Axial irradiance distribution throughout the wholespace behind an annular aperture

James E. Harvey and Andrey Krywonos

In many photonics and fiber-optics applications, the irradiance distribution in the very near field �z�D �0.25� behind a circular or annular aperture is of interest. We present the results of detailed calculationsof the irradiance distribution throughout the entire space behind an annular aperture. Included as aspecial case of the annular aperture is the circular aperture and the opaque circular disk. A log–log plotover many orders of magnitude in axial distance provides particular insight. The behavior throughoutthe Fresnel and Fraunhofer region is well known; however, we pay particular attention to the behaviorin the near field. A variety of subtle effects in the near field are presented and discussed. © 2002Optical Society of America

OCIS codes: 050.0050, 260.0260, 050.1220, 050.1960.

1. Introduction

The oscillatory nature of the axial irradiance distri-bution throughout the Fresnel region behind a circu-lar aperture is well known1:

E2�00; z� � E0�2 � 2 cos�k D2

8z��� 4E0 sin2�k

D2

16z� . (1)

This quantity oscillates rapidly between zero andfour times the incident irradiance throughout theFresnel region as illustrated in Fig. 1�a�. As theFraunhofer region is approached, the period of theseoscillations along the axis increases with z until theyeventually stop altogether, and the well-known 1�z2

dependence prevails for large z. Note that the lastmaximum occurs at z � D2�4�, which is precisely afactor of � less than the expression in the inequalitycommonly referred to as the Fraunhofer criterion2:

z ��k2

� x12 � y1

2�max. (2)

The authors are with the Center for Research and Education inOptics and Lasers, P.O. Box 162700, 4000 Central Florida Boule-vard, University of Central Florida, Orlando, Florida 32816. J. E.Harvey’s e-mail address is harvey@creol.ucf.edu.

Received 5 November 2001; revised manuscript received 4March 2002.

0003-6935�02�193790-06$15.00�0© 2002 Optical Society of America

3790 APPLIED OPTICS � Vol. 41, No. 19 � 1 July 2002

The oscillations that occur throughout the Fresnelregion are a manifestation of the constructive anddestructive interference between the light wave thatpropagates straight through the aperture and thelight wave diffracted from the edge of the aperture.

Figure 1�b� illustrates a similar axial irradiancedistribution resulting from an annular aperture.The axial irradiance again oscillates between zeroand four times the incident irradiance; however, thelast maximum is now shifted toward the aperture tothe location z � �D2 � d2��4�. The oscillating be-havior in this case can be thought of as the construc-tive and destructive interference between the twoedge-diffracted waves. It is interesting to note thatthe strength of the two diffracted waves on the axis ofa uniformly illuminated annular aperture is identicalas evidenced by the complete null observed whendestructive interference occurs, and this is indepen-dent of the obscuration ratio.

Figure 1�c� illustrates the markedly different axialirradiance distribution produced by a circular obscu-ration as previously reported by Sommerfeld3 andHarvey and Forgham.4 Sommerfeld pointed out theirony in the fact that, although there are multiplelocations of complete darkness on the optical axisbehind a uniformly illuminated circular aperture,there is no darkness anywhere �for z � 0� along theaxis of a circular opaque disk! In fact, the irradianceis everywhere the same as it would be in the absenceof a diffracting screen or aperture! This bright spotat the center of the geometric shadow of a circulardisk is caused by the same edge-diffracted wave thatcaused the oscillations in the diffraction pattern of its

complementary aperture. However, in this casethere is no undiffracted wave Fig. 1�a� or seconddiffracted wave Fig. 1�b� to cause interference andproduce a modulated axial irradiance distribution.

This constant axial irradiance distribution behinda circular obscuration is, of course, the infamous spotof Arago or Poisson’s bright spot that played such acrucial role in establishing the wave theory of light in1818 when Augustin Jean Fresnel submitted hisnow-famous essay on diffraction in a competitionsponsored by the French Academy.4

Over the next 80 years, scalar diffraction theorywas developed until, in 1896, Sommerfeld eliminatedthe need for imposing boundary conditions on boththe field and its normal derivative, thus removing theinconsistencies in the Fresnel–Kirchhoff formulation.The resulting Rayleigh–Summerfeld diffraction the-ory is a rigorous treatment limited only in that it is ascalar theory that neglects the fact that the variouscomponents of the electric and magnetic fields arecoupled through Maxwell’s equations. The follow-ing Rayleigh–Sommerfeld diffraction formula is validfor all observation planes provided that z �� �:

U2� x2, y2� �Ai� �

��

�

���

�

U1� x1, y1�

�exp�ikl �

lcos�n, l �dx1dy1. (3)

The quantity U1�x1, y1� is the complex amplitudedistribution emerging from the diffracting aperture , and l is the distance from an arbitrary point in thediffracting aperture to an arbitrary point in the ob-servation plane as illustrated in Fig. 2. Note thatwe incorporated the finite limits of the aperture in thedefinition of U1�x1, y1�, in accordance with the usualassumed �Kirchhoff � boundary conditions.

