Phase relations between focused bichromatic laser pulses in terahertz wave generation from gas plasma

Optics Communications 284 (2011) 2206–2209

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Phase relations between focused bichromatic laser pulses in terahertz wave generation from gas plasma Virgilijus Vaičaitis ⁎, Valerijus Smilgevičius, Vygandas Jarutis Laser Research Center, Vilnius University, Saulėtekio 10, Vilnius LT-10223, Lithuania

a r t i c l e

i n f o

Article history: Received 16 September 2010 Received in revised form 17 December 2010 Accepted 18 December 2010 Available online 1 January 2011 Keywords: Terahertz Gas Plasma Bichromatic femtosecond laser pulses

a b s t r a c t Terahertz emission from air excited by the tightly focused fundamental and second harmonic pulses of femtosecond Ti:Sapphire laser has been analyzed. It has been found that the curved phase fronts of the pump waves cancel each other resulting in the flat phase front of the nonlinear polarization at terahertz frequency. Also, in addition to the phase terms obtained using plane-wave approximation, we have found the yet unreported phase term, which in most cases can be as large as π/2. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The generation of terahertz (THz) radiation in gaseous media by the focused bichromatic femtosecond laser pulses [1,2] is simple, very efficient and well-established technique. However, despite the very high THz field strengths (of up to 400 kV/cm [3]) and extremely broadband (0.1–75 THz [4]) spectral widths achieved by this method, there still remain some uncertainties concerning the physical mechanism of the generation. Thus, immediately after the discovery, the phenomenological model of the four-wave rectification based on third-order optical nonlinearity of the gaseous media has been proposed [1,5,6]. On the other hand, Kim et al. [4,7] has proposed the model based on the microcurrent formation in laser-induced plasma. One of the main arguments supporting this model was the dependence of THz field strength on the relative phase δΦ between bichromatic pump fields (ω and 2ω): in the case of four-wave rectification THz field strength should reach its maximum for the zero phase difference, while the photocurrent model predicts maximal THz yield when δΦ=π/2. Thus, the detailed investigations of the phase relations between the bichromatic pump pulses may provide an important information to validate these models experimentally. Indeed, recently [8,9] it has been shown that the birefringence of the BBO crystal has to be taken into account in order to explain the experimental observations using the four-wave rectification model. On the other hand, recently [10–12] apart from the good agreement of

⁎ Corresponding author. E-mail address: [email protected] (V. Vaičaitis). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.12.063

the theoretically obtained and experimental data, the coherent control of THz wave polarization following from the plasma microcurrent model have been demonstrated. However, it seems that two different regimes of plasma formation by femtosecond laser pulses are taking place: first, when the low power tightly focused light pulses create the relatively small plasma region, and second, when the high power (of high energy and duration of less than 100 fs) and loosely focused femtosecond laser pulses are producing spatially extended plasma filaments. While in the first case the plasma is created within the Rayleigh area of the optical beam and can be considered as a point source, in the case of spatially extended plasma filaments the situation is more complicated. Thus, recently [13] it has been demonstrated explicitly that in this case due to the Gouy phase shift various parts of plasma filament are generating THz radiation of different phases, which can cause the destructive interference and reduce the total yield of generated emission. Therefore we present here the results of the theoretical analysis of phase relations between the fundamental and second harmonic waves assuming that the plasma is created only in the focal area of the tightly focused pump beam and thus, the THz radiation is generated from the point source. In contrast to the previous reports, where the plane-wave approximation has been used, we have found that the focusing introduces an additional, to the best of our knowledge, unreported phase shift of the THz emission. Though this phase shift may be strongly dependent on the distance between the focus and nonlinear crystal, in most cases it is almost exactly equal to π/2. In addition, it has been found that in the case of tightly focused pump pulses the curved phase fronts of the fundamental and second harmonic waves almost ideally cancel each other resulting in the flat phase front of nonlinear polarization at the terahertz frequency.

