M. Gu and X. S. Gan Vol. 13, No. 4/April 1996 /J. Opt. Soc. Am. A 771

Fresnel diffraction by circular andserrated apertures illuminated

with an ultrashort pulsed-laser beam

Min Gu* and X. S. Gan

Department of Physical Optics, University of Sydney, New South Wales, 2006, Australia

Received May 19, 1995; revised manuscript received October 11, 1995; accepted October, 11, 1995

Fresnel diffraction patterns by circular and serrated apertures under ultrashort pulsed-laser illumination areinvestigated in detail. A comparison of the diffraction patterns for pulsed and continuous wave illuminationsreveals that an ultrashort pulsed beam of 10 fs can result in a significant effect on the distribution of theintensity. As a result a uniform intensity distribution in the transverse direction of the beam can be achievedas a result of the interference of the different frequency components within the pulsed beam. The interferencefringes caused by the wavelets originating from the diffraction aperture can be further reduced when a serratedaperture is used. 1996 Optical Society of America

1. INTRODUCTIONIt is well known that the diffraction pattern of a finite-sized aperture in the near-field region is not uniformbecause of the interference of the wavelets contributedfrom points over the aperture.1,2 For example, a circu-lar aperture can produce concentric bright and dark rings(or fringes) in intensity, particularly near the edge of theaperture. The intensity modulation caused by the inter-ference fringes can affect the imaging performance be-cause the effective pupil function of the lens illuminatedby the near-field Fresnel pattern can change, leading toan alteration of the diffraction pattern in the focal regionof a lens3 and therefore to an alteration of imaging qual-ity and contrast.4 – 7 The nonuniform intensity distribu-tion is sometimes harmful. One example of this is thatnonlinear self-focusing may occur in the medium throughwhich an ultrashort pulsed beam propagates, as a resultof the high peak intensity of the modulation. Accord-ingly, the medium can be damaged. In laser fusion thenonuniform distribution of the wave front can also reducethe efficiency of the energy transfer from the laser beam tothe target.8,9 One of the methods proposed recently8 – 11

for producing a uniform distribution of the intensity in thetransverse plane is to use a serrated circular aperture,12

as shown in Fig. 1, in the path of the laser propagation.Another important factor associated with the diffrac-

tion pattern in the near-field region is the Fresnel num-ber N . The distribution of the diffraction pattern in atransverse plane is fully determined when the value ofN is given. As is known, N is inversely proportional tothe wavelength of the illumination and to the propagationdistance for a given size of the aperture. The recent de-velopment of the ultrashort technique has produced laserpulses as short as 10 fs.13 – 15 Such a short-pulsed beamis not a monochromatic wave but comprises a finite spec-trum. Consequently, the Fresnel number has a distri-bution for a given plane of observation, and the Fresneldiffraction pattern may be changed significantly. So farthere has been, to the best of the authors’ knowledge, no

0740-3232/96/040771-08$06.00

paper that discusses the effect of the ultrashort pulse onthe near-field diffraction pattern by circular and serratedapertures.

This paper reports on the first theoretical analysis ofthe Fresnel diffraction when an ultrashort pulsed beamis used for illumination. Both circular and serratedapertures are considered. In Section 2 the effect of anultrashort pulsed beam on the Fresnel diffraction patternby a circular aperture is examined. Section 3 gives a de-tailed description of the Fresnel diffraction by a serratedaperture under continuous wave (CW) and ultrashortpulsed-beam illumination. In Section 4, discussionsabout practical relevance are put forward.

2. FRESNEL DIFFRACTION BYA CIRCULAR APERTUREThe diffraction pattern of light waves in a plane placeda few wavelengths away from a diffraction screen canbe described approximately by the Fresnel diffractionformula.1,2 Under the paraxial approximation and for anincident illumination of a given frequency v, it can be ex-pressed as

U sx, y, z, vd ­i exps2i2pzyld

lzexp

"2

ip

lzsx2 1 y2d

#

3Z 1`

2`

Z 1`

2`

U1sx1, y1, vd

3 exp

"i2p

lzsx1x 1 y1yd

#

3 exp

"2

ip

lzsx1

2 1 y12d

#dx1dy1 , (1)

where l ­ 2pcyv is the wavelength of the illuminationlight. U1sx1, y1, vd and U sx, y, z, vd are the light fieldsin the diffraction plane and the observation plane, respec-tively. The distance between the two planes is denotedby z. The Cartesian coordinates in the planes are de-noted by x1, y1 and x, y. When the Kirchhoff boundary

