Generalized Bessel pulse beams - OSA Publishing

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J. Opt. Soc. Am. A / Vol. 19, No. 11 / November 2002

Colin J. R. Sheppard

Generalized Bessel pulse beams Colin J. R. Sheppard Optics Research Group, Department of Applied Sciences, Delft University of Technology, NL 2628 CJ Delft, The Netherlands, and Department of Physical Optics, School of Physics, University of Sydney, Sydney NSW 2006, Australia Received December 20, 2001; revised manuscript received April 25, 2002; accepted June 25, 2002 A generalization of type 3 ultrashort pulses (also known as pulse beams or isodiffracting pulses) is introduced. The Bessel beam form of this generalized beam consists of pulses that propagate in free space, without spreading, with a velocity that can be less than that of light. A model spectral distribution that is zero outside a finite range is investigated. © 2002 Optical Society of America OCIS codes: 140.7090, 260.1960, 270.5530, 280.3640, 320.5550.

1. INTRODUCTION When laser pulses become very short (less than a few cycles), they exhibit dispersion effects even in free-space propagation. This occurs because the various spectral components diffract differently. The details of how ultrashort pulses propagate depend on the relationship between the angular spectra of the spectral components. In general, the propagation behavior is very complicated. However, for Gaussian beams, a particular special relationship is that of isodiffracting pulses (also called type 3 pulses or pulse beams), first proposed by Heyman and Melamed.1 Sheppard and Gan2 explained that the spectral components of such pulses exhibit identical phase fronts, so that they are the form of pulsed beam produced by a mode-locked cavity. Isodiffracting pulses have also been discussed by Wang et al.,3 Porras,4 Caron and Potvliege,5 and Feng and Winful.6 The propagation characteristics are found to be much simpler than those of other types of pulsed beam. For continuous wave beams, the two important fundamental beams are the Gaussian beam and the Bessel beam. The focus wave mode (FWM) is a solution of the scalar or vector wave equation that results in localized pulses that propagate without spreading in free space.7 Recently Reivelt and Saari have described how FWMs can be generated.8 It has been shown that FWMs are a special case of a type 3 (isodiffracting) Bessel beam.9 The properties and behavior of type 3 pulsed beams of different forms can be understood by consideration of the three-dimensional (3-D) pupil function,10 showing the angular spectrum representation in k space.11 This approach can be used to show the angular spectrum content of various forms of the type 3 pulse beam, including Gaussian beams and Bessel beams. The 3-D pupil for the type 3 pulse is given by the product of a function of the wave number k (the spectral distribution) and a function of the difference between the wave number and the axial component of the wave number, k ⫺ k z . Type 3 pulses propagate in free space at the speed of light. A different form of localized pulse beam is the so-called X wave,12 which exhibits an alternative relationship between the spectral components. Thus FWMs and X waves are two distinct types of pulsed beam. 1084-7529/2002/112218-05$15.00

The aim of this paper is to generalize the concept of the type 3 pulsed beams to pulses that propagate in free space slower than the speed of light. In particular, we will concentrate on pulsed Bessel beams.

2. THREE-DIMENSIONAL PUPIL FOR THE GENERALIZED BESSEL PULSE BEAM A zero-order Bessel beam is generated by an axially symmetric angular spectrum of plane waves, propagating at an angle ␪ to the axis. The amplitude of a single spectral component of a zero-order Bessel pulsed beam can be written in cylindrical coordinates r, z as U 共 r, z 兲 ⫽ J 0 共 kr sin ␪ 兲 exp共 ikz cos ␪ 兲 exp共 ⫺ikct 兲 , (1) where J 0 is a Bessel function of the first kind of order zero, and in general ␪ varies with k ⫽ 2 ␲ /␭ ⫽ ␻ /c. Summing over the spectral components gives the timevarying amplitude, U 共 r, z, t 兲 ⫽

冕

⫹⬁

⫺⬁

f 共 k 兲 J 0 共 kr sin ␪ 兲 exp共 ikz cos ␪ 兲

⫻ exp共 ⫺ikct 兲 dk,

(2)

in which the spectral distribution f(k) is zero outside its passband. This equation can be written as U 共 r, z, t 兲 ⫽

冕

⬁

⫺⬁

冋冉冊册

f 共 k 兲 J 0 共 kr sin ␪ 兲 exp ⫺ikz

冋冉

⫻ exp ⫺ikc t ⫺

z

␤c

dk,

1

␤

⫺ cos ␪

冊册 (3)

where ␤ is a parameter that is independent of k. We now choose to let the relationship between ␪ and k be given by k

