Carrier-envelope phase control using linear electro-optic effect

O. Gobert; P.M Paul; J.F Hergott; O. Tcherbakoff; F. Lepetit; P. D 'Oliveira; F. Viala; M. Comte

doi:10.1364/OE.19.005410

1. Introduction

A laser pulse, corresponding to an electric field, is usually described by the product of a wave envelope and a carrier wave. The envelope propagates at the group velocity, which corresponds to the speed of propagation of the energy, whereas the carrier wave propagates at the phase velocity.

In a dispersive medium, phase and group velocities are different, inducing a slippage of the carrier frequency wave inside the envelope (Fig. 1 ). This slippage is usually of no concern for long pulses (a ns pulse at optical frequency contains of the order of 3 × 10⁵ optical cycles). For ultra-short pulses, which contain few optical cycles, physical phenomena induced in a medium can strongly depend on the electric field and not only on its envelope [1,2]. In this case, it is of prime importance to control the Carrier-Envelope Phase (CEP).

Fig. 1 Temporal drift of the carrier wave inside the pulse envelope from shot to shot.

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Laser systems emitting ultra-short pulses do not generate a train of pulses with the same CEP value. This is mainly due to environmental effects such as vibrations and thermal drift. One elegant way to get rid of these variations is to use an optical parametric amplifier in a specific configuration [3]. Different methods [4–6] also exist in order to obtain, through the use of a fast loop, a train of CEP stabilized pulses from mode locked-oscillators.

Laser systems used to generate ultra-short pulses with energy per pulse above ten µJ are based on the Chirped Pulse Amplification technique [7] (Fig. 2 ). In these systems, a mode-locked oscillator generates a train of ultra-short pulses (repetition rate of the order of 100 MHz, energy ~1 nJ, spectrum tens of nm wide); these pulses are temporally stretched from fs to ps range. The pulses are then amplified in media such as Ti:Sa which allows a gain well above 10⁶. After amplification, the pulses are compressed back to some tens of fs. Filamentation in a rare gas cell or propagation in a hollow waveguide filled with rare gas, in combination with the use of a chirped mirror or prism compressor can be used to obtain sub-ten fs pulses with energy per pulse above the mJ level [8–10].

Fig. 2 CPA laser system design.

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Different techniques already exist to stabilize the CEP of the amplified pulses of a CPA laser system seeded by a CEP stabilized mode locked oscillator. They are mainly based on a slow feedback loop containing a f-2f interferometer [11], a PID and a specific technique to correct the CEP. These techniques can be split into two categories. In the first one, the correction is made by modifying parameters inside the cavity of the mode-locked oscillator [4] which can have the disadvantage of inducing coupling between fast and slow loops, leading to an increase of noise in amplitude and phase. In the second one, the corrections are made outside the mode-locked oscillator, usually before the amplification.

Examples of those techniques are the use of a pair of wedges to modify the optical path in the dispersive element composing these wedges [12], the modification of one parameter of the compressor or of the stretcher (this parameter can be the distance between the gratings) [13], the use of an Acousto-Optic Programmable Dispersive Filter (AOPDF) [14,15] or of a 4f system with an adaptive phase modulator device [16].

In this paper, we consider a new method to control the CEP by using the linear electro-optic (EO) effect with a specific arrangement for the direction of the applied static electric field and the polarisation of the laser field relative to the crystal axes. We first present the conceptual idea of the method. The mathematical expression of the CEP variation as a function of the applied electric field is derived in the case of a uniaxial crystal like LiNbO₃. Experimental demonstration using an f-2f interferometer and comparison with the calculation is presented in the case of LiNbO₃. The problem of the practical CEP stabilization will not be considered in great detail in this paper, the aim being to demonstrate the feasibility of the EO CEP control.

2. Phase and group velocity in a dispersive medium

Carrier-Envelope Phase of an optical field after propagation in a dispersive material is related to the difference between phase and group velocity in the medium. Because of the wavelength dispersion of the refractive index, phase and group velocities have in general different values. In a crystal with a nonvanishing EO effect, the refractive index can be linearly modulated by an external electric field, leading to a variation of the CEP.

