu-tion in Fig. 8 confirm that the etalon ofFig. 6 can only partially separate twobeams of diameter D = 2 � 104� and angu-lar separation �� = 0.115° = 0.002 rad,even though the value of D (��/�) = 40 inthis case amply satisfies Ineq. (1). Figure8(a) shows the incident pattern of intensi-ty distribution of the superposed beamsupon arriving at the etalon. One of these

When the composite beam(whose intensity and phase distribu-tions are shown in Figs. 2(e, f)) ispropagated along the Z-axis, one ob-tains at various distances from theaperture the intensity patterns dis-played in Fig. 3.3 It is seen in thesepictures that the two constituentbeams continue to overlap at first,giving rise to interesting interferencepatterns. After a sufficient propaga-tion distance, however, the beamsseparate and go their own ways. Theassumed value of D �kx in this ex-ample is 4�, which satisfies Ineq. (1).

Separating two beamsby means of a lensIn the preceding section it wasdemonstrated that separating twobeams of a certain angular distance�� requires a minimum beam diam-eter D in accordance with Ineq. (1).It may be asked whether a similarlimitation exists on the propagationdistance z before the individualbeams can be resolved. Apparentlyno physical law limits the requireddistance z, although practical con-siderations seem to impose certainconstraints. In free space, the re-quired propagation distance is typi-cally less than or equal to the Rayleighrange, D2/�, but one can substantially re-duce this distance by employing a lens, asshown in Fig. 4. Here two overlappingbeams of diameter D and angular separa-tion �� are resolved after going throughan aberration-free lens. In the focal planeof the lens the center-to-center spacing ofthe focused spots is f ��, which must begreater than the Airy disk4 radius of ~0.6�/NA = 1.2 f �/D. Note that the resolv-ability criterion is independent of f andNA, requiring only that D (��/�) > 1.2,which is a statement of the uncertaintyprinciple in the present context. The re-quired propagation distance f in this ex-ample can be much less than that neededin the case of free-space propagation ofFig. 1. It must be emphasized that the un-certainty principle does not impose anyconstraints on z, the requirement for re-solvability being only a restriction on theproduct of D and ��.

An interesting feature of separatingtwo beams by means of a lens is the result-ing dependence of the focused spots onthe state of polarization. To reduce the re-quired propagation distance z, one may

Figure 3.Two overlapping plane waves depicted inFig. 2 propagate along the Z-axis. The various in-tensity patterns in frames (a) to (o) are obtained atz/(103�) = 1, 2, 3, 10, 20, 30, 40, 50, 60, 70, 80, 90,100, 125, and 150, respectively. Initially the beamsstrongly interfere with each other, but as propaga-tion proceeds, they separate and exhibit their indi-vidual identities.

Figure 4.Two identical beams of diameter D andangular separation �� may be isolated after goingthrough an aberration-free lens. In the focal plane,the distance between the focused spots is f ��,which must be greater than the Airy disk radius of1.2 f �/D if the individual spots are to be resolved.

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this case the beam diameterD turns out to be irrelevant,but the available propagationdistance z is critical for iso-lating the individual beams.

A straightforward methodof separating two beams ofdiffering wavelengths isshown in Fig. 9. This Mach-Zehnder interferometer4

splits each input beam intotwo equal halves, provides aseparate path for each half,then recombines the halvesinto a single beam at the out-put. For one of the wave-lengths, say, �1, the path-length difference �z betweenthe two arms of the device

may be an integer-multiple of �1, in whichcase the corresponding half-beams inter-fere constructively and emerge from oneexit channel of the interferometer. For theother wavelength, �2, the path-length dif-ference may be a half-integer-multiple of�2, in which case interference will be de-structive and the beam will emerge from adifferent exit channel of the device. There-fore, separability condition for this inter-ferometer is �z/�1 - �z/�2 = 1_

2 , or

�z �kz ≈ 2� �z ��/�2 = �. (2)

Figure 10 shows computed detector signalsS1, S2 of the system of Fig. 9 versus the in-put wavelength in the vicinity of � = 633nm.3 For the particular path-length differ-ence chosen in this example (�z = 1.266mm), it is observed that, in compliancewith Eq. (2), a pair of beams having �� =0.158 nm can be readily separated fromeach other.