2. On-Axis Irradiance throughout the Whole Space

The axial irradiance distribution for monochromatic,normally incident, uniform illumination �of ampli-tude A� upon an annular aperture of inner diameter

d and outer diameter D is given when we set x2 � y2 �0 in the Rayleigh–Sommerfeld diffraction integral ofEq �3�. The cosine obliquity factor can now be writ-ten as cos�n, l� � z�l; hence, changing to polar co-ordinates and factoring out the 2� from theazimuthal integration, we obtain

U2�0, 0; z� � � ikA�r1�d�2

D�2 exp�ikl �

lzl

r1dr1 , (4)

where

l2 � z2 � r12, r1dr1 � ldl. (5)

The change of variable also affects the limits of inte-gration; hence

U2�0, 0; z� � � ikAz�l�� z2�d2�4�1�2

� z2�D2�4�1�2 exp�ikl �

ldl. (6)

Letting

a � 1 � D2�4z2, b � 1 � d2�4z2, (7)

Eq �6� becomes

U2�0, 0; z� � � ikAz�l�z�b

z�a

l�1 exp�ikl �dl. (8)

Letting

u � l�1, dv � exp�ikl�dl,

du � � l�2, v � exp�ikl ��ik, (9)

and integrating by parts, we obtain

U2�0, 0; z� � � Azexp�ikl �

l � z�a

z�b� Az�

l�z�b

z�a

l�2

� exp�ikl �dl. (10)

Integrating by parts multiple times, it becomes clearthat the on-axis diffracted wave field produced by anannular aperture illuminated by a uniform ampli-tude plane wave is given by the following infinitesummation:

U2�0, 0; z� � � Azexp�ikl �

l �n�0

� n!�ikl �n� z�a

z�b. (11)

No explicit approximations were made in the abovecalculations; hence this solution is valid for all val-

Fig. 1. Axial irradiance distribution throughout the Fresnel re-gion that is due to diffraction from �a� a circular aperture, �b� anannular aperture, and �c� a circular obscuration.

Fig. 2. Geometric relationship between the diffracting apertureand the observation space.

1 July 2002 � Vol. 41, No. 19 � APPLIED OPTICS 3791

ues of z �� �. However, because k is such a largenumber for optical wavelengths, the contribution ofthe higher-order terms diminishes rapidly. The on-axis irradiance is thus accurately approximated withonly the first term of the above summation:

E2�0, 0; z� � U2�0, 0; z�2

� E0 z2�exp�ikz�a�

z�a�

exp�ikz�b�

z�b�2

, (12)

where E0 � A2 is the irradiance incident upon theannular aperture. Taking the squared modulus andrearranging, we have

E2�0, 0; z� � U2�0, 0; z�2

� E0a � bab

�2

�a �bcoskz��a � �b�� ,

(13)

where a and b are determined by the outer radius andinner radius of the annular aperture as given by Eqs.�7�. We have thus solved three cases of interest be-cause the annular aperture reduces to a clear circularaperture if d � 0 and an opaque circular obscurationif D � �.

Figure 3 illustrates a log–log plot of the axial irra-diance distribution throughout the whole space be-hind a circular aperture. We assumed illumination�� � 0.5 �m� by a unit amplitude normally incidentplane wave and used Eq. �13� to calculate the axialirradiance behind a 20-mm-diameter circular aper-ture from z � 0.2 mm to z � 200 km. The log–logplot is illustrated with the axial position expressed interms of aperture diameters. Immediately behindthe aperture the irradiance is the same as the inci-dent irradiance as expected.

The oscillations that are due to the interferencebetween the edge-diffracted wave and the direct illu-

mination start immediately behind the aperture, andits modulation increases throughout the near field,asymptotically approaching a peak value of fourtimes the incident irradiance as we approach an axialposition specified by the Fresnel criterion, usuallywritten as the following inequality2:

z3 ���

4�� x2 � x1�

2 � � y2 � y1�2max

2. (14)

We take the axial position indicated by replacing thegreater than sign with an equal to sign to be the nom-inal boundary between the near field and the Fresnelregion. Throughout the Fresnel region, the oscilla-tory behavior is as predicted by approximation �1�and illustrated in Fig. 1�a�. The period of the oscil-lations increases with increasing z, and the last peakoccurs at z � 104 D or 200 m, precisely a factor of �before reaching the axial position specified by �againsubstituting an equal sign� the Fraunhofer criterionexpressed in inequality �2�. After reaching theFraunhofer region, or far field, the axial irradiancedecreases steadily as 1�r2, which is readily apparentas a straight line with a slope of �2 on the log–logplot.