V. Vaičaitis et al. / Optics Communications 284 (2011) 2206–2209

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with the initial condition B2x ðr; z1 Þ = B1x ðr; z1 Þ. The solution is B2x ðr; zÞ = bðz1 ÞB2x ðr; zÞ, where

2. Theory In our theoretical modelling the first harmonic (FH) is a linearly polarized Gaussian beam, characterized by the complex amplitudes Ajx = ax Ajx and Ajy = ay Ajy . Here Apj (p = x, y) is the dimensionless complex function which describes the beam profile, subscripts x and y label polarizations, and j denote the beam in front of ( j = 0), inside ( j = 1) and beyond ( j = 2) the BBO crystal. By solving the parabolic diffraction equations for each polarization, one can check that all dimensionless amplitudes of the FH are given by the similar expressions, namely ! 1 1 r2 ; exp − sj ðzÞ sj ðzÞ w20

Apj ðr; zÞ =

j = 0;1;2;

ð1Þ

where w0 is the waist radius of the initial Gaussian beam. The complex parameters sj(z) are calculated as follows s0 ðzÞ = 1 + i

2z ; k1 w20

ð2Þ

s1 ðzÞ = s0 ðz0 Þ + i

2ðz−z0 Þ ; k1p w20

ð3Þ

s2 ðzÞ = s1 ðz1 Þ + i

2ðz−z1 Þ ; k1 w20

ð4Þ

where k1 is a wave number of the FH in the air, while k1p is a wave number for the FH polarization p inside the crystal, z0 and z1 are the positions of the entrance and the end of the crystal with respect to initial Gaussian beam waist (z = 0), l = z1 − z0 is the thickness of the crystal. Inside the nonlinear crystal the y component of the fundamental beam generates the second harmonic (SH) wave polarized along the x axis. Assuming perfect phase matching one can find the complex amplitude B1x of the SH inside the crystal by solving the nonlinear equation ∂B1x 1 2 Δ B = iσA1y ðzÞ + 2ik2x ⊥ 1x ∂z

ð5Þ

bðzÞ = σay

  k1y w20 s ð zÞ ln 1 ; s0 ðz0 Þ 2

ð6Þ

B1x ðr; zÞ =

! 1 2 r2 exp − ; s1 ðzÞ s1 ðzÞ w20

ð7Þ

4ðz−z0 Þ : k2x w20

ð8Þ

s1 ðzÞ = s0 ðz0 Þ + i

Here k2x is the wave number of SH inside the BBO crystal. The complex amplitude of the SH wave beyond the crystal can be evaluated by solving the linear equation ∂B2x 1 Δ B =0 + 2ik2 ⊥ 2x ∂z

! 1 2 r2 ðzÞexp − ðzÞ 2 ; s2 s2 w0

s2 ðzÞ = s1 ðz1 Þ + i

4ðz−z1 Þ : k2 w20

ð9Þ

ð10Þ

ð11Þ

Here k2 is the wave number of SH in the ambient air. Another important issue which should be taken into account is the linear phase shift of light propagating through the BBO crystal. Taking into account that in the crystal the x component of the FH is the extraordinary wave and the y component is the ordinary wave, the appropriate phase shifts are ϕ1x = 2πn(e)(λ1)l/λ1 and ϕ1y = 2πno(λ1)l/λ1. Here no and n(e) are the refraction indexes for the ordinary and extraordinary waves inside the BBO crystal. It is worth to note that in the case of perfect phase matching the linear phase shift of the SH wave ϕ2x obeys the relation ϕ2x = 2ϕ1y. For the thorough theoretical analysis of THz generation through the four-wave mixing one has to include the terms representing self modulation of FH, cross modulation of the SH due to FH, and other processes related with the laser-induced plasma generation and its impact on the propagation of femtosecond laser pulses in nonlinear medium, such as Kerr effect and plasma defocusing, transient photocurrent effect and plasma pressure force [14], etc. However, here we account only for the terms which are directly responsible for the THz generation. This simplified model allows us to demonstrate explicitly the most important phase relations between the FH and SH waves. Thus, the field amplitude Cx of the THz radiation can be determined from the equation

