- Email: [email protected]
Chirp-dependent spectral distribution for few-cycle pulses propagating through nano-semiconductor devices
Accepted Manuscript Chirp-dependent spectral distribution for few-cycle pulses propagating through nano-semiconductor devices
Chaojin Zhang, Chengpu Liu
PII: DOI: Reference:
S0375-9601(16)30501-1 http://dx.doi.org/10.1016/j.physleta.2016.07.062 PLA 23989
To appear in:
Physics Letters A
Received date: Revised date: Accepted date:
7 March 2016 26 July 2016 26 July 2016
Please cite this article in press as: C. Zhang, C. Liu, Chirp-dependent spectral distribution for few-cycle pulses propagating through nano-semiconductor devices, Phys. Lett. A (2016), http://dx.doi.org/10.1016/j.physleta.2016.07.062
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights • The propagation of an initially chirped incident few-cycle pulse through an ensemble of quantum wells. • The distribution characteristic of the transmitted spectrum sensitively depending on the incident laser parameters. • The intensity of high-frequency spectral components enhancing obviously due to the nonlinear propagation effects.
Chirp-dependent spectral distribution for few-cycle pulses propagating through nano-semiconductor devices Chaojin Zhang1, and Chengpu Liu2,* 1
School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, China 2
State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
*Electronic address: [email protected]
1
Abstract:
The propagation of an initially chirped incident few-cycle pulse through
an ensemble of quantum wells is numerically investigated. It is found that the distribution characteristic of the transmitted spectrum sensitively depends on the incident laser parameters, especially its positive or negative chirp property. As for the incident pulse with a positive initial chirp, beyond the obvious spectral blue-shift, the transmitted spectral distribution is discrete. In contrast, as for a negative initial chirp, the spectral distribution is continuous instead. In addition, the insensitivity of chirp-dependent spectral distribution to medium symmetry character is also tested and the intensity of high-frequency spectral components enhances obviously due to the nonlinear propagation effects.
PACS number(s): 42.65.Re, 42.50.Md
2
With the rapid development of ultrafast laser technology, the laser pulse has already been compressed to a few cycles [1], which is widely used in the field of extreme nonlinear optics [2], where lots of unexpected and new phenomena are disclosed [3-9], such as carrier-wave Rabi flopping [4, 5], third-harmonic generation in the disguise of second-harmonic generation [6], and so on. With a moderate intense laser as a driver and subsequent propagation in many kinds of media, soliton formation [7, 8], self-reflection [9], and transmittance [10-12] are investigated. As for the transmitted spectrum, its continuous or discrete feature can be switched by adjusting the target medium’s symmetry degree [10], the time-delay between two color fields [11], or changing the field’s spatial homogeneity [12]. Based on the fact that spectral distribution sensitively depends on the carrier-envelope phase (CEP), one can in turn obtain the CEP information of the input laser pulse [13-16]. As is well known, when the laser pulse propagates in a nonlinear medium, a negative intrinsic chirp is induced which is indicated by the occurrence of an obvious frequency shift [17-20]. One may question that, if a chirp is initially added to the incident pulse in order to compensate or enhance the negative intrinsic chirp after a certain distance propagation, is the spectral distribution property still continuous or discrete? To our knowledge, no detailed investigation is made, specifically when slowly envelope and rotating wave approximations are not taken into account for the description of the few-cycle pulse propagation. In this letter, we present a numerical investigation and the results demonstrate that the red- or blue-shift still exists. Moreover, the more important is that the continuous or discrete feature of the spectral
3
distribution can be switched via changing the sign of the initial chirp. This switch route is simple and convenient if compared with other ways [10-12]. In addition, the higher spectral components enhance obviously after suitable propagation distance, which may be important for the design and application of sub-wavelength optical devices. In the following, we will investigate the chirped pulse interacting with a quantum well ensemble. The quantum well is chosen because its well-width is easily adjusted and then the electronic or optical property correspondingly changes, which inversely affects the above extreme nonlinear optical phenomena. The type structure of the quantum well in this letter has been described in our previous work [10]. Consider a hyperbolic secant few-cycle pulse propagating along the z-direction and polarized along the x direction. The full-wave Maxwell-Bloch equations modelling the interaction between the quantum wells and the initially chirped pulse are solved by employing Yee's finite-difference time-domain (FDTD) discretization scheme [21] for Maxwell equation and the predictor–corrector algorithm or four-order Runge-Kutta method for Bloch equation describing the medium response [22-26]. The incident pulse first propagates in the free space; then in the medium front interface, it is partially reflected backwards and mostly penetrates into the medium; after penetration, the remained pulse further propagates for a certain distance, and finally enters the free space again. The incident chirped pulse at initial time, can be generally written as [27, 28] ª
