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Propagation properties of ultrashort chirped pulsed Gaussian beams in the free space
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Optik
Optics
Optik 117 (2006) 259–264 www.elsevier.de/ijleo
Propagation properties of ultrashort chirped pulsed Gaussian beams in the free space Qihui Zou, Baida Lu¨ Institute of Laser Physics & Chemistry, Sichuan University, Chengdu 610064, China Received 29 April 2005; accepted 28 September 2005
Abstract Based on the Rayleigh diffraction integral and complex analytical signal representation, the free-space analytical propagation equation and its Fourier spectrum for ultrashort chirped pulsed Gaussian beams with constant diffraction length are derived. The effect of chirp parameter on the spatiotemporal and spectral properties is illustrated with analytical formulas and numerical calculation results. It is shown that the axial spectra of ultrashort chirped pulsed Gaussian beams become broadened with increasing chirp parameter. For single optical cycle, the transversal intensity distribution is affected by increasing chirp parameter, but almost not affected for several optical cycles. Moreover, the positive or negative sign of the chirp parameter has no effect on the spectral distribution and intensity distribution. r 2005 Elsevier GmbH. All rights reserved. Keywords: Ultrashort pulsed Gaussian beam; Chirp; Complex analytical signal representation
1. Introduction There is much current interest in spatiotemporal propagation properties of ultrashort pulsed light beams because the rapid development of femtosecond laser sources is moving toward few- and single-cycle optical pulses. Much attention is therefore being paid to the problems of their propagation in vacuum, linear and nonlinear dispersive media, and optical systems etc. [1–4]. But the Gaussian pulse [1,3,4], Poisson pulse [2,5], hyperbolic secant pulse [6,7], and Lorentz pulse [6] are usually used as initial temporal pulse form. With the development of fiber-optic communication technology, the spatiotemporal and spectral properties of chirped pulse [1,8,9] propagating in vacuum and dispersive media are of great practical interests. The purpose of Corresponding author.
E-mail address: [email protected] (Q. Zou). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.09.006
the present paper is to study the propagation properties of ultrashort chirped pulsed Gaussian beams with constant diffraction length. In Section 2, based on the Rayleigh diffraction integral and complex analytical signal representation, the propagation expressions are derived, which enable us to study the propagation properties of few-cycle and even single-cycle chirped pulsed beams with constant diffraction length in free space. Numerical calculation results and comparison with the previous work are given in Section 3. Finally, Section 4 summarizes the main results obtained in this paper.
2. Propagation equations Firstly, let us consider the propagation of each Fourier component of an ultrashort pulse in free space, which obeys the Rayleigh diffraction
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From Eq. (7) the Fourier spectrum of E(r, t) reads as
integral [10] Eðr; oÞ ¼
1 2p
Z
E 0 ðr0 ; oÞ
S
ikR
q e qz R
d 2 r0 ,
(1)
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ jr r0 j ¼ ðx x0 Þ2 þ ðy y0 Þ2 þ z2 ,
(2)
r ¼ (x, y, z) and r0 ¼ (x0, y0, 0) are points at the output plane and input plane S, respectively. E0(r0, o) denotes the Fourier component of the pulse with frequency o at the input plane and k is wave number. Under the condition krb1, (r ¼ ðx2 þ y2 þ z2 Þ1=2 ), Eq. (1) simplifies to Z ik eikR qR 2 d r0 . Eðr; oÞ ¼ E 0 ðr0 ; oÞ (3) 2p S R qz R in Eq. (3) can be approximately expressed as Rrþ
x20 þ y20 2xx0 2yy0 . 2r
(4)
Replacing R in the exponential term in Eq. (3) by Eq. (4) and in other terms by r, Eq. (3) becomes Z Z ik cos y eikr Eðr; oÞ ¼ E 0 ðr0 ; oÞ 2p r S ik 2 2 exp ðx0 þ y0 2xx0 2yy0 Þ 2r dx0 dy0 ,
ð5Þ
where cos y ¼ z=r, y being the diffraction angle. Assume that the input plane (z ¼ 0) has a Gaussian pulsed beam [3,11] E 0 ðr0 ; oÞ ¼ AðoÞ expðkr20 =2lÞ,
(6)
where A(o) is the initial on-axis spectrum, r20 ¼ x20 þ y20 , and l is the diffraction length (Rayleigh length) which is assumed to be independent of the frequency o. By using complex analytical signal representation [12], from Eq. (5) the analytical signal for ultrashort chirped pulsed beams in the space-time domain turns out to be [11] il cos y r ilr sin2 y Eðr; tÞ ¼ P t þ , (7) r þ il c 2cðr þ ilÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where sin y ¼ r=r, r ¼ x2 þ y2 , c is the speed of light in vacuum, P(t) is complex analytical signal representation of initial pulse A(t) Z 1 2 PðtÞ ¼ pffiffiffiffiffiffi AðoÞeiot do, (8) 2p 0 where A(o) being Fourier transform of A(t), i.e., Z þ1 1 AðoÞ ¼ pffiffiffiffiffiffi AðtÞ expðiotÞdt. 2p 1