1996 Optical Society of America

772 J. Opt. Soc. Am. A/Vol. 13, No. 4 /April 1996 M. Gu and X. S. Gan

Fig. 1. Serrated aperture with maximum and minimum radii,a 1 da and a 2 da, respectively. The ratio of daya is chosento be 0.05, giving a range of the Fresnel number DN ­ 10 forN0 ­ 100.

conditions and l ,, r (where r ­p

x2 1 y2 1 z2) (Refs. 1and 2) are applied, U1sx1, y1, vd is given simply by thefield within the diffraction aperture.

Under the illumination of an ultrashort pulsed beam,the light field is not monochromatic but includes a spec-tral distribution around the central frequency v0. In thiscase the total field in an observation plane is a coher-ent superposition of the contribution from each frequencycomponent. Let us assume that the incident pulse is rep-resented by U0std. Its Fourier transform, i.e., the spec-trum of the pulse, is given by V sDvd, where Dv ­ v 2 v0

denotes the detuning frequency from the central fre-quency v0. The total diffraction field U sx, y, z, td of anultrashort pulsed beam is the Fourier transform of Eq. (1)multiplied by V sDvd with respect to v. The correspond-ing instantaneous intensity is the modulus squared ofU sx, y, z, td. The time-averaged intensity, a measurablequantity in practice, is thus given by

I sx, y, zd ­ CZ `

2`

jV sDvdU sx, y, z, vdj2dv , (2)

where C is a coefficient of normalization. In the fol-lowing numerical calculation C is obtained from the as-sumption that the on-axis intensity over the diffractionaperture is unity and from the fact that energy is con-served in a transverse plane. Equation (2) also impliesthat the same effect can be obtained if a broadband CWillumination is adopted.

As an example, let us suppose that the ultrashortpulsed beam has a Gaussian shape in time, so that thetemporal amplitude of the incident pulse can be generallyexpressed as

U0std ­ exps2iv0tdexp

√2

t2

T 2

!, (3)

where t represents the local time and l0 ­ 2pcyv0 is the

central wavelength. The parameter T is related to thefull width at half-maximum and Dt of U0std by way of thefollowing relation:

T ­Dt

2p

ln 2. (4)

The Fourier transform of the temporal pulse functiongives its distribution in the frequency domain:

V sDvd ­p

p T exp

"2

T 2sv 2 v0d2

4

#

­p

p T exp

"T 2v0

2

4

√v

v02 1

!2#. (5)

Applying Eqs. (1) and (2) in the case of a circular aper-ture of radius a, we can take the cylindrical symmetryinto account. For this purpose two normalized radial co-ordinates can be introduced:

r1 ­ r1ya (6)

r ­ rya , (7)

where r1 ­ sx12 1 y1

2d1/2 and r ­ sx2 1 y2d1/2 are the realradial coordinates in the diffraction and the observationplanes, respectively. So Eq. (1) can be expressed as

U s r, z, vd ­2pia2 exps2i2pzyldexps2iNr2d

lz

3Z 1

0U1s r1, vdJ0s2Nr1rd

3 exps2iNr12dr1dr1 . (8)

Here N is the Fresnel number defined as

N ­pa2

lz, (9)

which includes the constant p. It is noted that N is afunction of the distance z and the frequency v. To in-vestigate the effect of the pulse spectrum, one can rewriteN as

N ­a2v0

2zcv

v0­ N0

v

v0

, (10)

where

N0 ­pa2

l0z(11)

is the Fresnel number corresponding to the central fre-quency.

It is now clear that for a given position of the obser-vation plane, the Fresnel number N is different fordifferent frequency components within the pulse. Thetotal diffraction pattern is therefore the summation ofthe contributions from different values of the Fresnelnumber. Accordingly, one can expect that the diffrac-tion pattern under ultrashort pulsed beam illuminationwill differ from that under CW illumination.

M. Gu and X. S. Gan Vol. 13, No. 4/April 1996 /J. Opt. Soc. Am. A 773

Fig. 2. On-axis intensity distribution of the Fresnel diffraction pattern: (a) circular aperture with CW illumination; (b) circularaperture with pulsed illumination; (c) serrated aperture with CW illumination; (d) serrated aperture with pulsed illumination.