Ex (t = 0, z ) = E0 sec h «1.76 ¬
ª (z − z 0 ) (z − z 0 ) º § z − z0 · º +β tanh ¨ » cos «ω0 ¸» cτ 0 ¼ c © cσ ¹ ¼ ¬ 4
(1)
In contrast the general Fourier-limited pulse adopted for Maxwell-Bloch models, two coefficients șand V are adjusted to introduce the initial chirp. E0 is the electric field amplitude, Ȧ0 is the central angular frequency of laser pulse, and τ 0 is the pulse duration (full width at half maximum, FWHM) of the pulse intensity envelope without the chirp. In addition, the choice of z0 is just to ensure that the pulse penetrates negligibly into the medium at t=0 [23], here z0=15 ȝm, and the length of quantum well ensemble is L =120 um. In the following demonstration, laser parameters are set as follows (unless otherwise indicated): Ȧ0=0.4 fs-1, τ 0 =20fs, laser intensity I=3.1h010 W/cm2, and the chirp control parameters £=6.28 and V=48 fs [27]. In addition, the quantum well width is adjusted to meet the two-photon resonance condition (i. e., the transition frequency Ȧ12 =2Ȧ0). First, the chirp influence on the transmitted spectrum is investigated. The result is shown in Fig. 1. When the chirp parameter is £=0, the pulse has no initial chirp (Figs. 1(a) and 1(d)), the odd-order harmonic peaks are clearly shown and their distribution is discrete in the transmitted spectrum. The reason can be interpreted by the corresponding laser waveform (solid line) and population inversion (dashed line) near the front face. The population inversion occurs between the lower and upper states. And the populations are assumed initialized on the lower state which means the population inversion is negative [4, 5, 23]. As shown in Fig. 1(a), there are the strong interferences between the carriers (dashed line) which will induce the pulse shaping and discrete odd-order harmonic generation. This interpretation is also consistent with the previous works [8, 10, 11].
5
If a positive chirp is added to the incident pulse (for example £=6.28) in order to compensate the negative intrinsic chirp produced when the pulse propagates [17, 28], a significant blue shift for each harmonic peak clearly occurs (see Fig. 1(e)). At the same time, the laser waveform changes as well (Fig. 1(b)) and a large positive chirp brings about the increase of carrier oscillating number cycles under the laser envelope [20]. Moreover, the spectral intensity decreases obviously, as a result of population inversion decreasing nearly by half than that without chirp (dashed lines in Figs. 1(a) and 1(b)).
30
40
50 Time(fs)
60
-0.4
40
50 60 Time(fs)
-1 80
70
0.4
-0.4
-0.8 20
5
ω/ω0
6
7
8
9
Transmitted spectra (arb. units)
Transmitted spectra (arb. units) 4
0
30
40
3
4
50 60 Time(fs)
-1 80
70
(f)
0.001
0.000
3
(c)
0.0
(e)
0.001
1
β=-6.28
0.002
(d) Transmitted spectra (arb. units)
30
0.002
0.002
0.000 2
0
0.0
-0.8 20
-1 80
70
0.4
Electric field (arb. units)
-0.4
-0.8 20
0.8
The population inversion
0
0.0
Electric field (arb. units)
(a)
0.4
1
β=6.28
(b) The population inversion
Electric field (arb. units)
β=0
The population inversion
0.8
1
0.8
2
3
4
5
ω/ω0
6
7
8
9
0.001
0.000 2
5
6
ω/ω0
7
8
9
Fig. 1. (Color online) (a, b, c) The laser waveforms (solid line) and population inversions (dashed line) near the front face. (d, e, f) the corresponding transmitted spectral distributions with the indicated initial chirp£ parameters after a certain distance pulse propagation z=120 ȝm.
In contrast, when the incident pulse is with a negative chirp (such as £= -6.28, the same absolute value but opposite property) to enhance the negative intrinsic chirp,
6
this leads to the obvious decrease of carrier oscillating number cycles (Fig. 1(c)). Interestingly, due to the weak interference between the carriers (see dashed line in Fig. 1(c)), the spectral distribution becomes quasi-continuous with harmonic peaks being indistinguishable as shown in Fig. 1(f), which is consistent with the previous work [10, 11]. Moreover, the intensity of high-frequency component decreases by half than that without initial chirp, as a result of population inversion decreasing a half (dashed line in Fig. 1(c)). From the initial chirp expression (ijchirp=ß*tanh(t/ı)) in Eq. (1), we can understand the reason why the red- or blue-shift is dependent on the chirp property when the parameter ß changes. Moreover, due to the introduction of this special kind of chirp in Eq. (1) with the pulse duration fixing, the laser oscillation number is increased for initial positive chirp, which means the quantum trajectory number increasing also. In contrast, with the negative chirp initially been introduced, the laser oscillation number obviously reduces and the corresponding quantum trajectory number for high-frequency spectral component is also expected to be reduced, which is indirectly indicated in the population inversion modulation as shown in Fig. 1(c). Based on this consideration, it is not difficult to understand why under negative chirp, the sharpness of harmonics is reduced, i.e. the high-frequency spectral component seems smoother. Second, from the point of practical application, one choice of an optimal chirp value is preferred. The same problem but under different chirp control parameters is investigated. When V is fixed (see Eq. (1.1)), only the chirp parameter £ changes.