An alternative form of the uncertaintyrelation may be obtained in this case by in-voking the quantum mechanical relationbetween the magnitude k of the wave-vec-tor and the photon energy E = h, namely,k = 2�/� = 2�/c = E/-hc. For two beams ofwavelengths � and � + ��, co-propagatingin the Z direction, �kz = �E/-hc. Also �z =c�t , where c is the speed of light and �t isthe time needed for light to travel a dis-tance �z in free space. The product �z �kzis thus proportional to �E �t, with -h beingthe proportionality constant. One maythus reinterpret Eq. (2) as a statement ofthe time-versus-energy uncertainty. Whenthe observations are made in a transparentmedium of refractive index n > 1, the in-crease of the k-vector by a factor of n dic-tates a corresponding decrease in �z. This

beams propagates along thedirection that makes a 45° an-gle with the etalon’s surfacenormal, while the other devi-ates from this direction by ��= 0.115°. The reflected inten-sity profile depicted in Fig.8(b) contains mostly the latterbeam, plus a small fraction ofthe former. This is due to theimperfect transfer function ofthe etalon, which cannot fullytransmit the angular spec-trum of the 45° beam, nor canit fully reflect the spectrum ofthe 45.115° beam. Eitherbeam’s angular spectrum hasa width of ~�/D ≈ 0.003°,which would readily passthrough a narrow rectangular transferfunction, but is partially blocked by thesharply peaked transfer functions of theetalon (see Fig. 7(a)). The same argumentsapply to the transmitted intensity distrib-ution shown in Fig. 8(c) which, althoughprimarily composed of the 45° incidentbeam, still contains a fraction of the45.115° beam.3

To summarize the results of this andthe preceding sections, there are severalways of separating two overlapping beamsof the same wavelength and differingpropagation directions. Some of thesemethods may be more effective than oth-ers, but none could violate the uncertaintyrelation given by Ineq. (1). Moreover,Ineq. (1) remains valid even if the beamsare observed within a transparent medi-um of refractive index n > 1. For instance,in Fig. 1 if the region to the right of theaperture happens to be filled with such amedium, the angular separation �� of thebeams shrinks by a factor n upon enteringthe medium, but the length of the k-vectorincreases by the same factor, thus preserv-ing the magnitude of �kx. Similarly, in Fig. 4 if the index of the medium on theright-hand-side of the lens happens to ben, the focused spot diameters will be ntimes smaller, but their center-to-centerspacing will also be reduced by the samefactor, resulting once again in the preser-vation of Ineq. (1).

Co-propagating beamsof differing wavelengthsA problem of general interest in spec-troscopy is that of separating two beamsof slightly different wavelengths, �1 and�2, propagating in the same direction. In

46 Optics & Photonics News � Month 2001

Figure 6. Fabry-Pérot etalon designed for opera-tion at � = 633 nm, � = 45°. Dielectric mirrorseach contain six pairs of high/low-index layers (n1 = 2.0, d1 = 84.6 nm; n2 = 1.5, d2 = 119.6 nm).Both mirror substrates are glass (nsub = 1.5), andthe medium separating the mirrors is air (dair =55.95 m).The incidence angle on the etalon is inthe vicinity of � = 45°; within the substrate, how-ever, the angle of incidence on the stack is close to�� = 28.1255° (sin� = nsub sin��). The etalon canseparate two beams of identical � arriving throughan aperture of diameter D, but differing in propa-gation direction, namely, �1 = 45°, �2 = 45° + ��.One beam is reflected by the etalon while the oth-er is transmitted. Only s-polarized light is consid-ered here, although p-polarized beams exhibit sim-ilar behavior.

Figure 5.Total electric field intensity distribution(|E |2 = |Ex|2 + |Ey|2 + |Ez|2) at the focal plane of a0.99NA lens. (Rainbow colors: red = maximum,blue = minimum).The beam at the entrance pupil isthe superposition of two D = 500� beams of angu-lar separation �� = 0.23°, as shown in Fig. 2. In (a)the assumed polarization state of both incidentbeams is linear along the X-axis. In (b) the twobeams are linearly polarized at 45° to the X-axis,i.e., along the direction of separation of the spots. In(c) one of the beams is right-circularly polarized,while the other is left-circularly polarized. Frames(d)-(f) in the bottom row are the logarithmic ver-sions of frames (a)-(c) in the top row. Like an over-exposed photographic plate, a logarithmic plot re-veals weak regions of an intensity distribution.