Fig. 3. Intensity distribution in a vertical plane for N0 ­ 100: (a) circular aperture with CW illumination; (b) circular aperturewith pulsed illumination sDN ­ 29d; (c) serrated aperture sDN ­ 10d with CW illumination; (d) serrated aperture sDN ­ 10d withpulsed illumination sDN ­ 29d.

774 J. Opt. Soc. Am. A/Vol. 13, No. 4 /April 1996 M. Gu and X. S. Gan

For a circular aperture illuminated by a Gaussian-shaped pulse, Eq. (2) can be rewritten as

I s r, zd ­ CZ 1`

0

1N0

ÉN exp

"2

T 2v02

4

√NN0

2 1

!2#

3Z 1

0J0s2Nr1rdexps2iNr1

2dr1dr1

É 2

dN .

(12)

We first consider the Fresnel diffraction pattern underCW illumination, in which case V sDvd ­ dsDvd and N ­N0. The intensity distribution on the axis, i.e., for r ­ 0,can be derived analytically as

I sN0d ­ 4 sin2sN0y2d , (13)

which has been expressed explicitly as a function of theFresnel number N0 and is shown in Fig. 2(a). As ex-pected the intensity changes periodically as a function ofthe Fresnel number N0. The intensity distribution in aparticular observation plane, i.e., for N0 ­ 100, is shownin Fig. 3(a). The intensity oscillates to form a series ofconcentric fringes, which can be clearly observed in thecross section of the intensity as a function of the nor-malized transverse coordinate yya [Fig. 4(a)]. The inten-sity distribution in a horizontal plane including the z axisgives an overall behavior of the diffraction pattern at dif-ferential distances and is depicted in Fig. 5(a) as a func-tion of the normalized propagation distance Z defined as

Z ­1

N0­

l0zpa2

. (14)

Fig. 4. Cross section of the intensity distribution in a vertical plane for N0 ­ 100: (a) circular aperture with CW illumination;(b) circular aperture with pulsed illumination; (c) serrated aperture with CW illumination; (d) serrated aperture with pulsed illumination.

As a result, the intensity distribution gives a series ofaperiodic bright and dark spots along the z axis. Thespacing between two adjacent bright or dark spots in-creases with Z. Eventually the diffraction pattern ap-proaches the far-field distribution called the Fraunhoferdiffraction pattern described by the Airy function: thereis a large bright spot in the center.

To understand the effect of an ultrashort pulsed beamwe consider a pulse of temporal width 10 fs, which hasbeen generated in many research laboratories.13 – 15 Fora Gaussian-shaped beam the intensity distribution alongthe propagation axis can be calculated by

I sN0d ­ CZ 1`

0

1N0

Éexp

"2

T 2v02

4

√NN0

2 1

!2#

3 sinsNy2d

É 2

dN , (15)

which can be expressed as an analytical form:

I sN0d ­ 2

"1 2 exp

√2

N02

2T2v02

!cossN0d

#. (16)

It can be seen from Eq. (16) that when a long pulse isused, i.e., when T is large, Eq. (16) reduces to Eq. (13).The same effect can be achieved if a pulse of a short wave-length, i.e., of a high frequency, is used. Equation (16) isshown in Fig. 2(b) for a 10-fs pulse and l0 ­ 1 mm. Com-paring it with that of the CW-illumination case shown inFig. 2(a), we can find that if the aperture is illuminatedby an ultrashort pulsed beam, a significant change occurswhen N0 is large. It can be understood from Eq. (10) that

M. Gu and X. S. Gan Vol. 13, No. 4/April 1996 /J. Opt. Soc. Am. A 775

Fig. 5. Intensity distribution in a horizontal plane along the z axis. The plotted range of Z is from 0.005 to 2, corresponding to N0in the range from 200 to 2: (a) circular aperture with CW illumination; (b) circular aperture with pulsed illumination; (c) serratedaperture with CW illumination; (d) serrated aperture with pulsed illumination.

the spectral width DV (full width at half-maximum) of apulsed beam corresponds to a distribution of the Fresnelnumber DN :

DN ­ N0DV

v0

. (17)

Therefore it is concluded that for a given spectral widththe smaller the distance of the observation plane, thelarger the spreading of the Fresnel number.