7
As for a positive initial chirp, with the increment of £, the blue shift enhances but the spectral intensity decreases. In contrast, as for a negative chirp, beyond spectral intensity decreasing, the spectral distribution becomes more and more continuous (Fig. 2(b)). If the parameter£ (absolute value) is fixed (£= ±6.28) is fixed, only V changes. With the increment of V, for an initial positive chirp (Fig. 2(c)), the red shift enhances and spectral intensity also does. In contrast, for an initial negative chirp (Fig. 2(d)), with the increment of V, the spectral distribution becomes more and more discrete. 0.002
0.002
β=6.28 β=9.28 β=12.28
2
3
4
5
0.002
ω/ω0
6
7
8
3
4
5
0.002
ω/ω0
6
7
β=6.28
8
9
σ=28 σ=48 σ=78
(d)
0.001
0.000 2
σ=48 0.001
0.000 2
9
σ=28 σ=48 σ=78
(c) Transmitted spectra (arb. units)
Transmitted spectra (arb. units)
σ=48 0.001
0.000
β=-6.28 β=-9.28 β=-12.28
(b)
Transmitted spectra (arb. units)
Transmitted spectra (arb. units)
(a)
β=-6.28
0.001
3
4
5
ω/ω0
6
7
8
0.000 2
9
3
4
5
ω/ω0
6
7
8
9
Fig. 2. (Color online) The transmitted spectral distribution with (a), (b) V fixed but different£ and (c), (d) £ fixed but different V. The other parameters are same as those in Fig. 1.
8
0.001
3
4
5
ω/ω0
6
7
8
0.000 2
9
1
1
-1 400
The population inversion Electric field (arb. units)
450
Time(fs)
500
5
ω/ω0
6
7
-1 550
8
9
0.000 2
1
0
-1 400
Time(fs)
500
4
5
ω/ω0
6
-1 550
7
8
9
1
L=120um β=-6.28
(f)
0
450
3
1
L=120um β=6.28
(e)
0
0
4
1
L=120um β=0
(d)
3
0.001
The population inversion Electric field (arb. units)
0.000 2
24um 48um 120um β=-6.28
(c) Transmitted spectra (arb. units)
Transmitted spectra (arb. units)
Transmitted spectra (arb. units)
24um 48um 120um β=6.28
(b)
0.001
Electric field (arb. units)
0.002
0.002
24um 48um 120um β=0
(a)
The population inversion
0.002
0
-1 400
0
450
Time(fs)
500
-1 550
Fig. 3. (Color online) (a, b, c) The transmitted spectral distributions under different chirp values and simultaneously after different propagation distances. (d, e, f) The laser waveforms (solid line) and population inversions (dashed line) at L=120 um. The other parameters are same as those in Fig.
1.
The above demonstration indicates that one can easily obtain a continuous or discrete spectral distribution just via the adjustment to the initial chirp, not necessarily referring to breaking the inversion symmetry [10, 12] or using two-color or multi-color laser field [11]. This method here is simple and convenient for practical implementation. The Maxwell-Bloch model here is convenient for investigating the influence of propagation effects on the transmitted spectra. For input laser fields without chirp as shown in Fig. 3(a), the intensity of the discrete high-frequency spectral components will enhance significantly with the propagation lengths increasing. This enhancement effects can be further described by the laser waveform and population at L=120um as shown in Fig. 3(d). The laser waveform is reshaped 9
and its peak enhances due to the propagation effects, which induces the generating and enhancing effects of high-frequency spectral component. Similarly, this enhancement effects are also found when input laser fields with positive chirp as shown in Fig. 3(b) and Fig. 3(e). But seen from Fig. 3(c) and Fig. 3(f), besides the enhancement effects can also be found, the distributions of transmitted spectra become from discrete to continuous feature with the propagation lengths increasing. Importantly, this distribution changes are based on odd-order harmonic generations, which is unnecessary for even-order harmonic generation occurring and then are different reasons with that in the related Refs. [10-13]. (a)
β=0
Transmitted spectra (arb. units)
0.002
0.001
0.000 2
3
4
5
ω/ω0
6
0.002
8
9
0.002
(c)
β=-6.28
Transmitted spectra (arb. units)
Transmitted spectra (arb. units)
(b)
0.001
0.000 2
7
3
4
5
ω/ω0
6
7
8
0.001
0.000 2
9
β=6.28
3
4
5
ω/ω0
6
7
8
9
Fig. 4. (a,b,c) the corresponding transmitted spectral distributions for the different indicated initial chirp£ parameters with the quantum well system being asymmetric.