冉

1

␤

冊

⫺ cos ␪ ⫽ k c ,

(4)

where k c is a constant. In this case the first exponential function can be taken out of the integral, to give © 2002 Optical Society of America


Vol. 19, No. 11 / November 2002 / J. Opt. Soc. Am. A

U 共 r, z, t 兲 ⫽ exp共 ⫺ik c z 兲

冕

⬁

⫺⬁

The eccentricity of the generating ellipse is

f 共 k 兲 J 0 共 kr sin ␪ 兲

冋冉

⫻ exp ⫺ikc t ⫺

z

␤c

冊册

dk.

e ⫽ 共 1 ⫺ ␤ 2 兲 ⫺1/2. (5)

Introducing the local time t ⬘ , t⬘ ⫽ t ⫺

,

(6)

we see that ␤ c represents the group velocity of the pulse. Equation (4) describes the surface of revolution of a conic section in k space, corresponding to the 3-D pupil function. If ␤ is unity, the pulse travels at the speed of light, corresponding to the case that has been variously called a pulse beam, a type 3 pulse, or an isodiffracting pulse. The FWM is a special case (for a particular spectral distribution) of this isodiffracting pulse. The 3-D pupil function is then a paraboloid, with a minimum wave number (Fig. 1) k min ⫽ k c /2.

(7)

The angular spectrum contains backward-propagating components for k ⬍ kc .

(8)

The fact that the 3-D pupil function is represented by a paraboloid is equivalent to the observation that FWM solutions are products of a function that is a solution of the paraxial wave equation and a plane wave.13 If ␤ is less than unity, the 3-D pupil function is an prolate spheroid. The wave number has a minimum value k min ⫽

␤kc

U 共 r, z, t 兲 ⫽ exp共 ⫺ik c z 兲

冋

k max ⫽

1⫺␤

k max

冠再冋册冎冡

f共 k 兲J0

k共 1 ⫺ ␤ 兲

␤

k共 1 ⫹ ␤ 兲

␤

⫺ kc

册

1/2

r exp共 ⫺ikct ⬘ 兲 dk. (12)

If ␤ is greater than unity, representing a superluminal pulse, the 3-D pupil function is a hyperboloid of two sheets.

3. SPECTRAL DISTRIBUTION The generalized pulse beam for ␤ ⬍ 1 contains frequency components over a finite range of frequencies. So to model the behavior of the beam, we need to consider a frequency spectrum that is appropriately limited. A frequency distribution that is zero outside a finite range was described by Caron and Potvliege.5 However, here we choose to examine a different distribution that has a simple Fourier transform: 共 k 1z 0 兲n

1

f共 k 兲 ⫽

␲ 1/2k 1 2 n ⌫ 共 n ⫹

1 2

兲 I n共 k 1z 0 兲

冋冉 1⫺

k ⫺ k0 k1

冊册

2 n⫺1/2

⫻ exp关 ⫺共 k ⫺ k 0 兲 z 0 兴 , 共 k0 ⫹ k1兲 ⬎ k ⬎ 共 k0 ⫺ k1兲,

and a maximum value

␤kc

冕

k min

⫻ kc ⫺

(9)

1⫹␤

(11)

In this case, backward-propagating components are present if k ⬍ ␤ k c . From Eq. (4) the argument of the sine can be calculated, so that, after also using Eq. (6), one finds

z

␤c

2219

(13)

so that .

(10)

冕

k 1 ⫹k 0

k 1 ⫺k 0

f 共 k 兲 exp共 ⫺ikct ⬘ 兲 dk ⫽ exp共 ⫺ik 0 ct ⬘ 兲 ⫻

冋

k 1z 0 k 1 共 ct ⬘ ⫺ iz 0 兲

册

n

J n (k 1 共 ct ⬘ ⫺ iz 0 兲 ) I n共 k 1z 0 兲

,

(14)

where J n , I n are Bessel functions and modified Bessel functions of the first kind and order n, respectively. In this expression, k 0 is the central wave number, 2k 1 is the total spread in wave numbers, n is a parameter (not necessarily an integer) greater than ⫺ 21 that controls the width of the spectral distribution, and k 1 z 0 is a dimensionless parameter that controls the skewness of the distribution. An advantage of this distribution is that skewness can be chosen to be positive or negative. The modulus at t ⬘ ⫽ 0 is unity, the mean of the distribution is Fig. 1. Three-dimensional pupil function for generalized pulse beams. For ␤ ⫽ 1, the pupil function is a paraboloid of revolution. For ␤ less than unity, the pupil function is a prolate spheroid. For ␤ greater than unity, the pupil function is a hyperboloid of two sheets.