For a 10-fs pulse, DVyv0 is approximately 0.294 forl0 ­ 1 mm. Therefore DN becomes bigger than unity if

776 J. Opt. Soc. Am. A/Vol. 13, No. 4 /April 1996 M. Gu and X. S. Gan

N0 is larger than 3, leading to a pronounced effect on theintensity distribution as shown in Fig. 2(b). The diffrac-tion intensity on the axis remains nearly as a constantwith a magnitude twice as large as the on-axis incidentintensity when N0 . 35, approximately corresponding toDN . 10. This feature will be discussed further in Sec-tion 4, and it means that a bright spot may always beseen in this region.

The distribution of the diffraction intensity is plottedin Fig. 3(b) for N0 ­ 100, corresponding to the value ofDN ø 29. The effect of the pulsed beam on the Fresnelpattern near the edge of the aperture is different fromthat near the center: little change occurs on the patternnear the edge, but a significant change occurs in the cen-tral part compared with that in Fig. 3(a). Within a cer-tain region, e.g., r , 0.5 [Fig. 4(b)], the Fresnel fringesare almost washed out, leading to a circular band of uni-form intensity [except for the central bright spot describedby Eq. (16)]. In addition, the weak modulation of the in-tensity appearing in Fig. 4(a) almost disappears. All ofthese features can be explained by the fact that a pulsedbeam has a finite width of the spectrum and that theFresnel number is frequency dependent. The position ofthe concentric fringes in Fig. 3(a) is determined by theFresnel number. Therefore different frequency compo-nents represent different effective Fresnel numbers. Thesuperposition with respect to N in Eq. (12) may smooththe profile of the intensity and result in a uniform distri-bution of the intensity in the transverse plane. When theobservation point yya is close to 1, i.e., close to the edgeof the aperture, the path difference between the points onthe aperture and the observation point may be larger thanthe pulse length of the laser beam so that less pronouncedchange occurs near the edge of the aperture. Generallythe appearance of the uniform intensity pattern can be ex-plained by the fact that the pulsed illumination producesthe interference of different frequency components withinthe spectrum of the pulse.

Figure 5(b) is the intensity distribution in a horizontalplane along the z axis. Compared with that for CW illu-mination, it shows a much more uniform intensity alongthe transverse direction in the region where Z is smalland is a bright line on the axis, as expected.

3. FRESNEL DIFFRACTION BYA SERRATED APERTURENow we turn to the case in which the aperture is serrated.The radius of a serrated aperture may be expressed as afunction of the angular coordinate u (Ref. 9):

r ­ af1 1 a sinsm1udsinsm2udg , (18)

where a ­ daya is the ratio of the serrating amplitudeda to the average radius a; m1 and m2 are two periods ofthe variation of the serrated aperture with respect to u.A serrated aperture of a ­ 0.05, m1 ­ 50, and m2 ­ 5 isshown in Fig. 1. The condition corresponds to a range ofthe Fresnel number DN ­ 10 for N0 ­ 100.

For a serrated aperture Eq. (1) is no longer of cylindri-cal symmetry. If the aperture is illuminated by a CWbeam of v0, the diffracted light field can be described, interms of Eq. (1), by

U s r, u, z, v0d ­ia2 exps2i2pzyl0dexps2iN0r2d

l0z

3Z 2p

0

Z r/a

0expfi2N0r1r cossu1 2 udg

3 exps2iN0r12dr1dr1du1 , (19)

where two polar coordinates have been introduced inthe diffraction and observation planes. The modulussquared of Eq. (19) gives rise to the intensity of the Fres-nel diffraction pattern.

The on-axis intensity as a function of the Fresnel num-ber N0 is shown in Fig. 2(c). Note that the variation ofthe on-axis intensity exhibits an interference pattern pro-duced by two groups of wavelets and that the magnitudeof the interference pattern decreases with N0. Accordingto the Huygens–Fresnel principle,1 the diffraction pat-tern is produced by the interference of wavelets originat-ing from each point within the diffraction aperture. Theon-axis pattern can be considered approximately as the in-terference of the wavelets between the points on the edgeand the point at the center of the aperture. In the caseof a circular aperture, for a given point on the axis thedistance from the point to each point on the edge isthe same, so that the constructive interference occurs asshown in Fig. 2(a). However, for a serrated aperture thedistance differs from point to point on the edge, and there-fore the interference fringes become much weaker as aresult of the possible out-of-p phase difference, so thatno strong bright spot exists in the center and the inten-sity approaches unity for large N0. The larger the valueof N0 (i.e., the smaller the axial distance Z), the largerthe phase difference (i.e., the weaker the constructive in-terference). Also found in Fig. 2(c) is a beat structure,which is caused by the interference of the wavelets fromthe outermost points and the innermost points on the edgeof the serrated aperture.