10
As for the spectral distribution, after a detail investigation to laser chirp parameter and nonlinear propagation effects related to medium length, one could consider the influence of the medium symmetry characteristic. In fact, such a chirp-dependent phenomenon is not only limited to the symmetric quantum well system. When the quantum well is asymmetric [10], as shown in Fig. 4, the spectral distribution’s chirp-dependence is kept, expect for the occurrence of even-order harmonics (such as in Fig. 4(a)) effects and the corresponding spectra distribution features also occur.
In conclusion, we have investigated the chirped incident laser pulses propagating through an ensemble of quantum wells. The results show that the transmitted spectral distribution feature sensitively depends on the initial chirp and the compensation with the intrinsic chirp resulting from the nonlinear propagation effects. If the incident pulse being with a positive chirp, the spectral blue shifts and discrete distribution feature can be found. In contrast, when the incident pulse has negative chirp and simultaneously quantum well ensemble is symmetric, the transmitted spectral distribution is continuous instead and only odd order harmonic peaks exist, which are related to the weak carrier wave interference effect. In addition, the intensity of high-frequency spectral components enhances obviously due to the nonlinear propagation effects and non-sensitivity of chirp-dependent spectral distribution to medium symmetry character was also tested.
11
Acknowledgements The work is supported by National Natural Science Foundation of China (Grant No. 11374318) and Natural Science Foundation of Jiangsu (BK20161159) and Anhui Province (1508085QF140). C.J.Z. gratefully acknowledges the support of open fund of the state key laboratory of high field laser physics of SIOM and overseas training of excellent young teachers of universities in Jiangsu Province. C.P.L. is appreciated to the supports from the 100-talents Project of Chinese Academy of Sciences and Department of Human Resources and Social Security of China.
References: [1] T. Brabec and F. Krausz, “Intense few-cycle laser field: frontiers of nonlinear optics,” Rev. Mod. Phys. 72: 545-591 (2000). [2] M. Wegner “Extreme Nonlinear Optics” [M]. Heidelberg: Springer-Verlag, 2005. [3] D. J. Frantzeskakis, H. Leblond, and D. Mihalache, “Nonlinear optics of intense few-cycle pulses: An overview of recent theoretical and experimental developments,” Rom. J. Phys. 59: 767-784 (2014). [4] S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81: 3363-3366 (1998). [5] O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, “Signatures of carrier-wave Rabi flopping in GaAs,” Phys. Rev. Lett. 87: 057401 (2001). [6] T. Tritschler, O. D. Mücke, M. Wegener, U. Morgner, and F. X. Kärtner, “Evidence 12
for third-harmonic generation in disguise of second-harmonic generation in extreme nonlinear optics,” Phys. Rev. Lett 90: 217404 (2003). [7] H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523: 61-126 (2013) [8] H. Leblond, H. Triki, and D. Mihalache, “Theoretical studies of ultrashort-soliton propagation in nonlinear optical media from a general quantum model,” Rom. Rep. Phys. 65: 925-942 (2013). [9] Y. Chen, X.L. Feng, and C P Liu. “Doppler effects in the propagation of a few-cycle pulse through a dense medium,” Opt. Express 23:17046-17055 (2015). [10] C. Zhang, X. Song, W. Yang, and Z. Xu, “Carrier-envelope phase control of carrier-wave Rabi flopping in asymmetric semiparabolic quantum well,” Opt. Express 16: 1487-1496 (2008). [11] X. Song, S. Gong, S. Jin, and Z. Xu, “Formation of higher spectral components in a two-level medium driven by two-color ultrashort laser pulses,” Phys. Rev. A 69: 015801 (2004). [12] C. Zhang, C.P. Liu, and Z. Xu, “Control of higher spectral components by spatially inhomogeneous fields in quantum wells,” Phys. Rev. A 88: 035805 (2013). [13] C. Zhang, W. Yang, X. Song, and Z. Xu, “Control of harmonic generation in a quantum well with two-colour few-cycle pulses,” J. Phys. B: At. Mol. Opt. Phys. 42: 125604 (2009). [14] N. Cui, Y. Xiang, Y. Niu, S. Gong, “Coherent control of terahertz harmonic generation by a chirped few-cycle pulse in a quantum well,” New J. Phys. 12: 013009 (2010). 13