¯k ⫽ k 0 ⫺ k 1

I n⫹1 共 k 1 z 0 兲 I n共 k 1z 0 兲

,

and the variance of the spectral distribution is

(15)

2220


再冉冊冋册冎

␴ 2 ⫽ k 12 1 ⫺ n ⫹ ⫺

I n⫹1 共 k 1 z 0 兲 I n共 k 1z 0 兲

2

1


I n⫹1 共 k 1 z 0 兲

2 k 1z 0 I n共 k 1z 0 兲 2

.

(16)

which for z 0 ⫽ 0 reduces to k 1 2 / 关 2 共 n ⫹ 1 兲兴 .

(17)

The frequency distribution is illustrated in Fig. 2, the mean is shown in Fig. 3, and the variance is shown in Fig. 4.

Fig. 4. Variance of the spectral distribution as a function of k 1 z 0 , for different values of the parameter n.

4. AMPLITUDE DISTRIBUTION Substituting the spectral distribution from Eq. (13) into Eq. (12), one finds that the amplitude at a general point in space is U 共 r, z, t 兲 ⫽

exp共 ⫺ik c z 兲

␲ 1/2k 1 ⫻

共 k 1z 0 兲n 1 2

2 n⌫共 n ⫹

冕冋冉 k 0 ⫹k 1

k 0 ⫺k 1

1⫺


k ⫺ k0 k1

⫻ exp关 ⫺共 k ⫺ k 0 兲 z 0 兴 ⫻ J0

冋

冠再冋

⫻ kc ⫺

k共 1 ⫹ ␤ 兲

␤

⫺ kc

k共 1 ⫺ ␤ 兲

␤

冊册

2 n⫺1/2

册

册冎冡 1/2

r

2

⫻ exp共 ⫺ikct ⬘ 兲 dk.

(18)

This expression should be compared with the corresponding one for FWM solutions.9,14 We assume that the maximum frequency corresponds to an angular spectral component directed along the axis, whereas the minimum frequency occurs at a particular angle ␣, so that the pulsed beam can be generated by an optical system of a limited aperture. Then we have Fig. 2. Model spectral distribution (a) for different values of the parameter n, for a constant value of k 1 z 0 ⫽ 0, and (b) for different values of the parameter k 1 z 0 , for a constant value of n ⫽ 3.

k 0 ⫹ k 1 ⫽ k max ⫽

␤

k c⬘

1⫺␤

k 0 ⫺ k 1 ⫽ k min ⫽ k ␣ ⬘

(19)

giving k1 ⫽

k0 ⫽

kc

␤ 2 共 1 ⫺ cos ␣ 兲

2 共 1 ⫺ ␤ 兲共 1 ⫺ ␤ cos ␣ 兲 ⬘ k c ␤ (2 ⫺ ␤ 共 1 ⫹ cos ␣ 兲 ) 2 共 1 ⫺ ␤ 兲共 1 ⫺ ␤ cos ␣ 兲

.

(20)

The ratio of the maximum to minimum wave number is Fig. 3. Mean of the spectral distribution as a function of k 1 z 0 , for different values of the parameter n.

k max k␣

⫽1⫹

2␤ 1⫺␤

sin2

␣ 2

;

(21)


Vol. 19, No. 11 / November 2002 / J. Opt. Soc. Am. A

i.e., the frequency range is appreciable for small ␣ only if ␤ is close to unity. Thus it is not possible to generate short pulses that propagate at slow speeds from a system of small angular aperture. To evaluate the integral in Eq. (18), it is convenient to introduce the variable k ⫺ k0

p⫽

k1

,

(22)

冋

k 1z 0 k 1 共 ct ⬘ ⫺ iz 0 兲

␲ ⫻

冕

1/2

共 k 1z 0 兲n

2 ⌫共 n ⫹ n

1 2


共 1 ⫺ p 2 兲 n⫺1/2J 0

⫻ 共 v 关共 1 ⫺ p 兲共 p ⫹ a 兲兴

1/2

⫽ ⫺k 0 c ⫹ k 1 c

兲

(23)

N⫽

k 0 r sin 共 ␣ /2兲共 1 ⫺ ␤ 兲

v⫽

2 1/2

(24)

共 1 ⫺ ␤ 兲 ⫹ ␤ sin2 共 ␣ /2兲

共 3 ␤ ⫺ 1 兲 sin2 共 ␣ /2兲 ⫹ 2 共 1 ⫺ ␤ 兲共 1 ⫹ ␤ 兲 sin2 共 ␣ /2兲

.