(9)
^ oÞ ¼ 2HðoÞ il cos y AðoÞeikr Eðr; r þ il lkr sin2 y exp 2ðr þ ilÞ
ð10Þ
with H(o) being the Heaviside step function, i.e., ( 1 o40; HðoÞ ¼ (11) 0 op0: Suppose that real form of chirped Gaussian pulse [13] at r ¼ 0 is 2 t (12) AðtÞ ¼ exp 2 cosðoc t þ Ct2 Þ, T where oc is the carrier frequency, Tc is the corresponding oscillating period, T is the pulse duration, C is chirp parameter, and m ¼ T=T c is the number of oscillation within the pulse duration. On substituting Eq. (12) into Eqs. (9), (8) and (7), straightforward integral calculations lead to ! il cos y t00 2 exp 2 Eðr; tÞ ¼ r þ il T ( ! t00 2 exp iC 2 expðioc t00 Þ T " !# pffiffiffiffiffiffiffiffiffiffiffiffiffiffi oc T 1 iC 00 t þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ erf i T 2 1 iC ! t00 2 þ exp iC 2 expðioc t00 Þ T pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1 þ iC 00 oc T t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ erf i , ð13Þ T 2 1 þ iC where t00 ¼ t0 þ ilr sin2 y=½2cðr þ ilÞ, t0 ¼ t r=c is local time, erf(d) being error function. It is readily shown that Eq. (13) is consistent with Eq. (24) in Ref. [11] when the chirp parameter vanishes. Eq. (13) is the basic analytical formula obtained in present paper, which is not only applicable to Fresnel and Fraunhofer diffraction region, but also to large diffraction angle. On substituting Eq. (12) into Eqs. (9) and (10), the Fourier Spectrum for ultrashort chirped pulsed Gaussian beams yields ^ oÞ Eðr;
1 il cos y lkr sin2 y ¼ pffiffiffi HðoÞ expðikrÞ exp 2ðr þ ilÞ 2 r þ il 2 2 T T ðo þ oc Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 4ð1 iCÞ 1 iC 2 T T ðo oc Þ2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp . 4ð1 þ iCÞ 1 þ iC
ð14Þ
ARTICLE IN PRESS Q. Zou, B. Lu¨ / Optik 117 (2006) 259–264
From Eq. (14) the power spectrum of ultrashort chirped pulsed Gaussian beams reads as ^ oÞj2 jEðr;
T 2 l 2 cos2 y olr2 sin2 y 2 1=2 Þ exp ð1 þ C 2ðr2 þ l 2 Þ cðr2 þ l 2 Þ T 2 ðo oc Þ2 exp 2ð1 þ C 2 Þ T 2 ðo þ oc Þ2 þ exp 2ð1 þ C 2 Þ T 2 ðo2 þ o2c Þ þ 2 exp 2ð1 þ C 2 Þ 2 2 T ðo þ o2c ÞC f , cos 2ð1 þ C 2 Þ
¼
ð15Þ
ð16Þ
where ‘‘Re’’ denotes the real part, u1 ¼
lr2 ð1 þ C 2 Þsin2 y T 2 ðr2 þ l 2 Þoc c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2ð1 þ C 2 Þðr2 þ l 2 ÞTc
(16a)
u2 ¼
lr2 ð1 þ C 2 Þsin2 y þ T 2 ðr2 þ l 2 Þoc c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2ð1 þ C 2 Þðr2 þ l 2 ÞTc
(16b)
u3 ¼
ð1 þ C 2 Þ1=4 lr2 sin2 y pffiffiffi exp½if=2, 2ðr2 þ l 2 ÞTc
(16c)
F ¼ Co2c T 2 =2ð1 þ C 2 Þ f=2.
If chirp parameter vanishes (C ¼ 0), from Eq. (16), we obtain pffiffiffiffiffiffi 2 T 2 o2c 2pTl cos2 y exp I 0 ðrÞ ¼ 2 4ðr2 þ l 2 Þ
2 ½1 erf ðu4 Þ exp u4
þ ½1 erf ðu5 Þ exp u25
ð18Þ þ 2½1 erf ðu6 Þ exp u26 , where
where f ¼ tan1 C. It is easily shown that the power spectrum of ultrashort chirped pulsed Gaussian beams is irrelevant to the sign of the chirp parameter. According to Eq. (15) and Parseval’s theorem, the spatial intensity of ultrashort chirped pulsed Gaussian beams yields pffiffiffiffiffiffi 2 2pTl cos2 y T 2 o2c IðrÞ ¼ exp 2ð1 þ C 2 Þ 4ðr2 þ l 2 Þ ½1 erf ðu1 Þ expðu21 Þ þ ½1 erf ðu2 Þ expðu22 Þ þ 2ð1 þ C 2 Þ1=4 Re½expðu23 Þ expðiFÞð1 erf ðu3 ÞÞ ,
261
(16d)
On-axis (y ¼ 01) intensity can reduce to pffiffiffiffiffiffi 2 2pTl I 0 ðzÞ ¼ 1 þ ð1 þ C 2 Þ1=4 2ðz2 þ l 2 Þ T 2 o2c CT 2 o2c f cos exp . ð17Þ 2ð1 þ C 2 Þ 2ð1 þ C 2 Þ 2 From Eq. (17), we see that the axial intensity is irrelevant to the sign of the chirp parameter. When the initial pulse duration contains several optical cycles, onaxis intensity approximately simplify pffiffiffiffifficould ffi I 00 ðzÞ 2pTl 2 =2ðz2 þ l 2 Þ, that is, axial intensity is irrelevant to the chirp parameter for few-cycle ultrashort chirped pulsed Gaussian beams.
u4 ¼
lr2 sin2 y T 2 ðr2 þ l 2 Þoc c pffiffiffi , 2ðr2 þ l 2 ÞTc
(18a)
u5 ¼
lr2 sin2 y þ T 2 ðr2 þ l 2 Þoc c pffiffiffi , 2ðr2 þ l 2 ÞTc
(18b)
lr2 sin2 y u6 ¼ pffiffiffi . 2ðr2 þ l 2 ÞTc
(18c)
When pulses are not chirped (C ¼ 0), on-axis intensity can be gained from Eq. (18) pffiffiffiffiffiffi I 0 ðzÞ ¼ 2pTl 2 =ð2z2 þ 2l 2 Þ, (19) From eq. (19) it is readily seen that the axial intensity for few-cycle pulsed beams is the same as that of pulsed beams without chirp.