[15] C. Zhang, W. Yang, X. Song, and Z. Xu, “Carrier-envelope phase dependence of the spectra of reflected few-cycle laser pulses in the presence of a static electric field,” Phys. Rev. A 79: 043823 (2009). [16] C. Zhang and C. P. Liu, “Control of the carrier-envelope-phase effect in the transmitted spectra in quantum wells via terahertz waves,” Euro. Phys. J. D 69:134 (2015) [17] Z. Chang, A. Rundquist, H. Wang, I. Christov, H. C. Kapteyn, and M. M. Murnane, “Temporal phase control of soft-x-ray harmonic emission,” Phys. Rev. A 58: R30-R33 (1998). [18] D. G. Lee, J.H. Kim, K. H. Hong, and C. H. Nam, “ Coherent control of high-order harmonics with chirped femtosecond laser pulses”, Phys. Rev. Lett. 87: 243902. [19] M. Klaus and J. K. Shaw, “Influence of pulse shape and frequency chirp on stability of optical solitons,” Opt. Commun. 197: 491-500 (2001). [20] X. Song, S. Gong, W. Yang, S. Jin, X. Feng, and Z. Xu, “Coherent control of spectra effects with chirped femtosecond laser pulse,” Opt. Commun. 236: 151-157 (2004). [21] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14: 302-307 (1996). [22] R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52: 3082-3094 (1995). [23] V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full
14
Maxwell-Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83: 544-547 (1999). [24] J. Xiao, Z. Wang, and Z. Xu, “Area evolution of a few-cycle pulse laser in a two-level-atom medium,” Phys. Rev. A 65: 031402(R) (2002). [25] C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92: 047402 (2004). [26] Y. Chen, X. Feng, Z. Xu, and C. Liu, “Abnormal enhancement against interference inhibition for few-cycle pulses propagating in dense media,” Opt. Commun. 364:123-128 (2016). [27] Y. Niu, Y. Xiang, Y. Qi, and S. Gong, “Single attosecond pulse generation from multicycle nonlinear chirped pulses,” Phys. Rev. A 80: 063818 (2009). [28] P. Li, X. Zhou, G. Wang, and Z. Zhao, “Isolated sub-30-as pulse generation of an He+ ion by an intense few cycle chirped laser and its high-order harmonic pulses,” Phys. Rev. A 80: 053825 (2009). [29]
J. H. Kim, D. G. Lee, H. J. Shin, and C. H. Nam, “Wigner time-frequency distribution of high-order harmonics”, Phys. Rev. A 63: 063403 (2001).
[30] M. Oberthaler and R. A. HĘpfel, “Special narrowing of ultrashort laser pulses by self-phase modulation in optical fibers,” Appl. Phys. Lett. 63: 1017-1019 (1997).
15
Chaojin Zhang, Chengpu Liu
PII: DOI: Reference:
S0375-9601(16)30501-1 http://dx.doi.org/10.1016/j.physleta.2016.07.062 PLA 23989
To appear in:
Physics Letters A
Received date: Revised date: Accepted date:
7 March 2016 26 July 2016 26 July 2016
Please cite this article in press as: C. Zhang, C. Liu, Chirp-dependent spectral distribution for few-cycle pulses propagating through nano-semiconductor devices, Phys. Lett. A (2016), http://dx.doi.org/10.1016/j.physleta.2016.07.062
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights • The propagation of an initially chirped incident few-cycle pulse through an ensemble of quantum wells. • The distribution characteristic of the transmitted spectrum sensitively depending on the incident laser parameters. • The intensity of high-frequency spectral components enhancing obviously due to the nonlinear propagation effects.
Chirp-dependent spectral distribution for few-cycle pulses propagating through nano-semiconductor devices Chaojin Zhang1, and Chengpu Liu2,* 1
School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, China 2
State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
*Electronic address: [email protected]
1
Abstract:
The propagation of an initially chirped incident few-cycle pulse through
an ensemble of quantum wells is numerically investigated. It is found that the distribution characteristic of the transmitted spectrum sensitively depends on the incident laser parameters, especially its positive or negative chirp property. As for the incident pulse with a positive initial chirp, beyond the obvious spectral blue-shift, the transmitted spectral distribution is discrete. In contrast, as for a negative initial chirp, the spectral distribution is continuous instead. In addition, the insensitivity of chirp-dependent spectral distribution to medium symmetry character is also tested and the intensity of high-frequency spectral components enhances obviously due to the nonlinear propagation effects.