∂Cx ð3Þ  2 i2 ϕ −ϕ iΔkζ ; ∝ ∑ iχ B A e ð 1j 1y Þ e ∂ζ j = x;y xxjj 2x 2j

ð12Þ

where Δk = 2k1 − k2 is the phase mismatching in the air, and ζ = z − z1. It is worth to note that denoting refraction indexes of the ambient air for the FH and SH waves as n1 and n2 respectively, for n1 ≈ n2 we obtain



s2 ðzÞ + s2 ðzÞ = 2 + i

with the initial condition B1x ðr; z0 Þ = 0. Here σ is a coefficient proportional to the second order permeability χ(2). Under the nondepleted pump approximation, the solution is B1x ðr; zÞ = bðzÞB1x ðr; zÞ, where 2

B2x ðr; zÞ =

  λ1 ðz−z1 Þ 1 1 − ≈2: n1 n2 πw20

ð13Þ



As a result, in Eq. (12) arg(B2x A22j) ( j = x, y) becomes independent of r. Physically it means that the polarization term at the terahertz frequency represents almost ideally plane waves (see Fig. 1). This may explain, why the plane-wave approximation used in many previous models has led to a good agreement with the experimental results. However there is a significant difference from the plane-wave model. Thus, the phase front of the FH inside the nonlinear crystal is not flat and this, due to the nonlinear interaction, introduces an additional phase shift of the SH wave. This phase shift depends on the crystal position with respect to the FH beam waist, and certainly has the impact on the THz emission. Therefore, assuming that the THz radiation is generated mainly in the confocal region, where s2 ≈ 1, near the beam optical axis (r = 0), i.e., where plasma is created and χ(3) is significant, one may estimate the contribution to the yield of THz emission as ! n1y ln1 L ln 1 + i Q: Cx ∝i n1 d n1y ðLd + iz0 Þ

ð14Þ

2208

V. Vaičaitis et al. / Optics Communications 284 (2011) 2206–2209

b

c 2

2

1

1

1

0

-1

r/w0

2

r/w0

r/w0

a

0

-1

-2 -4

-2

0

2

4

-2 -4

0

-1

-2 -2

z/Ld

0

2

4

-4

z/Ld

-2

0

2

4

z/Ld

Fig. 1. Phase fronts of the slowly varying amplitudes beyond the crystal for (a) the FH beam (arg(A2y)), (b) the SH beam (arg(B2x)), and (c) THz polarization source term (arg(B2x A22y)).

Here the asterisk means complex conjugated, Ld = k1w20/2 is the diffraction length of the FH in the ambient air and −iΔkz1

Q =e

2 2 i2 ϕ −ϕ ð3Þ   iΔkz′ ∑ ay aj e ð 1j 1y Þ ∫R χ xxjj z′ e dz′;

j = x;y

ð15Þ

where R in the integral denotes confocal region of the pump. It is worth to note that in the plane-wave approximation, i.e. when Ld ≫ |z0|, the Eq. (14) takes the form CPW x ∝lQ. Therefore the impact of the Gaussian beam profile to the THz generation can be analyzed by considering the dimensionless quantity ! n1y Ld Cx ln1 η = PW = i ln 1 + i : n1 l n1y ðLd + iz0 Þ Cx

ð16Þ

Note that in the case of tight focusing, i.e., when |z0| ≫ Ld one obtain ! n1y Ld ln1 ln 1 + η=i ; n1 l n1y z0