3. Numerical calculation and analysis Numerical calculations were carried out using Eqs. (13), (15) and (16) to quantitatively illustrate the effect of the chirp parameter on the spatiotemporal and spectral properties for ultrashort chirped pulsed Gaussian beams. In the following calculations we take oc ¼ 2:36 fs1 , l ¼ 10 mm, and r ¼ 5l. Fig. 1 gives the temporal pulse forms of real electric field Re[E(r, t)] for the chirp parameter C ¼ 5, and m ¼ 2. From Fig. 1 we see that on-axis temporal pulse is positively chirped [7,14], and the trailing edge of the offaxis positively chirped pulse oscillation. However, as compared with pulsed beams (C ¼ 0) the on-axis temporal pulse form is chirped [11]. Fig. 2 shows temporal pulse shapes of the ultrashort chirped pulsed Gaussian beams for different diffraction angles y ¼ 01 and 1.01. From Fig. 2 we see that the on-axis pulse shapes are distorted, the trailing edge and the leading edge of the on-axis chirped pulse become asymmetric, and the trailing edge of the off-axis positively chirped pulse shape develops oscillatory tail, while the leading edge of the off-axis negatively chirped pulse shape develops oscillatory tail. By Eq. (15), as the chirp parameter and diffraction angle are smaller, the power spectrum of ultrashort chirped pulsed Gaussian beams could be approximately
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×10-1
4.5
θ=0°
1.0 0.5 0 − 40
−30
−20
−10
0
10
20
30
×10-2
4.0
40
− 0.5
2
real (E) (arbitrary units)
1.5
|E(r,t)| (arbitrary units)
2.0
−1.0
C=−6
3.0 2.5 2.0 1.5 1.0 0.5
−1.5
0 −20
−2.0
−12
local time t´ (fs) 6
(a)
×10-3
2.5
2
−30
−20
−10
0
10
20
30
40
−2 −4 −6
|E(r,t)|2 (arbitrary units)
4
0 − 40
−4
4
12
20
local time t´ (fs)
θ=1.0° real (E) (arbitrary units)
C=6
θ=0°
3.5
θ=1.0°
C=6
2.0 C=−6
1.5 1.0 0.5 0 −20
local time t´ (fs) (b)
(a)
×10-4
−12
−4
4
12
20
local time t´ (fs) (b)
Fig. 1. The temporal pulse forms of ultrashort chirped pulsed Gaussian beams for chirp parameter C ¼ 5. (a) y ¼ 01, (b) y ¼ 1:01.
Fig. 2. The temporal pulse shapes of ultrashort chirped pulsed Gaussian beams for chirp parameters C ¼ 6 and 6. (a) y ¼ 01, (b) y ¼ 1:01.
reduced to T 2 l 2 cos2 y ð1 þ C 2 Þ1=2 2ðr2 þ l 2 Þ olr2 sin2 y T 2 ðo oc Þ2 exp exp . ð20Þ 2ð1 þ C 2 Þ cðr2 þ l 2 Þ ^ oÞj2 =qojx¼y¼0 ¼ 0 the By using Eq. (20) and qjEðr; position of maximum power spectrum is determined by
^ oÞj2 jEðr;
oc o ¼ ð1 þ C 2 Þlr2 sin2 y=½T 2 cðr2 þ l 2 Þ.
(21)
From Eq. (21) we see that on-axis power spectra have no shift, spectral redshift increases with increasing diffraction angle, and the spectral redshift increases by C2 times due to the chirp parameter. Fig. 3 illustrates the ^ oÞj2 =jEðr; ^ oÞj2 of normalized power spectra jEðr; max chirped pulsed Gaussian beams at different diffraction angles y ¼ 01, 0.51, and 1.01 for (a) C ¼ 0, (b) C ¼ 1, ^ oÞj2 is given by Eq. (15). As can be seen, the where jEðr; spectra are redshifted for ya01, the spectral redshift increases with increasing diffraction angle y, and the spectral redshift increases rapidly due to the chirp parameter. For example, when C ¼ 0, the spectral redshift (oc o) is 0, 0.06, 0.34 and 0.88 fs1 for y ¼ 01, 0.21, 0.51 and 0.81, respectively. When C ¼ 1, the
spectral redshift is 0, 0.11, 0.69 and 1.77 fs1 for y ¼ 01, 0.21, 0.51 and 0.81, respectively. When C ¼ 2, the spectral redshift is 0, 0.28, 1.72 and 2.35 fs1 for y ¼ 01, 0.21, 0.51 and 0.81, respectively. These calculation results agree very well with results calculated with Eq. (21). From Eq. (15) on-axis power spectrum approximately reduces to a Gaussian function when chirp parameter is small. From Eq. (20), the spectral half-bandwidth (at 1/e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi intensity point) is given by 2ð1 þ C 2 Þ=T, hence, onaxis spectral bandwidth increases by (1+C2)1/2 times due to the chirp parameter [8]. Fig. 4 gives normalized on-axis power spectra of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 0, 1, 2 and 4. As can be seen, on-axis spectra almost have no shift, and on-axis spectral distributions remain Gaussian distribution when the chirp parameter is smaller, but the on-axis spectral bandwidth increases with increasing chirp parameter. However, on-axis spectral distributions are no longer Gaussian distribution with increasing chirp parameter. Because the power spectrum of ultrashort chirped pulsed Gaussian beams is irrelevant to the sign of the
ARTICLE IN PRESS Q. Zou, B. Lu¨ / Optik 117 (2006) 259–264
263
1.2
1.2 θ = 0°
0.8
θ = 0.5°
0.6
θ = 1.0°
normalized power spectra
normalized power spectra
C=4 1.0
0.4 0.2 0
1.0 C=2 0.8 0.6 C=0
0.4
C=1
0.2 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
frequency (fs-1) (a)
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
frequency (fs-1)
1.0
θ = 0°
0.8
θ = 0.5°
Fig. 4. Normalized on-axis power spectra of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 0, 1, 2 and 4.