On the axis the amplitude is simply

冋冉

U 共 0, z, t 兲 ⫽ exp ⫺ik 0 c t ⫺

I n⫹1 共 k 1 z 0 兲

z v cp

冊册冋

J n (k 1 共 ct ⬘ ⫺ iz 0 兲 )

␴c

␲␴

k 1z 0 k 1 共 ct ⬘ ⫺ iz 0 兲

(25)

册

n

k 0 ⫺ k 1 ⫽ k min ⫽

,

␤c 1 ⫺ ␤ 共 k c /k 0 兲

.

(27)

k0

2 共 1 ⫺ ␤ 兲共 1 ⫺ ␤ cos ␣ 兲

␤ (2 ⫺ ␤ 共 1 ⫹ cos ␣ 兲 )

,

k1 ⫽

2 ⫺ ␤ 共 1 ⫹ cos ␣ 兲 1 ⫹ 共 1 ⫺ 2 ␤ 兲 cos ␣

c.

k0 ⫽

(28)

(32)

.

(33)

␤ 1⫺␤

␤ 1⫹␤

kc ,

kc ,

(34)

␤2 1 ⫺ ␤2

␤ 1 ⫺ ␤2

kc ,

kc ,

(35)

and the wave consists of both forward- and backwardpropagating components, as in a FWM.7 We find that in this case the amplitude reduces to

so that the carrier phase velocity is v cp ⫽

,

so that

We can derive a relationship between ␣ and k c /k 0 for a given value of ␤: ⫽

(31)

The FWM consists of an angular spectrum that includes components traveling in all directions, both forward and backward. Consider now, therefore, the generalized FWM, which is the special case of the generalized Bessel pulse beam in which the maximum and minimum values for the spectral distribution correspond to angles of zero and ␲ rad, respectively, i.e.,

where the component of the phase velocity v cp that is due to the carrier wave (the carrier phase velocity) is

kc

.

¯k

(26)

I n共 k 1z 0 兲

v cp ⫽

I n共 k 1z 0 兲

2

k 0 ⫹ k 1 ⫽ k max ⫽

⫻

(30)

5. GENERALIZED FOCUS WAVE MODE

and the constant a is given by a⫽

⫹ ....

so that the number N of oscillations per pulse is

where the optical coordinate v is given by 2

2

The pulse length, defined following Caron and Potvliege5 as the full width at which the amplitude has dropped to one-half, can also be estimated from the quadratic term in Eq. (30) as T⫽

⫻ exp关 ⫺ik 1 p 共 ct ⬘ ⫺ iz 0 兲兴 dp,

共 ␴ ct ⬘ 兲 2

At t ⬘ ⫽ 0, the total phase time gradient is thus [by use of Eq. (15)]

⳵t

1

⫺1

I n共 k 1z 0 兲

⳵␾

exp关 ⫺i 共 k 0 ct ⬘ ⫹ k c z 兲兴

J n (k 1 共 ct ⬘ ⫺ iz 0 兲 )

⫽ 1 ⫹ i 共 k 0 ⫺ ¯k 兲 ct ⬘ ⫺

so that U 共 r, z, t 兲 ⫽

册

n

2221

(29)

We find that the carrier phase velocity is always greater than or equal to the speed of light but that the product of the carrier phase and group velocities is always less than or equal to c 2 . However, the Bessel function in Eq. (26) also contributes to the phase velocity for nonzero values of k 1 z 0 . The Bessel function can be expanded about t ⬘ ⫽ 0 as a Maclaurin series, which, by use of the derivative theorem of Fourier transforms, can be written as

U 共 r, z, t 兲 ⫽

exp关 ⫺ik 0 共 ct ⫺ ␤ z 兲兴

␲ 1/2 ⫻

冕

␲

共 k 1z 0 兲n

2 n⌫共 n ⫹

1 2


sin2n ␥ J 0 (k 0 r 共 1 ⫺ ␤ 2 兲 1/2 sin ␥ )

0

⫻ exp关 ⫺ik 1 共 ct ⬘ ⫺ iz 0 兲 cos ␥ 兴 d␥ .