An additional bit of insight into the above behavioris obtained if we recall that the light from adjacentFresnel zones in a circular aperture interferes de-structively, whereas light from either even or oddFresnel zones interferes constructively.5 By consid-ering the number of Fresnel zones subtended by acircular aperture of fixed size as we move along theoptical axis, we can assign a unique Fresnel numberN to the above diffracting aperture for each axialposition in Fig. 3:

N �D2

4�z. (15)

This Fresnel number is a critical parameter usedextensively in the design and analysis of laser cavi-ties and resonators.6

Because of the above inverse relationship betweenthe Fresnel number and the axial position, at largedistances from a circular aperture the Fresnel num-ber is small �less than unity� throughout the entireFraunhofer region. As we move closer to the aper-ture, passing from the Fraunhofer region into theFresnel region, the axial irradiance increases, reach-ing a maximum at a Fresnel number of exactly unity.A further reduction in z causes the aperture to consistof more than one Fresnel zone, and destructive inter-ference starts to diminish the axial irradiance. Itreaches a minimum �zero� at the axial position thatcorresponds to precisely two Fresnel zones. The os-cillatory behavior in Fig. 3 is thus completely consis-tent with a qualitative discussion of destructive�constructive� interference of an even �odd� number ofFresnel zones. Figure 4 illustrates a portion of theabove irradiance distribution with a few discreteFresnel numbers indicated.

Figure 5 illustrates a similar log–log plot of the

Fig. 3. Axial irradiance distribution illustrated throughout thewhole space behind a circular aperture illuminated with a unitamplitude plane wave.

3792 APPLIED OPTICS � Vol. 41, No. 19 � 1 July 2002

axial irradiance distribution throughout the wholespace behind an annular aperture with an obscura-tion ratio of 0.5. The on-axis irradiance starts atzero immediately behind the aperture, and again themodulation that is due to the interference betweenthe two edge-diffracted waves increases throughoutthe near field, asymptotically approaching a peakvalue of four times the incident irradiance as we ap-proach an axial position specified by the Fresnel cri-terion. Note that the minima of the oscillationsincrease initially, then reach a maximum value andfall to zero before the axial location that satisfies theFresnel criterion is reached.

3. On-Axis Irradiance in the Near Field

From Eq. �13� we calculate that, although the spot ofArago has a constant irradiance of unity throughoutthe Fresnel region as illustrated in Fig. 1, it starts outat zero immediately behind the obscuration and in-creases to 90% of its eventual value at an axial loca-tion of only z � 1.5d. In addition to showing thisnear-field behavior of the axial irradiance distribu-tion behind the circular obscuration, Fig. 6 illus-

trates, on a linear scale, the envelope of theoscillations in the axial irradiance distribution for theother two apertures throughout the near field; i.e.,for those z values that do not satisfy the Fresnelcriterion.

These nondimensional plots remain unchanged forvariations in the diameter D of the circular apertureand the annular aperture �ε � 0.5�. The circularobscuration has a diameter equal to that of the cir-cular aperture. The value of the wavelength doesnot affect the envelope of the oscillations; however,it does affect the frequency of the oscillations, withthe number of oscillations decreasing with increasingwavelength. Note that, for the circular aperture,the axial irradiance is equal to E0 at z � 0, and thepeak value of the oscillations approaches 4E0 at z �1.5D. Likewise, for the annular aperture the axialirradiance is equal to zero at z � 0, and the peakvalue of the oscillations approaches 4E0 at z � 1.5D.The minima of the oscillations approach zero in bothcases. This behavior is consistent with what weknow to be true at z � 0 and that predicted by ap-proximation �1� in the Fresnel region.

4. On-Axis Irradiance in the Very Near Field

Figure 7�a� shows the nature of the actual oscillationsin the axial irradiance distributions in the very nearfield �z�D � 0.25� for the case of the circular apertureof diameter D � 0.2 mm. Small apertures such asthese are more likely to be encountered in photonicsand fiber-optics applications. Note that the on-axisirradiance is equal to the incident irradiance at z � 0;however, interference effects between the edge-diffracted wave and the directly transmitted wavecauses an oscillatory modulation that initially in-creases linearly as z increases. Likewise, Fig. 7�b�shows that the axial irradiance directly behind the

Fig. 4. Axial irradiance distribution with Fresnel numbers indi-cated for a limited range of axial positions throughout the Fresnelregion.

Fig. 5. Axial irradiance distribution illustrated throughout thewhole space behind an annular aperture �ε � 0.5� illuminated witha unit amplitude plane wave.