θ = 1.0°
0.6
1.2
0.4 0.2 0 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
frequency (fs-1)
(b)
Fig. 3. Normalized power spectra of ultrashort chirped pulsed Gaussian beams for different values of y ¼ 01, 0.51, and 1.01. (a) C ¼ 0, (b) C ¼ 1.
normalized intensity I(r)/Imax
normalized power spectra
1.2
C=0 C=2
1.0
m=1
C=− 4
0.8
C=4 0.6
C=6
0.4 0.2 0
0.08
0.04
0
0.12
0.16
0.20
(mm)
4. Concluding remarks In this paper, based on the Rayleigh diffraction integral and complex analytical signal representation,
Fig. 5. Normalized transversal intensity distributions of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 0, 2, 4, 4 and 6. 1.2 C=2
normalized intensity I(r)/Imax
chirp parameter, according to Parseval’s theorem, so the spatial intensity is irrelevant to the sign of the chirp parameter. Fig. 5 shows normalized transversal intensity (I(r)/Imax) distributions of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 0, 2, 4, 4 and 6, where Imax is the on-axis intensity for C ¼ 0. The calculation parameters we take are m ¼ 1 and z ¼ 10 mm. As shown in Fig. 5, the transversal intensity distributions nearly remain unchanged when the chirp parameter are small, on-axis intensity decreases and the transversal intensity distributions broaden with increasing chirp parameter, but the sign of the chirp parameter has no effect on the transversal intensity distribution. Fig. 6 illustrates normalized transversal intensity distributions of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 2, 4 and 8. The calculation parameters are m ¼ 3 and z ¼ 100 mm. As can be seen, the three curves of transversal intensity distributions are nearly overlapped, therefore, the increase of the chirp parameter almost has no effect on the transversal intensity distribution for few-cycle pulsed beams.
m =3
1.0 C=4
0.8 0.6 0.4
C=8
0.2 0 0
0.1
0.2
0.3
0.4
0.5 0.6 (mm)
0.7
0.8
0.9
1.0
Fig. 6. Normalized transversal intensity distributions of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 2, 4 and 8.
the free-space analytical propagation equation and its Fourier spectrum for ultrashort chirped pulsed Gaussian beams with constant diffraction length are derived. The effect of the chirp parameter on the spatiotemporal
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and spectral properties is illustrated with analytical formulas and numerical calculation results. It is shown that the trailing edge and leading edge of the on-axis pulse shape become asymmetric, and the trailing edge of the off-axis positive chirped pulse develops oscillatory tail, while the leading edge of the off-axis negative chirped pulse develops oscillatory tail. The off-axis spectral redshift increases by about C2 times under the conditions that the chirp parameter and diffraction angle are small. The axial spectral bandwidth increases by (1+C2)1/2 times due to the chirp parameter. For single-cycle pulse, the transversal intensity distribution is affected by the increase of the chirp parameter, but almost not affected for few-cycle pulse. Moreover, the sign of the chirp parameter has no effect on the spectral distribution and spatial intensity distribution.
Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 10574097.
References [1] I.P. Christov, Propagation of femtosecond light pulses, Opt. Commun. 53 (1985) 364–366. [2] M.A. Porras, Nonsinusoidal few-cycle pulsed light beams in free space, J. Opt. Soc. Am. B 16 (1999) 1468–1474.