The transverse intensity distribution for N0 ­ 100 isshown in Fig. 3(c). The sharp sparks that appear inFig. 3(a) become less strong, as can be seen clearly fromFig. 4(c). The Fresnel diffraction fringes for the serratedaperture look weaker and more irregular than those of thecircular aperture case; this results from the weaker con-structive interference that is due to the use of a serratedaperture, as discussed above. In Fig. 5(c) the axial dis-tribution of the intensity is depicted. As expected, somebright and dark spots can be seen when Z is small butweaker in contrast in comparison with Fig. 5(a).

If illumination is an ultrashort pulsed beam of a Gauss-ian temporal shape, we can combine Eq. (1) and Eq. (2)to get the intensity distribution at a distance z from thediffraction plane:

I s r, u, zd ­ CZ `

0

1N0

ÉN exp

"2

v02

4a

√NN0

2 1

!2#

3Z 2p

0

Z r/a

0expfi2Nr1r cossu1 2 udg

3 exps2iNr12dr1dr1du1

É 2

dN . (20)

In Fig. 2(d) the on-axis intensity as a function of theFresnel number N0 is presented. Comparing it with

M. Gu and X. S. Gan Vol. 13, No. 4/April 1996 /J. Opt. Soc. Am. A 777

Fig. 2(c) for the CW illumination case, one can find thatthe intensity variation is smaller when pulse illuminationis applied. Especially in the region of N0 . 35 the inten-sity remains as a constant value of unity, which suggeststhat there is no bright spot on the axis.

A three-dimensional view of the transverse intensitydistribution is shown in Fig. 3(d). As we can see theintensity distribution is much more uniform than that inFig. 3(c), and the magnitude of the fringes near the edgeof the aperture is much smaller [see the cross section ofthe transverse intensity distribution shown in Fig. 4(d)].In addition to the reduction of the interference fringes byuse of a serrated aperture, pulse illumination can reducethe interference fringes further.

According to the transverse intensity distribution as afunction of the normalized propagation distance Z pre-sented in Fig. 5(d), the intensity is quite uniform alongthe transverse direction without a bright spot in the cen-ter when Z is small. Comparing it with that for the ser-rated aperture under CW illumination, we can concludethat the pulse illumination reduces the constructive inter-ference effect and therefore significantly suppresses thefringes of the Fresnel diffraction pattern to give a moreuniform front of the beam.

We can also compare figures for the serrated aperture[Figs. 2(d), 3(d), 4(d), and 5(d)] with those for the cir-cular aperture with pulse illumination [Figs. 2(b), 3(b),4(b), and 5(b)]. In the case of a serrated aperture theon-axis intensity varies around unity, whereas in the cir-cular aperture case it varies around a constant of 2. Thisfeature results in the absence of the bright spot at the cen-ter of the beam in the case of the serrated aperture. Inaddition, in the transverse direction the Fresnel fringesappear in smaller magnitude for the serrated aperture togive a more uniform beam. The constructive interferenceeffect can be reduced further by use of a serrated apertureilluminated by a pulsed beam.

4. DISCUSSIONIn terms of Fig. 5 we find that there is no differenceamong the four cases in the region of the far field andthat significant change happens when the observationplane is not large. In the far-field region the Fresnelnumber is small; the superposition is not strong becauseDN is small. As a result the ultrashort pulsed illumina-tion may not cause pronounced change in the diffractionpattern. This conclusion is consistent with the previouswork on the diffraction field of a pulse beam in the fo-cal region of a lens,16,17 which corresponds to the far-fielddiffraction pattern by a circular aperture. On the otherhand, the constructive interference of the wavelets fromthe points on the edge of the serrated aperture becomesstronger in the far-field region so that the diffractionfringes are pronounced.

It should be pointed out that Eq. (1) cannot be usedto calculate the diffraction pattern when the observationplane is placed very close to the diffraction apertures.In this case a generalized Fresnel diffraction theory isneeded.18 According to the condition z3 . 25a2yl de-scribed in Ref. 1, Eq. (1) is still a good approximation forN0 ­ 200 if lya , 3.93 3 1024.