We consider a laser pulse whose carrier angular frequency is ω₀, propagating in a homogeneous dispersive medium of length L which is non-centrosymmetric and exhibits a Pockels effect. The phase delay T_φ and the group delay T_g are defined as follows:

T_{ϕ} = \frac{n (ω_{0})}{c} L; T g = \frac{n_{g} (ω_{0})}{c} L

n(ω₀) and n_g(ω₀) are the refractive index and the group refractive index, c is the speed of light in vacuum. The delay due to the difference between the group and phase velocities can be written:

T_{g} - T_{ϕ} = [n_{g} (ω_{0}) - n (ω_{0})] \frac{L}{c}

To obtain the expression of n_g(ω₀), one can use the following usual relations where k = nω₀/c, is the wave vector modulus and v_g(ω₀) the group velocity at pulsation ω₀:

\frac{1}{v_{g} (ω_{0})} = \frac{n_{g} (ω_{0})}{c} = {\frac{\partial k}{\partial ω} |}_{ω_{0}} = \frac{{n(ω}_{0})}{c} + \frac{ω_{0}}{c} {\frac{\partial n}{\partial ω} |}_{ω_{0}}

Using the wavelength λ₀ instead of ω₀, (3) leads to:

n_{g} (λ_{0}) = {n(λ}_{0}) - λ_{0} {\frac{\partial n}{\partial λ} |}_{λ_{0}}

The difference between group and phase delay which is directly connected to the CEP, can be written:

T_{g} - T_{ϕ} = - λ_{0} {\frac{\partial n}{\partial λ} |}_{λ_{0}} \frac{L}{c}

3. Changing the CEP by the linear EO effect

The purpose of this paragraph is to establish the expression of the CEP change of an ultrashort laser pulse at the exit of an EO medium when an electric field is applied in a specific geometry. Calculations will be restricted to the case of a uniaxial crystal and more specifically to the case of LiNbO₃ (uniaxial crystal, point group 3m).

3.1 EO effect

A static electric field E applied on the medium generates a variation of the refractive index due to the Pockels effect. In a solid, the relationship between the refractive index and the applied electric field can be written:

Δ {(1 / n^{2})}_{ij} = \sum_{k} r_{ijk} E_{k}

Where Δ(1/n²)_ij is the second-rank tensor describing the change in relative permittivity, E_k is the k-th component of the electric field, and i, j, k = x, y, z. The term r_ijk, the linear EO coefficient tensor, is a third-rank tensor with 27 elements.

For the sake of simplicity, we will consider here the case of LiNbO₃ which is uniaxial but analogous results can be obtained in biaxial crystals. Due to the symmetry and using the reduced-subscript notation, the EO effect in lithium niobate can be described by only four independent coefficients (r₅₁, r₂₂, r₁₃ and r₃₃). The measured values of these coefficients depend on the mechanical constraints imposed on the crystal. In the free condition (unclamped crystal), the r coefficient value includes a contribution from the secondary EO effect which is the result of the applied electric field causing a strain in the crystal through the inverse piezoelectric effect. This electrically induced strain then causes a change in the crystal's refractive index through the photoelastic effect [17]. This secondary contribution is inseparable from the primary linear EO component. The relationship between the clamped coefficient r^S and the unclamped coefficient r^T can be written:

r_{ij}^{T} = r_{ij}^{S} + \sum_{k = 1}^{6} pik dkj

As a consequence, the phase of the light passing through the crystal is modified by both the change in the refractive index due to the EO effect and the change in the crystal length ΔL due to the inverse piezoelectric effect.

3.2 Geometry of the interaction

We consider an optical pulse propagating in the crystal along the Oz direction (Fig. 3 ). The optical electric field is supposed to be linearly polarized along the Ox direction as well as the applied static electric field. We assume that the crystal is oriented so that X-axis corresponds to the optical axis.

Fig. 3 The EO configuration chosen for the interaction

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Expressions of the ordinary n_o(E,λ₀) and extraordinary n_e(E,λ₀) refractive indices when applying an electric field E are, to first order, given by:

n_{o} (E, λ_{0}) = n_{o} (λ_{0}) - \frac{1}{2} n_{o}^{3} (λ_{0}) r_{13}^{T} (λ_{0}) E

n_{e} (E, λ_{0}) = n_{e} (λ_{0}) - \frac{1}{2} n_{e}^{3} (λ_{0}) r_{33}^{T} (λ_{0}) E

In these equations, r₁₃ ^T(λ₀) and r₃₃ ^T(λ₀) are the unclamped EO coefficients of LiNbO₃ at wavelength λ₀.