(36)

Note that although the integral has the form of an integral over plane waves, the variable ␥ is not equal to the physical angle ␪. The integral in Eq. (36) can be recog-

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nized as a spherical harmonic: The coordinate transformation has changed the prolate spheroid into a sphere. Along on the axis, we have U 共 0, z, t 兲 ⫽ exp关 ⫺ik 0 共 ct ⫺ ␤ z 兲兴 ⫻

冋

J n (k 1 共 ct ⬘ ⫺ iz 0 兲 ) I n共 k 1z 0 兲

k 1z 0 k 1 共 ct ⬘ ⫺ iz 0 兲

册

n

.

(37)

The Bessel function represents a pulse traveling at a velocity ␤ c, whereas the complex exponential factor represents a carrier wave propagating at a phase velocity c/ ␤ , i.e., greater than the speed of light. The case for n ⫽ 1/2 and k 1 z 0 ⫽ 0 is particularly interesting. Then the integral in Eq. (36) can be evaluated to give for the amplitude, with ␳ the spherical radius,

with angular spectra identically zero outside an assumed range. A special case in which the angular spectrum is present over a complete sphere, analogous to the FWM, was also described. This can be regarded as equivalent to the field of a moving, complex source–sink pair. The author can be reached at the address on the title page or by e-mail, [email protected].

REFERENCES 1. 2. 3.

U 共 r, z, t ⬘ 兲 ⫽ exp关 ⫺ik 0 共 ct ⫺ ␤ z 兲兴 ⫻

sin关 k 0 (共 1 ⫺ ␤ 2 兲 r 2 ⫹ ␤ 2 c 2 t ⬘ 2 )1/2兴 k 0 (共 1 ⫺ ␤ 2 兲 r 2 ⫹ ␤ 2 c 2 t ⬘ 2 )1/2

4.

.

5.

(38) This represents a fictitious simple source, together with a sink, traveling along the z axis with a velocity ␤ c. In an analogous way, Eq. (36) represents a multipole pair, at a complex displacement, traveling with a velocity ␤ c.

6. 7. 8.

6. DISCUSSION The focus wave modes7 represent pulses that travel at the speed of light without spreading. These have been identified as being a particular summation over Bessel beams of varying frequency.8,9 The original form of the FWM consisted of angular spectral components traveling at all directions in space, although by appropriate choice of the spectral distribution some of these can be made negligibly small.14 In this paper we have generalized these waves to pulses that travel at speeds slower than that of light. These pulses are rigorous solutions of the scalar wave equation. A model spectral distribution was introduced that allows study of a wide range of different forms of pulsed beam. This allows generation of pulsed beams

9. 10. 11. 12.

13. 14.

E. Heyman and T. Melamed, ‘‘Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,’’ IEEE Trans. Antennas Propag. 42, 518–525 (1994). C. J. R. Sheppard and X. Gan, ‘‘Free-space propagation of femto-second light pulses,’’ Opt. Commun. 133, 1–6 (1997). Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, ‘‘Space-time profiles of an ultrashort pulsed beam,’’ IEEE J. Quantum Electron. 33, 566–573 (1997). M. A. Porras, ‘‘Ultrashort pulsed Gaussian light beams,’’ Phys. Rev. E 58, 1086–1093 (1998). C. F. R. Caron and R. M. Potvliege, ‘‘Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,’’ J. Mod. Opt. 45, 1881–1892 (1999). S. Feng and H. G. Winful, ‘‘Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,’’ Phys. Rev. E 61, 862–873 (2000). J. N. Brittingham, ‘‘Focus wave modes in homogeneous Maxwell’s equations: transverse electric modes,’’ J. Appl. Phys. 54, 1179–1189 (1983). K. Reivelt and P. Saari, ‘‘Optical generation of focused wave modes,’’ J. Opt. Soc. Am. A 17, 1785–1790 (2000). C. J. R. Sheppard, ‘‘Bessel pulse beams and focus wave modes,’’ J. Opt. Soc. Am. A 18, 2594–2600 (2001). C. J. R. Sheppard and M. D. Sharma, ‘‘Spatial frequency content of focused ultra-short pulsed beams,’’ J. Opt. A Pure Appl. (to be published). C. W. McCutchen, ‘‘Generalized aperture and the threedimensional diffraction image,’’ J. Opt. Soc. Am. 54, 240– 244 (1964). J. Lu and J. F. Greenleaf, ‘‘Nondiffracting X waves—exact solutions to free-space scalar wave equations and their finite aperture realizations,’’ IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992). P. A. Be´langer, ‘‘Packetlike solutions of the homogeneous wave equation,’’ J. Opt. Soc. Am. A 1, 723–724 (1984). R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, ‘‘Aperture realizations of exact solutions to homogeneous-wave equations,’’ J. Opt. Soc. Am. A 10, 75–87 (1993).