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January 2002 � Optics & Photonics News 47

finition, kz = 2�/� and, therefore, �kz =2���/�2. Figure 12 shows the abovebeams arriving at an incidence angle 0° ≤ �< 90° on a grating of period P. The N th

diffracted order emerges from the gratingat an angle ��, in accordance with Bragg’slaw,4,5

sin�� = sin� +N � /P, (3a)

which yields,

cos�� ��� = (N /P)��. (3b)

Now, the emergent beam diameter is D� =D |cos��/cos�|. Since the lens is expected toresolve the two wavelengths, Ineq. (1) re-quires that |���| ≥ �/D�, which leads to|cos�����| ≥ � cos�/D, which in turn leads

is consistent with the reduced speed oflight in the medium of index n, whichyields the same travel time �t for theshorter propagation distance �z/n. Need-less to say, �E = h� is independent of n.

Wavelength discriminationusing a Fabry-Pérot etalonThe etalon of Fig. 6 may also be used toseparate co-propagating beams of slightlydifferent wavelengths, say, � and � + ��.Figure 11 shows computed plots of reflec-tion and transmission coefficients versus� for a resonator having an air gap dair =55.95 m. From Eq. (2) at � = 633 nm,considering that �z = 2daircos(45°) =79.125 m, we find �� = 2.53 nm, inagreement with the peak-to-valley dis-tance in the simulated results of Fig. 11.The figure, however, indicates the feasibil-ity of resolving beams with a smaller �� aswell; this is due to the high finesse of theFabry-Pérot etalon. In other words, multi-ple back and forth reflections within theetalon’s cavity build up an optical fieldwhose amplitude is G times stronger than

Figure 9.The Mach-Zehnder interferometer canbe used to separate two beams of differing wave-lengths, �1 and �2.The beams have identical diame-ters and arrive in the same direction. The twobeams are split equally at the first 50/50 splitter,travel the two arms of the device, and are recom-bined at the second 50/50 splitter.The lengths ofthe two arms of the interferometer differ by �z.When �z/�1 – �z/�2 = 1⁄ 2, constructive interfer-ence at the second beam-splitter for one of thetwo wavelengths coincides with destructive inter-ference for the other.The beams are thus separat-ed at the second splitter, one is captured by detec-tor 1, the other by detector 2. The 45° mirrors(three in each arm) have a reflectivity of 90%, re-sulting in an overall system transmission of 73%.The 50/50 splitters are identical, each consisting ofa glass substrate coated with a six-layer dielectricstack as follows:

(Substrate, nsub = 1.5) / (d1 = 30 nm, n1 = 2.64) / (d2 = 140 nm, n2 = 1.76) / (d3 = 50 nm, n3 = 2.64) /(d4 = 105 nm, n4 = 1.76) / (d5 = 60 nm, n5 = 2.64) /(d6 = 100 nm, n6 = 1.76) / Air

Although the above stack works for both p- and s-polarized light, its splitting ratio is much closer to50/50 for s-light than for p-light. In our simulationsthe polarization state of the incident beam wasfixed at s.

Figure 8.Two overlapping beams of uniform am-plitude and circular cross-section (� = 633 nm,D =2 x 104�) arrive at the etalon of Fig. 6. One beamtravels at � = 45° relative to the etalon’s surfacenormal, the other at � = 45.115°. (a) Intensity dis-tribution of the superposed beams at the entranceaperture. (b) Reflected intensity distribution, con-sisting mainly of the second beam plus a small frac-tion of the first. (c) Transmitted intensity distribu-tion, consisting mostly of the first beam plus asmall fraction of the second.

Figure 7. Computed reflection and transmissioncoefficients versus the incidence angle � for theetalon of Fig. 6 at � = 633 nm for an s-polarizedplane wave. The magnitude and phase of the re-flection and transmission coefficients are definedthrough the relations rs = |rs| exp(i�rs) and ts = |ts| exp(i�ts).At � = 633 nm the stack is tuned tofully transmit at � = 45°. A small deviation from45° incidence causes a sharp drop in |ts| and acorresponding rise in |rs|.

that of the incident beam. (In the presentexample, G is 3.0 for s-light and 1.94 for p-light.) The effective �z is thus G times theeffective gap width, resulting in a corre-sponding increase in the resolution of thedevice.