3.3 Calculating the variation of CEP when the electric field is applied

The variation of the difference between phase and group delay ΔT when applying the electric field has the following form:

\begin{matrix} Δ T = (T_{g} - T_{ϕ}) [E] - (T_{g} - T_{ϕ}) [0] \\ = (\frac{L + ΔL}{c}) [n_{g} ({E,λ}_{0}) - n ({E,λ}_{0})] - \frac{L}{c} [n_{g} ({0,λ}_{0}) - n ({0,λ}_{0})] \end{matrix}

As n(0,λ₀) = n_e and using Eq. (5), one obtains:

Δ T = - λ_{0} \frac{(L+ΔL)}{c} [{\frac{\partial n_{e} (E,λ)}{\partial λ} |}_{λ_{0}}] + λ_{0} \frac{L}{c} [{\frac{\partial n_{e}}{\partial λ} |}_{λ_{0}}]

Using 8.2 gives:

Δ T = - λ_{0} \frac{(L+ΔL)}{c} [{\frac{\partial n_{e}}{\partial λ} |}_{λ_{0}} - \frac{3}{2} n_{e}^{2} r_{33} E {\frac{\partial n_{e}}{\partial λ} |}_{λ_{0}} - \frac{1}{2} n_{e}^{3} E {\frac{\partial r_{33}}{\partial λ} |}_{λ_{0}}] + λ_{0} \frac{L}{c} [{\frac{\partial n_{e}}{\partial λ} |}_{λ_{0}}]

Which leads to:

(T_{g} - T_{ϕ}) [E] - (T_{g} - T_{ϕ}) [0] = \frac{λ_{0} E}{c} [(\frac{3}{2} n_{e}^{2} r_{33}^{T} {\frac{\partial n_{e}}{\partial λ} |}_{λ_{0}} + \frac{n_{e}^{3}}{2} {\frac{\partial r_{33}^{T}}{\partial λ} |}_{λ_{0}}) (L + Δ L) - d_{32} \frac{\partial n_{e}}{\partial λ_{0}} L]

Where we used the following relation between the change in the crystal length ΔL due to the inverse piezoelectric effect:

ΔL = d_{32} EL

d₃₂ is the piezoelectric constant. In the case of LiNbO₃, d₃₂ is more than 50 times lower [18] than the EO coefficient r₃₃ which justifies neglecting all the terms involving ΔL in Eq. (12). Finally, one obtains the variation Δφ _CEP of the CEP, after applying the electric field E:

Δ ϕ_{CEP} \approx 2 π [\frac{3}{2} n_{e}^{2} (λ_{0}) r_{_{33}}^{T} (λ_{0}) {\frac{\partial n e}{\partial λ} |}_{λ_{0}} + \frac{n_{e}^{3} (λ_{0})}{2} {\frac{\partial r_{_{33}}^{T}}{\partial λ} |}_{λ_{0}}] LE

This phase change is proportional to the length of the crystal and to the electric field applied. It can be easily plotted taking into account the wavelength dispersion of the extraordinary index [19] and the wavelength dispersion of the EO coefficient [20] for LiNbO₃.

4. Practical realization

4.1 Experimental results

The experimental demonstration was made within the IMPULSE laboratory which is a common R&D laboratory associating CEA Saclay and Amplitude Technologies. The laser source was developed in collaboration with Amplitude Technologies and is a classical CEP stabilised Ti:S CPA system which delivers up to 2.5 mJ 35 fs pulses with a shot-to-shot CEP RMS noise after amplification of 320 mrad (over a period of one hour). The EO system (LiNbO₃) was inserted between the stretcher and the regenerative amplifier. A train of pulses of 0.6 nJ energy, at a repetition rate of 75 MHz, was sent into the crystal (mean power 45 mW). We used a 5% MgO doped LiNbO₃ crystal (provided by CASTECH Inc.) in the configuration described above. The length of the crystal in the z direction was 40mm and the width was 4mm in the X direction where the field is applied (see Fig. 3). Gold was deposited on each side of the crystal normal to the X direction. Using MgO doped crystal limits the photorefractive effects and permits higher incident laser power without beam degradation [21]. The simplified experimental set-up is given on Fig. 4 . A sine wave signal from a low voltage pulse generator is amplified in a high speed, High Voltage (HV) amplifier and applied on the crystal. The CEP variation as a function of the HV voltage is directly measured through the use of an in-house developed fast f-2f interferometer allowing multi-kHz shot to shot measurements.

Fig. 4 Experimental set-up (see text for the CEP characteristics of the laser beam).