ð17Þ

which explicitly shows that Cx has an additional ± π/2 phase shift in comparison to the plane-wave model. In general, this additional phase shift δϕ is determined by the expression δϕ = arg(η). Therefore, when |z0| ≫ Ld and z0 b 0, which is the case in most of the experiments, δϕ = − π/2 (see Fig. 2). Note, that the point where δϕ = 0 is at the coordinate z ≠ 0 since as shown in Fig. 3, the focal positions of SH and FH beams are slightly different due to the dispersion of the BBO crystal and air. Note also that despite the fact that the graph presented in Fig. 2 looks like the Gouy phase shift of focused Gaussian beam, in fact δϕ is a combination of the FH and SH Gouy phases integrated over the entire region of the nonlinear interaction (see the Eq. (15)). Specifically, the Gouy phase shift for the FH and SH beams is almost identical (arctan(z/Ld1) ≈ arctan(z/Ld2)), and decreases to zero in the focus (at the point z = 0). On the other hand, in the focus δϕ is about ± π/2 and approaches to zero only when the nonlinear crystal is placed within the confocal region of the pump, i.e. when |z0| ≪ Ld and the plane wave approximation is valid. However, the nonlinear response of the medium is proportional to the combination of the FH and SH complex amplitudes (B*A2), which implies that the net phase shift of the nonlinear polarization δϕ is ∼ 2arctan(z0/Ld1) − arctan(z0/

Ld2) ≈ arctan(z0/Ld1). Here Ld1 and Ld2 are the diffraction lengths of the FH and SH beams, respectively. 3. Experiment For the experiments a regeneratively amplified 1 kHz repetition rate femtosecond Ti:Sapphire laser system delivering single pulses (with pulse duration of about 120 fs) of 0.8 mJ maximal energy at 804 nm central wavelength has been used. The laser pulses were focused into the ambient air by the lens of about 15 cm focal length through the 100 μm-thick nonlinear BBO crystal (ooe phase-matching type, see Fig. 3). THz radiation generated in air was collimated and focused into the 0.5 mm-thick 110-cut ZnTe crystal for the electrooptical characterization based on the polarization rotation of the probe pulse by the transient electric field. As a probe beam a small fraction of the pump separated by the thin beam splitter (BS) was used. In order to optimize THz output we have rotated polarization of the fundamental beam by the rotary half-wave plate inserted into the beam path. Since during the experiment the optical axis of the nonlinear BBO crystal was kept fixed in the xz plane of the coordinate system, only the x-polarized SH wave was generated for any FH polarization angle (the maximal second harmonic generation efficiency took place at zero rotation angle of the fundamental beam polarization). Therefore

δΦ π 2

π 4

-10

-5

5

10

z0/Ld

π 4

π 2 Fig. 2. Dependence of the additional phase shift δϕ on the normalized distance between the focus and nonlinear BBO crystal.

V. Vaičaitis et al. / Optics Communications 284 (2011) 2206–2209

L

2209

BBO

BS FH

SH

Pump

THz

z0

Probe Fig. 3. Experimental setup.

corrections have to be made at high pump pulse energies due to the deviations from the spatial isotropy and influence of higher-order optical nonlinearities of the gaseous media [8,15] since even at the moderate excitation levels light intensity in the focus may exceed 1014 W/cm2 [1,9]), at which higher order polarizations could be comparable or even exceed the third-order one [16,17]. Also, as it was mentioned above, at high pump power levels many additional nonlinear effects including the formation of extended plasma filament and its impact on the THz yield should be taken into account. Nevertheless, the results of our analysis could be important in developing the more sophisticated models of THz wave generation from the gaseous media.

5

ETHz,x (a.u.)

4 3 2 1 0

2π /Δk

-1 20

40

60

80

100

z0 (mm) Fig. 4. THz peak field versus BBO-to-plasma distance. Solid and dashed lines represent our model and plane wave approximation, respectively. Points are the experimentally obtained data.

Acknowledgement This work was supported by the Research Council of Lithuania project “ConTeX” (contract No. AUT-05/2010). References

the FH beam could be considered as a superposition of the two orthogonally polarized (x and y) beams. THz output also was the superposition of the two orthogonally polarized components, however, the comparison of the theoretically and experimentally obtained data was performed for the x-polarized THz wave since according to all reports it is generated through the dominant third order nonlinear susceptibility tensor component χ(3) xxxx (there is no general agreement on the role of other χ(3) components, see, for example [8,9]). One can see from Fig. 4 that the experimental data well correspond to our predictions. 4. Conclusions In conclusion, the good agreement between theoretically and experimentally obtained data indicates that the THz generation from the gaseous media excited by focused bichromatic pump pulses can be interpreted in terms of four-wave rectification. However, some

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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