[3] G.P. Agrawal, Far-field diffraction of pulsed optical beams in dispersive media, Opt. Commun. 167 (1999) 15–22. [4] M.A. Porras, Propagation of single-cycle pulsed light beams in dispersive media, Phys. Rev. A 60 (1999) 5069–5073. [5] M.A. Porras, R. Borghi, M. Santarsiero, Few-opticalcycle Bessel-Gauss pulsed beams in free space, Phys. Rev. E 62 (2000) 5729–5737. [6] T. Brabec, F. Krausz, Nonlinear optical pulse propagation in the single-cycle regime, Phys. Rev. Lett. 78 (1997) 3282–3285. [7] B. Lu¨, Z. Liu, Propagation properties of ultrashort pulsed Bessel beams in dispersive media, J. Opt. Soc. Am. A 20 (2003) 582–587. [8] G.P. Agrawal, Fiber-optical Communication Systems, third Ed., Tsinghua University Press, 2004. [9] G.P. Agrawal, Spectrum-induced changes in diffraction of pulsed optical beams, Opt. Commun. 157 (1998) 52–56. [10] Z. Wang, Z. Zhang, Z. Xu, Spectral and temporal properties of ultrashort light pulse in the far zone, Opt. Commun. 123 (1996) 5–50. [11] B. Lu¨, Z. Liu, Spatial and temporal properties of ultrashort pulses propagating in free space, J. Mod. Opt. 50 (2003) 815–823. [12] T. Melamed, L.B. Felsen, Pulsed-beam propagation in lossless dispersive media I. Theory, J. Opt. Soc. Am. A 15 (1998) 1268–1276. [13] Z. Yang, W. Hu, X. Fu, The production and elimination of the spatial singularity for ultrashort chirped pulsebeam propagation in free space, Acta Phys. Sinica 52 (2003) 1920–1924. [14] A.E. Siegman, Laser, University Science Books Mill Valley, California, 1986 (Chapter 9).
Optik
Optics
Optik 117 (2006) 259–264 www.elsevier.de/ijleo
Propagation properties of ultrashort chirped pulsed Gaussian beams in the free space Qihui Zou, Baida Lu¨ Institute of Laser Physics & Chemistry, Sichuan University, Chengdu 610064, China Received 29 April 2005; accepted 28 September 2005
Abstract Based on the Rayleigh diffraction integral and complex analytical signal representation, the free-space analytical propagation equation and its Fourier spectrum for ultrashort chirped pulsed Gaussian beams with constant diffraction length are derived. The effect of chirp parameter on the spatiotemporal and spectral properties is illustrated with analytical formulas and numerical calculation results. It is shown that the axial spectra of ultrashort chirped pulsed Gaussian beams become broadened with increasing chirp parameter. For single optical cycle, the transversal intensity distribution is affected by increasing chirp parameter, but almost not affected for several optical cycles. Moreover, the positive or negative sign of the chirp parameter has no effect on the spectral distribution and intensity distribution. r 2005 Elsevier GmbH. All rights reserved. Keywords: Ultrashort pulsed Gaussian beam; Chirp; Complex analytical signal representation
1. Introduction There is much current interest in spatiotemporal propagation properties of ultrashort pulsed light beams because the rapid development of femtosecond laser sources is moving toward few- and single-cycle optical pulses. Much attention is therefore being paid to the problems of their propagation in vacuum, linear and nonlinear dispersive media, and optical systems etc. [1–4]. But the Gaussian pulse [1,3,4], Poisson pulse [2,5], hyperbolic secant pulse [6,7], and Lorentz pulse [6] are usually used as initial temporal pulse form. With the development of fiber-optic communication technology, the spatiotemporal and spectral properties of chirped pulse [1,8,9] propagating in vacuum and dispersive media are of great practical interests. The purpose of Corresponding author.
E-mail address: [email protected] (Q. Zou). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.09.006
the present paper is to study the propagation properties of ultrashort chirped pulsed Gaussian beams with constant diffraction length. In Section 2, based on the Rayleigh diffraction integral and complex analytical signal representation, the propagation expressions are derived, which enable us to study the propagation properties of few-cycle and even single-cycle chirped pulsed beams with constant diffraction length in free space. Numerical calculation results and comparison with the previous work are given in Section 3. Finally, Section 4 summarizes the main results obtained in this paper.
2. Propagation equations Firstly, let us consider the propagation of each Fourier component of an ultrashort pulse in free space, which obeys the Rayleigh diffraction
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From Eq. (7) the Fourier spectrum of E(r, t) reads as
integral [10] Eðr; oÞ ¼
1 2p
Z
E 0 ðr0 ; oÞ
S
ikR
q e qz R
d 2 r0 ,
(1)
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ jr r0 j ¼ ðx x0 Þ2 þ ðy y0 Þ2 þ z2 ,
(2)
r ¼ (x, y, z) and r0 ¼ (x0, y0, 0) are points at the output plane and input plane S, respectively. E0(r0, o) denotes the Fourier component of the pulse with frequency o at the input plane and k is wave number. Under the condition krb1, (r ¼ ðx2 þ y2 þ z2 Þ1=2 ), Eq. (1) simplifies to Z ik eikR qR 2 d r0 . Eðr; oÞ ¼ E 0 ðr0 ; oÞ (3) 2p S R qz R in Eq. (3) can be approximately expressed as Rrþ