Two parameters, m1 and m2, are used in Eq. (18) for the

serrated aperture considered in this paper. Thereforethe edge of the serrated aperture can be recognized tobe random, so that the results presented in Figs. 2–5should be less sensitive to the serrating pattern. In fact,the serrating amplitude a is important. As an example,we consider a serrated aperture consisting of a seriesof identical sawtooth functions. For the nth sawtoothfunction the radius of the aperture can be expressed as

r ­ a

"1 1 a

√mu

p1 1 2 2n

!#,

2sn 2 1dpm

# u #2np

m,

(21)

where m represents the total number of sawtooth func-tions around the edge of the aperture. The on-axis in-tensity of the diffracted light can be expressed, in termsof Eq. (19), as

I sN0d ­ C

É2p 2 m

Z 2p/m

0exps2iN0r2ddu

É 2

, (22)

which clearly shows the interference between the directbeam represented by the first term and the wavelets fromthe edge of the aperture corresponding to the second term.Equation (22) reduces to Eq. (13) for a circular aperture.Note that the contribution from the wavelets from theedge has a p-phase change compared with the directbeam. Under the condition of Eq. (21), Eq. (22) can bederived, when a is small, as

I sN0d ­

(1 2 2 cossN0d

sins2aN0d2aN0

1

"sins2aN0d

2aN0

# 2),

(23)

which gives behavior similar to that shown in Fig. 2(c).The first two terms in Eq. (23) give two periods, 2p andpya. The former corresponds to a fast periodic variationresulting from the interference between the direct beamfrom the center of the aperture and the wavelets fromthe points of radius a on the edge, and the latter gives aslow periodic change caused by the interference betweenthe innermost and outermost points on the edge. Thesmaller the serrating amplitude, the larger the secondperiod. Eventually it approaches infinity for a circularaperture, as expected.

According to Figs. 2(b) and 2(d), the intensity ap-proaches a constant when N0 is larger than 35. Forsimplicity we use Eq. (16) to consider the effect of a pulsedbeam. The second term of Eq. (16) almost approacheszero when the argument in the exponent becomes largerthan 5, so that we have

N02 . 10sv0T d2, (24)

which gives N0 . 35 for our conditions. Equation (24)together with Eq. (11) may be used as a formula for esti-mating where a uniform distribution of the intensity oc-curs when the illumination is a pulsed beam; it can beexpressed explicitly as

z ,a2

p40 cT

, (25)

where c is the speed of light in vacuum.

778 J. Opt. Soc. Am. A/Vol. 13, No. 4 /April 1996 M. Gu and X. S. Gan

Equation (24) can be rewritten, in terms of T ­ pyDV

and Eq. (17), as

DN . 3p . (26)

However, when DN . p, a noticeable change in thediffraction pattern can occur in comparison with thediffraction pattern for a circular aperture under CWillumination.

To obtain the practical significance of the figures shownin this paper, we consider two apertures. The averageradius of the first aperture is 3 mm under illuminationof l ­ 0.633 mm, which may correspond to the case inan optical microscope. The real distance z correspondingto the Fresnel number larger than 20 (or smaller than0.05 in Z) is shorter than 2.2 m. From our calculations(Fig. 5) the Fresnel diffraction in this region may changeappreciably when a pulsed beam or a serrated aperture (orboth) is used. Therefore if an optical system is operatedwithin this range of the distance, experimental resultsmay be affected by an ultrashort pulsed beam or by aserrated aperture.

The second example is an aperture of radius 50 mmilluminated by a beam of wavelength 1 mm. In this casethe range when the Fresnel diffraction pattern is appre-ciably affected by an ultrashort pulsed beam or by a ser-rated aperture is expanded approximately up to 390 m.In other words, in a high-power laser system, e.g., inlaser fusion, a uniform distribution of the intensity canbe achieved in this region by use of optical systems ofserrated apertures.