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The results obtained are given on Fig. 5 which plots the measured change of the CEP as a function of the applied voltage. It clearly shows a linear dependency as predicted by the model. The measured slope of ~3.5 rad/kV for the crystal used is in good agreement with the theoretical value of 3.1 rad/kV given by Eq. (14).

Fig. 5 CEP shift as a function of applied voltage

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As refractive indices, EO coefficients and their wavelength dispersion for LiNbO₃ are known with a good accuracy, the difference (about 10%) is probably due to the precision of the f-2f measurements. The oscillator of the laser source we used is CEP stabilized but this is not the case of the amplifier. The f-2f measurements being made after amplification (see Fig. 1), the accuracy of the phase determination is limited by the CEP noise on the laser. A CEP RMS noise of 320 mrad as mentioned before is quite compatible with the 10% difference observed between calculated and measured values in the studied range.

Finally, Fig. 6 shows periodic CEP phase sweep visible on f-2f interferometer fringes when a modulated high voltage is applied on the EO device respectively with a sine (a), a sawtooth (b) and a square function (c).

Fig. 6 CEP sweep on f-2f interferometer fringes for different periodic modulated applied high voltage.

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4.2 Evaluation of the second derivative of the spectral phase

Applying a high voltage to the crystal in the above configuration changes the group velocity dispersion, and can be used to modify the CEP of a laser pulse. It is essential, however, to verify that the other main parameters of the laser pulse are unchanged. In particular, we have to evaluate the possible effects on the temporal laser pulse shape. The spectral phase ϕ(ω) induced by the crystal on the laser pulse is defined by the following relation:

ϕ (ω) = {ωT}_{ϕ} = \frac{n_{e} (E)}{c} ωL

Where n_e(E) is given by Eq. (8.2). As the wavelength dispersion of n_e and r₃₃ are known, the second derivative of ϕ(ω) can be numerically calculated and its square root compared to the pulse duration τ.

Figure 7 plots the parameter η = ([d²ϕ/dω²]^1/2)/τ as a function of the laser wavelength λ, for a pulse duration τ = 5 fs and a calculated CEP shift of Δφ _CEP = 2π radians (which corresponds to an applied voltage of about 1985 V in our configuration). For wavelengths between 0.6 and 1-µm, the parameter η being lower than 1, no significant change of the temporal laser pulse shape or duration is expected even for sub-ten fs short pulses.

Fig. 7 Calculated parameter η (see text) as a function of the wavelength λ, for a laser pulse duration of 5 fs and a calculated CEP shift of 2π radians

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Conclusion

We have presented a new, simple and very promising method to control the Carrier-Envelope Phase (CEP) of ultra short laser pulses by using the linear EO Effect. Experimental results obtained on a CEP stabilized Chirped Pulse Amplification based laser are in good agreement with calculations and show that CEP phase shifts greater than π radian can be obtained by applying less than 1 kV voltage on a 40 mm long LiNbO₃ crystal without altering the main characteristics of the pulse. The feasibility of controlling CEP shifts with the EO effect being demonstrated, which is the object of the paper, the next step should concern the application of the method to CEP stabilization. Even if this objective clearly needs major efforts, we believe it to be achievable. Performances of stabilizing devices depend crucially on two aspects which concern the quality of the measurement and the correction system. The correction device described in this paper makes possible on one hand large phase shifts (if necessary two or more crystals can be used to improve the amplitude of the CEP correction) and is characterized on the other hand by a very short time response compared to most present equivalent systems. This will probably make possible high repetition rate shaping or CEP stabilization in ultra short laser systems with performances comparable or perhaps better than what is currently obtained. This concerns obviously the CPA laser system slow loop stabilization as illustrated in Fig. 1 but may also apply to the fast loop correction. The subject is currently being studied by our laboratory.

Acknowledgements

The authors acknowledge the financial support from the Conseil Général de l'Essonne (ASTRE program), the ANR-09-BLAN-0031-01 ATTO-WAVE and from the European Community (grant agreement PIAPP-GA-2008-218053).

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Carrier-envelope phase control using linear electro-optic effect

Abstract

1. Introduction

2. Phase and group velocity in a dispersive medium

3. Changing the CEP by the linear EO effect

3.1 EO effect

3.2 Geometry of the interaction

3.3 Calculating the variation of CEP when the electric field is applied

4. Practical realization

4.1 Experimental results

4.2 Evaluation of the second derivative of the spectral phase

Conclusion

Acknowledgements

References and links

Cited By

Figures (7)

Equations (16)

Optics Express