x20 þ y20 2xx0 2yy0 . 2r
(4)
Replacing R in the exponential term in Eq. (3) by Eq. (4) and in other terms by r, Eq. (3) becomes Z Z ik cos y eikr Eðr; oÞ ¼ E 0 ðr0 ; oÞ 2p r S ik 2 2 exp ðx0 þ y0 2xx0 2yy0 Þ 2r dx0 dy0 ,
ð5Þ
where cos y ¼ z=r, y being the diffraction angle. Assume that the input plane (z ¼ 0) has a Gaussian pulsed beam [3,11] E 0 ðr0 ; oÞ ¼ AðoÞ expðkr20 =2lÞ,
(6)
where A(o) is the initial on-axis spectrum, r20 ¼ x20 þ y20 , and l is the diffraction length (Rayleigh length) which is assumed to be independent of the frequency o. By using complex analytical signal representation [12], from Eq. (5) the analytical signal for ultrashort chirped pulsed beams in the space-time domain turns out to be [11] il cos y r ilr sin2 y Eðr; tÞ ¼ P t þ , (7) r þ il c 2cðr þ ilÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where sin y ¼ r=r, r ¼ x2 þ y2 , c is the speed of light in vacuum, P(t) is complex analytical signal representation of initial pulse A(t) Z 1 2 PðtÞ ¼ pffiffiffiffiffiffi AðoÞeiot do, (8) 2p 0 where A(o) being Fourier transform of A(t), i.e., Z þ1 1 AðoÞ ¼ pffiffiffiffiffiffi AðtÞ expðiotÞdt. 2p 1
(9)
^ oÞ ¼ 2HðoÞ il cos y AðoÞeikr Eðr; r þ il lkr sin2 y exp 2ðr þ ilÞ
ð10Þ
with H(o) being the Heaviside step function, i.e., ( 1 o40; HðoÞ ¼ (11) 0 op0: Suppose that real form of chirped Gaussian pulse [13] at r ¼ 0 is 2 t (12) AðtÞ ¼ exp 2 cosðoc t þ Ct2 Þ, T where oc is the carrier frequency, Tc is the corresponding oscillating period, T is the pulse duration, C is chirp parameter, and m ¼ T=T c is the number of oscillation within the pulse duration. On substituting Eq. (12) into Eqs. (9), (8) and (7), straightforward integral calculations lead to ! il cos y t00 2 exp 2 Eðr; tÞ ¼ r þ il T ( ! t00 2 exp iC 2 expðioc t00 Þ T " !# pffiffiffiffiffiffiffiffiffiffiffiffiffiffi oc T 1 iC 00 t þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ erf i T 2 1 iC ! t00 2 þ exp iC 2 expðioc t00 Þ T pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1 þ iC 00 oc T t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ erf i , ð13Þ T 2 1 þ iC where t00 ¼ t0 þ ilr sin2 y=½2cðr þ ilÞ, t0 ¼ t r=c is local time, erf(d) being error function. It is readily shown that Eq. (13) is consistent with Eq. (24) in Ref. [11] when the chirp parameter vanishes. Eq. (13) is the basic analytical formula obtained in present paper, which is not only applicable to Fresnel and Fraunhofer diffraction region, but also to large diffraction angle. On substituting Eq. (12) into Eqs. (9) and (10), the Fourier Spectrum for ultrashort chirped pulsed Gaussian beams yields ^ oÞ Eðr;
1 il cos y lkr sin2 y ¼ pffiffiffi HðoÞ expðikrÞ exp 2ðr þ ilÞ 2 r þ il 2 2 T T ðo þ oc Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 4ð1 iCÞ 1 iC 2 T T ðo oc Þ2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp . 4ð1 þ iCÞ 1 þ iC
ð14Þ
ARTICLE IN PRESS Q. Zou, B. Lu¨ / Optik 117 (2006) 259–264
From Eq. (14) the power spectrum of ultrashort chirped pulsed Gaussian beams reads as ^ oÞj2 jEðr;
T 2 l 2 cos2 y olr2 sin2 y 2 1=2 Þ exp ð1 þ C 2ðr2 þ l 2 Þ cðr2 þ l 2 Þ T 2 ðo oc Þ2 exp 2ð1 þ C 2 Þ T 2 ðo þ oc Þ2 þ exp 2ð1 þ C 2 Þ T 2 ðo2 þ o2c Þ þ 2 exp 2ð1 þ C 2 Þ 2 2 T ðo þ o2c ÞC f , cos 2ð1 þ C 2 Þ
¼
ð15Þ
ð16Þ
where ‘‘Re’’ denotes the real part, u1 ¼
lr2 ð1 þ C 2 Þsin2 y T 2 ðr2 þ l 2 Þoc c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2ð1 þ C 2 Þðr2 þ l 2 ÞTc
(16a)
u2 ¼
lr2 ð1 þ C 2 Þsin2 y þ T 2 ðr2 þ l 2 Þoc c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2ð1 þ C 2 Þðr2 þ l 2 ÞTc
(16b)
u3 ¼
ð1 þ C 2 Þ1=4 lr2 sin2 y pffiffiffi exp½if=2, 2ðr2 þ l 2 ÞTc
(16c)
F ¼ Co2c T 2 =2ð1 þ C 2 Þ f=2.
If chirp parameter vanishes (C ¼ 0), from Eq. (16), we obtain pffiffiffiffiffiffi 2 T 2 o2c 2pTl cos2 y exp I 0 ðrÞ ¼ 2 4ðr2 þ l 2 Þ
2 ½1 erf ðu4 Þ exp u4
þ ½1 erf ðu5 Þ exp u25
ð18Þ þ 2½1 erf ðu6 Þ exp u26 , where
where f ¼ tan1 C. It is easily shown that the power spectrum of ultrashort chirped pulsed Gaussian beams is irrelevant to the sign of the chirp parameter. According to Eq. (15) and Parseval’s theorem, the spatial intensity of ultrashort chirped pulsed Gaussian beams yields pffiffiffiffiffiffi 2 2pTl cos2 y T 2 o2c IðrÞ ¼ exp 2ð1 þ C 2 Þ 4ðr2 þ l 2 Þ ½1 erf ðu1 Þ expðu21 Þ þ ½1 erf ðu2 Þ expðu22 Þ þ 2ð1 þ C 2 Þ1=4 Re½expðu23 Þ expðiFÞð1 erf ðu3 ÞÞ ,
261
(16d)
On-axis (y ¼ 01) intensity can reduce to pffiffiffiffiffiffi 2 2pTl I 0 ðzÞ ¼ 1 þ ð1 þ C 2 Þ1=4 2ðz2 þ l 2 Þ T 2 o2c CT 2 o2c f cos exp . ð17Þ 2ð1 þ C 2 Þ 2ð1 þ C 2 Þ 2 From Eq. (17), we see that the axial intensity is irrelevant to the sign of the chirp parameter. When the initial pulse duration contains several optical cycles, onaxis intensity approximately simplify pffiffiffiffifficould ffi I 00 ðzÞ 2pTl 2 =2ðz2 þ l 2 Þ, that is, axial intensity is irrelevant to the chirp parameter for few-cycle ultrashort chirped pulsed Gaussian beams.