PACS number(s): 42.65.Re, 42.50.Md
2
With the rapid development of ultrafast laser technology, the laser pulse has already been compressed to a few cycles [1], which is widely used in the field of extreme nonlinear optics [2], where lots of unexpected and new phenomena are disclosed [3-9], such as carrier-wave Rabi flopping [4, 5], third-harmonic generation in the disguise of second-harmonic generation [6], and so on. With a moderate intense laser as a driver and subsequent propagation in many kinds of media, soliton formation [7, 8], self-reflection [9], and transmittance [10-12] are investigated. As for the transmitted spectrum, its continuous or discrete feature can be switched by adjusting the target medium’s symmetry degree [10], the time-delay between two color fields [11], or changing the field’s spatial homogeneity [12]. Based on the fact that spectral distribution sensitively depends on the carrier-envelope phase (CEP), one can in turn obtain the CEP information of the input laser pulse [13-16]. As is well known, when the laser pulse propagates in a nonlinear medium, a negative intrinsic chirp is induced which is indicated by the occurrence of an obvious frequency shift [17-20]. One may question that, if a chirp is initially added to the incident pulse in order to compensate or enhance the negative intrinsic chirp after a certain distance propagation, is the spectral distribution property still continuous or discrete? To our knowledge, no detailed investigation is made, specifically when slowly envelope and rotating wave approximations are not taken into account for the description of the few-cycle pulse propagation. In this letter, we present a numerical investigation and the results demonstrate that the red- or blue-shift still exists. Moreover, the more important is that the continuous or discrete feature of the spectral
3
distribution can be switched via changing the sign of the initial chirp. This switch route is simple and convenient if compared with other ways [10-12]. In addition, the higher spectral components enhance obviously after suitable propagation distance, which may be important for the design and application of sub-wavelength optical devices. In the following, we will investigate the chirped pulse interacting with a quantum well ensemble. The quantum well is chosen because its well-width is easily adjusted and then the electronic or optical property correspondingly changes, which inversely affects the above extreme nonlinear optical phenomena. The type structure of the quantum well in this letter has been described in our previous work [10]. Consider a hyperbolic secant few-cycle pulse propagating along the z-direction and polarized along the x direction. The full-wave Maxwell-Bloch equations modelling the interaction between the quantum wells and the initially chirped pulse are solved by employing Yee's finite-difference time-domain (FDTD) discretization scheme [21] for Maxwell equation and the predictor–corrector algorithm or four-order Runge-Kutta method for Bloch equation describing the medium response [22-26]. The incident pulse first propagates in the free space; then in the medium front interface, it is partially reflected backwards and mostly penetrates into the medium; after penetration, the remained pulse further propagates for a certain distance, and finally enters the free space again. The incident chirped pulse at initial time, can be generally written as [27, 28] ª
Ex (t = 0, z ) = E0 sec h «1.76 ¬
ª (z − z 0 ) (z − z 0 ) º § z − z0 · º +β tanh ¨ » cos «ω0 ¸» cτ 0 ¼ c © cσ ¹ ¼ ¬ 4
(1)
In contrast the general Fourier-limited pulse adopted for Maxwell-Bloch models, two coefficients șand V are adjusted to introduce the initial chirp. E0 is the electric field amplitude, Ȧ0 is the central angular frequency of laser pulse, and τ 0 is the pulse duration (full width at half maximum, FWHM) of the pulse intensity envelope without the chirp. In addition, the choice of z0 is just to ensure that the pulse penetrates negligibly into the medium at t=0 [23], here z0=15 ȝm, and the length of quantum well ensemble is L =120 um. In the following demonstration, laser parameters are set as follows (unless otherwise indicated): Ȧ0=0.4 fs-1, τ 0 =20fs, laser intensity I=3.1h010 W/cm2, and the chirp control parameters £=6.28 and V=48 fs [27]. In addition, the quantum well width is adjusted to meet the two-photon resonance condition (i. e., the transition frequency Ȧ12 =2Ȧ0). First, the chirp influence on the transmitted spectrum is investigated. The result is shown in Fig. 1. When the chirp parameter is £=0, the pulse has no initial chirp (Figs. 1(a) and 1(d)), the odd-order harmonic peaks are clearly shown and their distribution is discrete in the transmitted spectrum. The reason can be interpreted by the corresponding laser waveform (solid line) and population inversion (dashed line) near the front face. The population inversion occurs between the lower and upper states. And the populations are assumed initialized on the lower state which means the population inversion is negative [4, 5, 23]. As shown in Fig. 1(a), there are the strong interferences between the carriers (dashed line) which will induce the pulse shaping and discrete odd-order harmonic generation. This interpretation is also consistent with the previous works [8, 10, 11].