5. CONCLUSIONUse of a pulsed illumination can reduce the construc-tive interference effect of the wavelets from the apertureand therefore can suppress the interference fringes in theFresnel diffraction pattern. In particular, the Fresneldiffraction pattern by a circular aperture illuminated bya 10-fs pulse is more uniform than that for CW illumina-tion. The range in which the Fresnel diffraction patternchanges is inversely proportional to the pulse width anddirectly proportional to the square of the aperture radius[Eq. (25)]. The interference effect can also be reduced byserration of the edge of the diffraction aperture. In thiscase the central bright spot, normally appearing in theFresnel diffraction pattern for a circular aperture, fadesout. The combination of a serrated aperture and ultra-short pulsed-laser illumination provides a quite uniformdistribution of the beam intensity within a certain dis-tance from the diffraction plane. The uniform intensityis useful in optical imaging through random media, non-linear optical imaging, and time-resolved imaging as wellas in laser fusion. It may also be helpful when one triesto compensate for the high-order dispersion to obtain anultrashort pulse of 10 fs or less.13

ACKNOWLEDGMENTSThe authors acknowledge support from the AustralianResearch Council and thank C. J. R. Sheppard for his en-

couragement and discussions. Reference 10 was broughtto our attention by Liejia Qian of the Shanghai Institute ofOptics and Fine Mechanics, Chinese Academy of Sciences.

Permanent address, Department of Applied Physics,Victoria University of Technology, P.O. Box 14428,MCMC, Victoria 8001, Australia

REFERENCES1. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,

New York, 1968).2. M. Born and E. Wolf, Principles of Optics (Pergamon, New

York, 1980).3. P. Jacquinot and M. B. Roizen-Dossier, “Apodisation,” in

Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam,1964), Vol. 3.

4. C. J. R. Sheppard, “Imaging in optical systems of finiteFresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).

5. T. Wilson and C. J. R. Sheppard, “Imaging with finite valuesof Fresnel number,” J. Opt. Soc. Am. 72, 1639–1641 (1982).

6. S. F. Gibson and F. Lanni, “Diffraction by a circular apertureas a model for three-dimensional optical microscopy,” J. Opt.Soc. Am. A 6, 1357–1367 (1989).

7. A. Arimoto, “Intensity distribution of aberration-free diffrac-tion pattern due to circular aperture in large F-number op-tical systems,” Opt. Acta 23, 245–250 (1976).

8. C. Bibeau, D. R. Speck, R. B. Ehrlich, C. W. Laumann, D. T.Kyrazis, M. A. Henesian, J. K. Lawson, M. D. Perry, P. J.Wegner, and T. L. Weiland, “Power, energy, and temporalperformance of the Nova laser facility with recent improve-ments to the amplifier system,” Applied Opt. 31, 5799–5809(1992).

9. J. Hunt and D. R. Speck, “Recent and future performance ofthe Nova laser system,” Opt. Eng. 28, 461–486 (1989).

10. J. M. Auerbach and V. P. Karpenko, “Serrated-apertureapodisers for high-energy systems,” Appl. Opt. 33,3179–3183 (1994).

11. X. M. Deng, D. Fan, and Qian Liejia, “Serrated apertureand its application in high power lasers,” in Collection ofTheses on High Power Laser and Plasma Physics (NationalLaboratory on High Power Laser and Physics, Shanghai,1993), pp. 45–47.

12. N. George and G. M. Morris, “Diffraction by serrated aper-tures,” J. Opt. Soc. Am. 70, 6–17 (1980).

13. J. Zhou, G. Taft, C. Huang, M. M. Murnane, and H. C.Kapteyn, “Pulse evolution in a broad-bandwidth Ti: sapphirelaser,” Opt. Lett. 19, 1149–1151 (1994).

14. A. Stingl, G. Spielman, and F. Krausz, “Generation of 11-fspulses from a Ti: sapphire laser without the use of prisms,”Opt. Lett. 19, 204–206 (1994).

15. M. S. Pshenichnikov, W. P. deBoeij, and D. A. Wiersma,“Generation of 13-fs, 5-mW pulses from a cavity-damped Ti:sapphire laser,” Opt. Lett. 19, 572–574 (1994).

16. M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spa-tial and temporal transformation of femtosecond laserpulses by lenses and lens systems,” J. Opt. Soc. Am. B9, 1158–1159 (1992).

17. M. Gu and C. J. R. Sheppard, “Analysis of confocal mi-croscopy under ultrashort light-pulse illumination: com-ments,” J. Opt. Soc. Am. A 11, 2742–2743 (1994).

18. C. J. R. Sheppard and M. Hrynevych, “Diffraction by acircular aperture: a generalization of Fresnel diffractiontheory,” J. Opt. Soc. Am. A 9, 274–281 (1992).