u4 ¼
lr2 sin2 y T 2 ðr2 þ l 2 Þoc c pffiffiffi , 2ðr2 þ l 2 ÞTc
(18a)
u5 ¼
lr2 sin2 y þ T 2 ðr2 þ l 2 Þoc c pffiffiffi , 2ðr2 þ l 2 ÞTc
(18b)
lr2 sin2 y u6 ¼ pffiffiffi . 2ðr2 þ l 2 ÞTc
(18c)
When pulses are not chirped (C ¼ 0), on-axis intensity can be gained from Eq. (18) pffiffiffiffiffiffi I 0 ðzÞ ¼ 2pTl 2 =ð2z2 þ 2l 2 Þ, (19) From eq. (19) it is readily seen that the axial intensity for few-cycle pulsed beams is the same as that of pulsed beams without chirp.
3. Numerical calculation and analysis Numerical calculations were carried out using Eqs. (13), (15) and (16) to quantitatively illustrate the effect of the chirp parameter on the spatiotemporal and spectral properties for ultrashort chirped pulsed Gaussian beams. In the following calculations we take oc ¼ 2:36 fs1 , l ¼ 10 mm, and r ¼ 5l. Fig. 1 gives the temporal pulse forms of real electric field Re[E(r, t)] for the chirp parameter C ¼ 5, and m ¼ 2. From Fig. 1 we see that on-axis temporal pulse is positively chirped [7,14], and the trailing edge of the offaxis positively chirped pulse oscillation. However, as compared with pulsed beams (C ¼ 0) the on-axis temporal pulse form is chirped [11]. Fig. 2 shows temporal pulse shapes of the ultrashort chirped pulsed Gaussian beams for different diffraction angles y ¼ 01 and 1.01. From Fig. 2 we see that the on-axis pulse shapes are distorted, the trailing edge and the leading edge of the on-axis chirped pulse become asymmetric, and the trailing edge of the off-axis positively chirped pulse shape develops oscillatory tail, while the leading edge of the off-axis negatively chirped pulse shape develops oscillatory tail. By Eq. (15), as the chirp parameter and diffraction angle are smaller, the power spectrum of ultrashort chirped pulsed Gaussian beams could be approximately
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×10-1
4.5
θ=0°
1.0 0.5 0 − 40
−30
−20
−10
0
10
20
30
×10-2
4.0
40
− 0.5
2
real (E) (arbitrary units)
1.5
|E(r,t)| (arbitrary units)
2.0
−1.0
C=−6
3.0 2.5 2.0 1.5 1.0 0.5
−1.5
0 −20
−2.0
−12
local time t´ (fs) 6
(a)
×10-3
2.5
2
−30
−20
−10
0
10
20
30
40
−2 −4 −6
|E(r,t)|2 (arbitrary units)
4
0 − 40
−4
4
12
20
local time t´ (fs)
θ=1.0° real (E) (arbitrary units)
C=6
θ=0°
3.5
θ=1.0°
C=6
2.0 C=−6
1.5 1.0 0.5 0 −20
local time t´ (fs) (b)
(a)
×10-4
−12
−4
4
12
20
local time t´ (fs) (b)
Fig. 1. The temporal pulse forms of ultrashort chirped pulsed Gaussian beams for chirp parameter C ¼ 5. (a) y ¼ 01, (b) y ¼ 1:01.
Fig. 2. The temporal pulse shapes of ultrashort chirped pulsed Gaussian beams for chirp parameters C ¼ 6 and 6. (a) y ¼ 01, (b) y ¼ 1:01.
reduced to T 2 l 2 cos2 y ð1 þ C 2 Þ1=2 2ðr2 þ l 2 Þ olr2 sin2 y T 2 ðo oc Þ2 exp exp . ð20Þ 2ð1 þ C 2 Þ cðr2 þ l 2 Þ ^ oÞj2 =qojx¼y¼0 ¼ 0 the By using Eq. (20) and qjEðr; position of maximum power spectrum is determined by
^ oÞj2 jEðr;
oc o ¼ ð1 þ C 2 Þlr2 sin2 y=½T 2 cðr2 þ l 2 Þ.