5
If a positive chirp is added to the incident pulse (for example £=6.28) in order to compensate the negative intrinsic chirp produced when the pulse propagates [17, 28], a significant blue shift for each harmonic peak clearly occurs (see Fig. 1(e)). At the same time, the laser waveform changes as well (Fig. 1(b)) and a large positive chirp brings about the increase of carrier oscillating number cycles under the laser envelope [20]. Moreover, the spectral intensity decreases obviously, as a result of population inversion decreasing nearly by half than that without chirp (dashed lines in Figs. 1(a) and 1(b)).
30
40
50 Time(fs)
60
-0.4
40
50 60 Time(fs)
-1 80
70
0.4
-0.4
-0.8 20
5
ω/ω0
6
7
8
9
Transmitted spectra (arb. units)
Transmitted spectra (arb. units) 4
0
30
40
3
4
50 60 Time(fs)
-1 80
70
(f)
0.001
0.000
3
(c)
0.0
(e)
0.001
1
β=-6.28
0.002
(d) Transmitted spectra (arb. units)
30
0.002
0.002
0.000 2
0
0.0
-0.8 20
-1 80
70
0.4
Electric field (arb. units)
-0.4
-0.8 20
0.8
The population inversion
0
0.0
Electric field (arb. units)
(a)
0.4
1
β=6.28
(b) The population inversion
Electric field (arb. units)
β=0
The population inversion
0.8
1
0.8
2
3
4
5
ω/ω0
6
7
8
9
0.001
0.000 2
5
6
ω/ω0
7
8
9
Fig. 1. (Color online) (a, b, c) The laser waveforms (solid line) and population inversions (dashed line) near the front face. (d, e, f) the corresponding transmitted spectral distributions with the indicated initial chirp£ parameters after a certain distance pulse propagation z=120 ȝm.
In contrast, when the incident pulse is with a negative chirp (such as £= -6.28, the same absolute value but opposite property) to enhance the negative intrinsic chirp,
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this leads to the obvious decrease of carrier oscillating number cycles (Fig. 1(c)). Interestingly, due to the weak interference between the carriers (see dashed line in Fig. 1(c)), the spectral distribution becomes quasi-continuous with harmonic peaks being indistinguishable as shown in Fig. 1(f), which is consistent with the previous work [10, 11]. Moreover, the intensity of high-frequency component decreases by half than that without initial chirp, as a result of population inversion decreasing a half (dashed line in Fig. 1(c)). From the initial chirp expression (ijchirp=ß*tanh(t/ı)) in Eq. (1), we can understand the reason why the red- or blue-shift is dependent on the chirp property when the parameter ß changes. Moreover, due to the introduction of this special kind of chirp in Eq. (1) with the pulse duration fixing, the laser oscillation number is increased for initial positive chirp, which means the quantum trajectory number increasing also. In contrast, with the negative chirp initially been introduced, the laser oscillation number obviously reduces and the corresponding quantum trajectory number for high-frequency spectral component is also expected to be reduced, which is indirectly indicated in the population inversion modulation as shown in Fig. 1(c). Based on this consideration, it is not difficult to understand why under negative chirp, the sharpness of harmonics is reduced, i.e. the high-frequency spectral component seems smoother. Second, from the point of practical application, one choice of an optimal chirp value is preferred. The same problem but under different chirp control parameters is investigated. When V is fixed (see Eq. (1.1)), only the chirp parameter £ changes.
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As for a positive initial chirp, with the increment of £, the blue shift enhances but the spectral intensity decreases. In contrast, as for a negative chirp, beyond spectral intensity decreasing, the spectral distribution becomes more and more continuous (Fig. 2(b)). If the parameter£ (absolute value) is fixed (£= ±6.28) is fixed, only V changes. With the increment of V, for an initial positive chirp (Fig. 2(c)), the red shift enhances and spectral intensity also does. In contrast, for an initial negative chirp (Fig. 2(d)), with the increment of V, the spectral distribution becomes more and more discrete. 0.002
0.002
β=6.28 β=9.28 β=12.28
2
3
4
5
0.002
ω/ω0
6
7
8
3
4
5
0.002
ω/ω0
6
7
β=6.28
8
9
σ=28 σ=48 σ=78
(d)
0.001
0.000 2
σ=48 0.001
0.000 2
9
σ=28 σ=48 σ=78
(c) Transmitted spectra (arb. units)
Transmitted spectra (arb. units)
σ=48 0.001
0.000
β=-6.28 β=-9.28 β=-12.28
(b)
Transmitted spectra (arb. units)
Transmitted spectra (arb. units)
(a)
β=-6.28
0.001
3
4
5
ω/ω0
6
7
8
0.000 2
9
3
4
5
ω/ω0
6
7
8
9
Fig. 2. (Color online) The transmitted spectral distribution with (a), (b) V fixed but different£ and (c), (d) £ fixed but different V. The other parameters are same as those in Fig. 1.