Spectral analysis using a diffraction gratingConsider two co-propagating beams ofwavelengths � and � + ��, where it is as-sumed for convenience that ��> 0. Thesebeams travel along the Z-axis and passthrough an aperture of diameter D. By de-

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References1. R. P. Feynman, R. B. Leighton, and M. Sands,The Feyn-

man Lectures on Physics,Addison-Wesley, Reading,Massachusetts (1964).

2. A. E. Siegman, Lasers, University Science Books, Cali-fornia (1986).

3. The simulations reported in this article were per-formed by DIFFRACT™, and MULTILAYER™; bothprograms are products of MM Research, Inc.,Tucson,Arizona.

4. M. Born and E.Wolf, Principles of Optics, 6th edi-tion, Pergamon Press, Oxford, 1980.

5. M.V. Klein, Optics,Wiley, New York (1970).

OPN contributing editor Masud Mansuripur <[email protected]> is a professor of Optical Sciences atthe University of Arizona in Tucson and president ofMM Research, Inc.

to |N /P | �� ≥ �cos�/D. In other words,

D /cos� ≥ (�/��) |P/N |. (4a)

From Eq. (3a) it is clear that |N �/P | ≤ 2,namely, |P/N | ≥ 1_

2�. Inequality (4a) maythus be written as follows:

D /cos� ≥ 1_2�2/��. (4b)

Inequality (4b) places a lower bound forresolvability not on the beam diameter D,but on the illuminated length of the grat-ing, D/cos�, in the direction perpendicularto the grooves.

Next we examine the propagation dis-tance from the center of the entranceaperture to the focal plane of the lens.With reference to Fig. 12, the shortest pos-sible distance from the entrance apertureto the grating center is �z1 = 1_2D tan�. Sim-ilarly, the shortest possible distance fromthe grating to the lens center (ignoring thepossibility that the lens might block theincident beam) is �z2 = 1_

2D�|tan��| =1_2D|sin��|/cos�. The smallest feasible focallength for the lens is f = 1_

2D�, correspond-ing to NA = 1. Therefore, the shortest dis-

48 Optics & Photonics News � January 2002

Figure 12. Two beams of wavelengths � and�+��, propagating in the same direction Z, arriveat an aperture of diameter D.The beams propagatea distance �z1 from the center of the aperture toa grating of period P, shining on the grating at anangle �. One of the diffracted orders (the N th or-der) leaves the grating at an angle ��, travels a dis-tance �z2 (from the center of the grating to thecenter of the lens), then enters a lens of focallength f and numerical aperture NA ≈ 1. Emergingfrom the grating, the two wavelengths deviate fromeach other by an angle ���, thus forming separatefocused spots at the focal plane of the lens. Fromthe entrance aperture to the focal plane, the totalpropagation distance is �z = �z1 + �z2 + f.

Figure 11. Computed plots of amplitude reflec-tion and transmission coefficients versus � for theFabry-Pérot etalon depicted in Fig. 6. The air gapand the incidence angle are fixed at dair = 55.95 m and � = 45°. The incident beam is p-polarized in (a) and s-polarized in (b).

Figure 10. Computed detector signals S1 and S2versus the input wavelength � in the Mach-Zehn-der interferometer of Fig. 9. The assumed path-length difference between the two arms of the device is �z = 1.266 mm. In the vicinity of � = 633 nm the adjacent peaks of S1 and S2 are sepa-rated by �� = 0.158 nm, in agreement with Eq. (2).

tance �z from the center of the entranceaperture to the focal plane of the lens isgiven by,

�z = �z1 + �z2 + f (5a)= 1_

2(D/cos�) (sin� + |sin��| + |cos��| ).

Since sin� ≥ 0, and |sin��| + |cos��| ≥ 1 forany ��, Eq. (5a) yields,

�z ≥ 1_2D/cos�. (5b)

Combining Ineqs. (4b) and (5b) thenyields �z ≥ 1_

4�2/��, that is,

�z �kz ≥ 1_2�. (6)

Note that the initial beam diameter D inthis example is not restricted at all, where-as the propagation distance �z is requiredto be greater than a certain minimum,1_4�2/��, to ensure resolvability of thewavelengths � and � + ��.

AcknowledgmentI am grateful to Ewan M. Wright of theOptical Sciences Center for insightful sug-gestions.