(21)
From Eq. (21) we see that on-axis power spectra have no shift, spectral redshift increases with increasing diffraction angle, and the spectral redshift increases by C2 times due to the chirp parameter. Fig. 3 illustrates the ^ oÞj2 =jEðr; ^ oÞj2 of normalized power spectra jEðr; max chirped pulsed Gaussian beams at different diffraction angles y ¼ 01, 0.51, and 1.01 for (a) C ¼ 0, (b) C ¼ 1, ^ oÞj2 is given by Eq. (15). As can be seen, the where jEðr; spectra are redshifted for ya01, the spectral redshift increases with increasing diffraction angle y, and the spectral redshift increases rapidly due to the chirp parameter. For example, when C ¼ 0, the spectral redshift (oc o) is 0, 0.06, 0.34 and 0.88 fs1 for y ¼ 01, 0.21, 0.51 and 0.81, respectively. When C ¼ 1, the
spectral redshift is 0, 0.11, 0.69 and 1.77 fs1 for y ¼ 01, 0.21, 0.51 and 0.81, respectively. When C ¼ 2, the spectral redshift is 0, 0.28, 1.72 and 2.35 fs1 for y ¼ 01, 0.21, 0.51 and 0.81, respectively. These calculation results agree very well with results calculated with Eq. (21). From Eq. (15) on-axis power spectrum approximately reduces to a Gaussian function when chirp parameter is small. From Eq. (20), the spectral half-bandwidth (at 1/e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi intensity point) is given by 2ð1 þ C 2 Þ=T, hence, onaxis spectral bandwidth increases by (1+C2)1/2 times due to the chirp parameter [8]. Fig. 4 gives normalized on-axis power spectra of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 0, 1, 2 and 4. As can be seen, on-axis spectra almost have no shift, and on-axis spectral distributions remain Gaussian distribution when the chirp parameter is smaller, but the on-axis spectral bandwidth increases with increasing chirp parameter. However, on-axis spectral distributions are no longer Gaussian distribution with increasing chirp parameter. Because the power spectrum of ultrashort chirped pulsed Gaussian beams is irrelevant to the sign of the
ARTICLE IN PRESS Q. Zou, B. Lu¨ / Optik 117 (2006) 259–264
263
1.2
1.2 θ = 0°
0.8
θ = 0.5°
0.6
θ = 1.0°
normalized power spectra
normalized power spectra
C=4 1.0
0.4 0.2 0
1.0 C=2 0.8 0.6 C=0
0.4
C=1
0.2 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
frequency (fs-1) (a)
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
frequency (fs-1)
1.0
θ = 0°
0.8
θ = 0.5°
Fig. 4. Normalized on-axis power spectra of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 0, 1, 2 and 4.
θ = 1.0°
0.6
1.2
0.4 0.2 0 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
frequency (fs-1)
(b)
Fig. 3. Normalized power spectra of ultrashort chirped pulsed Gaussian beams for different values of y ¼ 01, 0.51, and 1.01. (a) C ¼ 0, (b) C ¼ 1.
normalized intensity I(r)/Imax
normalized power spectra
1.2
C=0 C=2
1.0
m=1
C=− 4
0.8
C=4 0.6
C=6
0.4 0.2 0
0.08
0.04
0
0.12
0.16
0.20
(mm)
4. Concluding remarks In this paper, based on the Rayleigh diffraction integral and complex analytical signal representation,
Fig. 5. Normalized transversal intensity distributions of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 0, 2, 4, 4 and 6. 1.2 C=2
normalized intensity I(r)/Imax
chirp parameter, according to Parseval’s theorem, so the spatial intensity is irrelevant to the sign of the chirp parameter. Fig. 5 shows normalized transversal intensity (I(r)/Imax) distributions of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 0, 2, 4, 4 and 6, where Imax is the on-axis intensity for C ¼ 0. The calculation parameters we take are m ¼ 1 and z ¼ 10 mm. As shown in Fig. 5, the transversal intensity distributions nearly remain unchanged when the chirp parameter are small, on-axis intensity decreases and the transversal intensity distributions broaden with increasing chirp parameter, but the sign of the chirp parameter has no effect on the transversal intensity distribution. Fig. 6 illustrates normalized transversal intensity distributions of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 2, 4 and 8. The calculation parameters are m ¼ 3 and z ¼ 100 mm. As can be seen, the three curves of transversal intensity distributions are nearly overlapped, therefore, the increase of the chirp parameter almost has no effect on the transversal intensity distribution for few-cycle pulsed beams.
m =3
1.0 C=4
0.8 0.6 0.4
C=8
0.2 0 0
0.1
0.2
0.3
0.4
0.5 0.6 (mm)
0.7
0.8
0.9
1.0
Fig. 6. Normalized transversal intensity distributions of ultrashort chirped pulsed Gaussian beams for different values of C ¼ 2, 4 and 8.
the free-space analytical propagation equation and its Fourier spectrum for ultrashort chirped pulsed Gaussian beams with constant diffraction length are derived. The effect of the chirp parameter on the spatiotemporal
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and spectral properties is illustrated with analytical formulas and numerical calculation results. It is shown that the trailing edge and leading edge of the on-axis pulse shape become asymmetric, and the trailing edge of the off-axis positive chirped pulse develops oscillatory tail, while the leading edge of the off-axis negative chirped pulse develops oscillatory tail. The off-axis spectral redshift increases by about C2 times under the conditions that the chirp parameter and diffraction angle are small. The axial spectral bandwidth increases by (1+C2)1/2 times due to the chirp parameter. For single-cycle pulse, the transversal intensity distribution is affected by the increase of the chirp parameter, but almost not affected for few-cycle pulse. Moreover, the sign of the chirp parameter has no effect on the spectral distribution and spatial intensity distribution.
Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 10574097.
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