8
0.001
3
4
5
ω/ω0
6
7
8
0.000 2
9
1
1
-1 400
The population inversion Electric field (arb. units)
450
Time(fs)
500
5
ω/ω0
6
7
-1 550
8
9
0.000 2
1
0
-1 400
Time(fs)
500
4
5
ω/ω0
6
-1 550
7
8
9
1
L=120um β=-6.28
(f)
0
450
3
1
L=120um β=6.28
(e)
0
0
4
1
L=120um β=0
(d)
3
0.001
The population inversion Electric field (arb. units)
0.000 2
24um 48um 120um β=-6.28
(c) Transmitted spectra (arb. units)
Transmitted spectra (arb. units)
Transmitted spectra (arb. units)
24um 48um 120um β=6.28
(b)
0.001
Electric field (arb. units)
0.002
0.002
24um 48um 120um β=0
(a)
The population inversion
0.002
0
-1 400
0
450
Time(fs)
500
-1 550
Fig. 3. (Color online) (a, b, c) The transmitted spectral distributions under different chirp values and simultaneously after different propagation distances. (d, e, f) The laser waveforms (solid line) and population inversions (dashed line) at L=120 um. The other parameters are same as those in Fig.
1.
The above demonstration indicates that one can easily obtain a continuous or discrete spectral distribution just via the adjustment to the initial chirp, not necessarily referring to breaking the inversion symmetry [10, 12] or using two-color or multi-color laser field [11]. This method here is simple and convenient for practical implementation. The Maxwell-Bloch model here is convenient for investigating the influence of propagation effects on the transmitted spectra. For input laser fields without chirp as shown in Fig. 3(a), the intensity of the discrete high-frequency spectral components will enhance significantly with the propagation lengths increasing. This enhancement effects can be further described by the laser waveform and population at L=120um as shown in Fig. 3(d). The laser waveform is reshaped 9
and its peak enhances due to the propagation effects, which induces the generating and enhancing effects of high-frequency spectral component. Similarly, this enhancement effects are also found when input laser fields with positive chirp as shown in Fig. 3(b) and Fig. 3(e). But seen from Fig. 3(c) and Fig. 3(f), besides the enhancement effects can also be found, the distributions of transmitted spectra become from discrete to continuous feature with the propagation lengths increasing. Importantly, this distribution changes are based on odd-order harmonic generations, which is unnecessary for even-order harmonic generation occurring and then are different reasons with that in the related Refs. [10-13]. (a)
β=0
Transmitted spectra (arb. units)
0.002
0.001
0.000 2
3
4
5
ω/ω0
6
0.002
8
9
0.002
(c)
β=-6.28
Transmitted spectra (arb. units)
Transmitted spectra (arb. units)
(b)
0.001
0.000 2
7
3
4
5
ω/ω0
6
7
8
0.001
0.000 2
9
β=6.28
3
4
5
ω/ω0
6
7
8
9
Fig. 4. (a,b,c) the corresponding transmitted spectral distributions for the different indicated initial chirp£ parameters with the quantum well system being asymmetric.
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As for the spectral distribution, after a detail investigation to laser chirp parameter and nonlinear propagation effects related to medium length, one could consider the influence of the medium symmetry characteristic. In fact, such a chirp-dependent phenomenon is not only limited to the symmetric quantum well system. When the quantum well is asymmetric [10], as shown in Fig. 4, the spectral distribution’s chirp-dependence is kept, expect for the occurrence of even-order harmonics (such as in Fig. 4(a)) effects and the corresponding spectra distribution features also occur.
In conclusion, we have investigated the chirped incident laser pulses propagating through an ensemble of quantum wells. The results show that the transmitted spectral distribution feature sensitively depends on the initial chirp and the compensation with the intrinsic chirp resulting from the nonlinear propagation effects. If the incident pulse being with a positive chirp, the spectral blue shifts and discrete distribution feature can be found. In contrast, when the incident pulse has negative chirp and simultaneously quantum well ensemble is symmetric, the transmitted spectral distribution is continuous instead and only odd order harmonic peaks exist, which are related to the weak carrier wave interference effect. In addition, the intensity of high-frequency spectral components enhances obviously due to the nonlinear propagation effects and non-sensitivity of chirp-dependent spectral distribution to medium symmetry character was also tested.
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Acknowledgements The work is supported by National Natural Science Foundation of China (Grant No. 11374318) and Natural Science Foundation of Jiangsu (BK20161159) and Anhui Province (1508085QF140). C.J.Z. gratefully acknowledges the support of open fund of the state key laboratory of high field laser physics of SIOM and overseas training of excellent young teachers of universities in Jiangsu Province. C.P.L. is appreciated to the supports from the 100-talents Project of Chinese Academy of Sciences and Department of Human Resources and Social Security of China.
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