Fig. 6. Curves indicate the envelope of the oscillations in the axialirradiance distribution in the near field behind a circular and anannular aperture. The axial irradiance distribution behind a cir-cular obscuration �of diameter equal to that of the circle and an-nulus� is also illustrated.

1 July 2002 � Vol. 41, No. 19 � APPLIED OPTICS 3793

annular aperture is zero, then starts to oscillate be-cause of interference effects between the two edge-diffracted waves.

The specific nature of the oscillations in the axialirradiance distribution behind an annular aperturevaries with the obscuration ratio. As illustrated inFig. 8, the frequency of the oscillations decreases andthe depth of modulation increases as the obscurationratio increases.

The above calculations were again based on a

wavelength of � � 0.5 �m. Note from Fig. 8�d� that,even in the very near field, the minima of these os-cillations approach zero as the obscuration ap-proaches unity. This is intuitive behavior becausethe obliquity angles are approaching the same value,and the strength of the two diffracted wave fields aretherefore close to the same value as the obscurationratio approaches unity.

In Fig. 7�a� above, the axial position ranges fromzero to 0.010 mm �10 �m� when a wavelength of 0.5

Fig. 7. �a� Axial irradiance distribution in the very near field behind a circular aperture. �b� Axial irradiance distribution in the verynear field behind an annular aperture �ε � 0.5� and a circular obscuration of the same diameter.

Fig. 8. Axial irradiance distribution in the very near field behind an annular aperture with obscuration ratio �a� ε � 0.20, �b� ε � 0.35,�c� ε � 0.65, and �d� ε � 0.80.

3794 APPLIED OPTICS � Vol. 41, No. 19 � 1 July 2002

�m is used. Recall that the standard form of theRayleigh–Sommerfeld diffraction formula is validonly for z �� �. If one considers a factor of 10 toqualify as much greater, the calculated values for theirradiance might be suspect for axial positions lessthat 0.005 mm �5 �m�. On the other hand, if a factorof 3 qualifies as being much greater, then only thosecalculated values for axial positions less than 0.002are suspect; however, the value of unity at the ex-treme axial position of zero is of course correct.Hence the range of suspect values is small.

Also, one can be assured that, although the axialirradiances differ drastically for a clear circular ap-erture and its complementary aperture �a circularobscuration�, the amplitudes sum to the incident am-plitude, as required by Babinet’s principle.

5. Summary and Conclusion

We first reviewed the well-known axial irradiancedistribution throughout the Fresnel and Fraunhoferregions behind an annular aperture �including thespecial cases of a circular aperture and a opaquecircular obscuration�. The oscillatory behavior inthe axial irradiance distribution behind a circularaperture was explained as an interference effect be-tween the directly transmitted radiation and theedge-diffracted wave �two edge-diffracted waves forthe annular aperture�. The nonintuitive behavior ofthe spot of Arago or Poisson’s bright spot behind anopaque circular disk, and its role in establishing thewave theory of light in 1818, was also reviewed.

We then calculated the axial irradiance distribu-tion throughout the whole space behind an annularaperture by using the Rayleigh–Sommerfeld diffrac-tion integral that is valid to within a few wavelengths

of the aperture and presented the results in a log–logplot that yield new insight and understanding. Themaxima and minima of these oscillations were alsorelated to the Fresnel number of the aperture at theobservation point along the axis.

We directly compared the axial irradiance distri-bution in the near field behind a circular aperture, anannular aperture, and a circular obscuration by su-perposing these plots on a single figure. In all threecases, they approached the well-known behavior asthe observation distance approached the Fresnel cri-terion, and they approached the necessary values ofzero or unity as the observation distance approachedzero.

Finally, the axial irradiance distribution in thevery near field behind an annular aperture was stud-ied in detail as a function of the obscuration ratio andcompared with that of a circular obscuration. Thisdetailed behavior was calculated for rather small ap-ertures �D � 0.20 mm�, as it is of potential interest ina variety of photonics and fiber-optics applications.

References1. J. M. Stone, Radiation and Optics �McGraw-Hill, New York,

1963�, Chap. 10, p. 204.2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed.

�McGraw-Hill, New York, 1996�, Chap. 4, pp. 69–74.3. A. Sommerfeld, Optics: Lectures on Theoretical Physics �Aca-

demic, P, New York, 1954�, Vol. 4, Chap. 5, p. 215.4. J. E. Harvey and J. L. Forgham, “The Spot of Arago: new

relevance for an old phenomenon,” Am. J. Phys. 52, 243–247�1984�.

5. M. Born and E. Wolf, Principles of Optics, 6th ed. �Pergamon,Oxford, UK, 1980�, Chap. 8, pp. 370–375.

6. A. E. Siegman, Lasers �University Science, Mill Valley, Calif.,1986�, p. 735.

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