# Linear Electro Optic Effect for High Repetition Rate Carrier Envelope Phase Control of Ultra Short Laser Pulses

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## Abstract

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## 1. Introduction

_{3}—lithium niobate) [17,18] and successfully applied it to the CEP control of a titanium-sapphire CPA laser [19] with stabilization performances of the order of those obtained with classical methods. Compared to the main equivalent CEP shifters (as will be detailed at the end of this paper), the major advantages of EO CEP shifters are related to the fact that they do not need mechanical displacements and especially to their high correction bandwidth (>10 kHz). In this paper, we introduce another original method, still based on the use of the EO effect, but in a more complex optical scheme and which relies on the use of an EO dispersive prism pair. While experimental demonstration of CEP shifts and CEP control [17,18,19] of CPA amplified pulses with a “longitudinal” shifter (simple rubidium titanyle phosphate (RTP) or LiNbO

_{3}rectangular slab) has been proven, no experimental results will be given here concerning the prism pair EO shifter, which will be the topic of a future paper.

## 2. EO CEP Shifter

#### 2.1. Theory

_{0}, propagating in a homogeneous dispersive medium of length, L, which is non-centrosymmetric and exhibits a Pockels effect. This is, for example, the case [17,20] in LiNbO

_{3}—uniaxial crystal, crystalline class 3 m or RTP-biaxial crystal orthorhombic crystalline class mm2—when choosing appropriate directions for the linearly polarized laser field and the applied static electric field. Figure 1 illustrates how the direction of propagation and the polarization of light are to be chosen in practice and how the voltage is to be applied on the crystal in two interesting cases, LiNbO

_{3}and RTP. X, Y and Z are the principal dielectric axes, which are parallel to the crystallographic axes. With these choices, the variation of the corresponding electric field-dependant refractive index, Δn, is given as a linear function of the EO coefficient, r, and of the electric field, E. We only consider here the case where one can neglect the change in the crystal length, ΔL, due to the inverse piezoelectric effect, as is the case in LiNbO

_{3}and RTP [20,21,22].

**Figure 1.**Geometry of the interaction for LiNbO

_{3}and simple rubidium titanyle phosphate (RTP). X, Y and Z are the principal dielectric axes (parallel to the crystallographic axes).

_{0}) coefficient (unclamped EO coefficient at wavelength λ

_{0}) and can be written in a scalar form as:

#### 2.2. Experiments

_{3}using a femtosecond 800 nm Ti:S laser source and an f-2f interferometer. The results show a good agreement between theory and experiments. Another approach using spectral interferometry with a broadband laser source [18] was applied to measure CEP shifts and confirmed again, with a better accuracy, the above theory. Stabilization of the CEP (slow loop) of a Ti:S femtosecond laser source was finally demonstrated in [19] with very promising results. The CEP-stable 20W range kHz laser and the EO device for CEP control arrangement is given in Figure 2. In this case, a LiNbO

_{3}crystal longitudinal EO shifter was used.

**Figure 2.**Carrier-envelope-phase (CEP)-stable 20 W range kHz laser and LiNbO

_{3}electro-optic (EO) CEP shifter.

**Figure 3.**Shot to shot (red dots) and 10 ms averaged (grey dots) measurement of stabilized CEP drift over 10 min of amplified pulses at 3 W output (

**a**) without slow feedback control, (

**b**) with EO feedback loop over 25 min leading to RMS CEP noise of, respectively, 320 and 130 mrad and (

**c**) over 7 min of amplified pulses at 20 W output leading to RMS CEP noise of respectively 440 and 250 mrad.

## 3. Prism Pair CEP Shifter

#### 3.1. Introduction

_{g}(Δτ

_{g}= 0 for an isochronous system) and the electric field-induced dispersion ΔФ

_{2}(ΔФ

_{2}= 0 for an iso-dispersive system) may play a role. Furthermore, lowering the applied voltage necessary to obtain the same CEP shift is also clearly of practical interest. Equation 3 shows that the two ways to lower the electric field are to increase the (effective) crystal length or to find a material with optimized characteristics (for example, with a higher EO coefficient).

_{3}or RTP crystals. We investigated a combination of two CEP shifters with different crystals, but this set-up proved to be ineffective, as it led to very low CEP shifts when a constant group delay (isochronous system) or a constant dispersion condition (iso-dispersive system) were sought.

_{3}for example) on which an electric field is applied (see Figure 4). In this configuration, the polarization of the optical electric field is parallel to the “static” electric field. This leads to an S polarization on the prism surfaces and, thus, requires an anti-reflection coating in order to reduce optical losses. The other point, which will be clarified hereafter, concerns the homogeneity of the “static” field in the prisms.

#### 3.2. Theory

_{op}is the optical path for a ray of angular frequency, ω, and c is the speed of light in vacuum.

#### 3.2.1. General Configuration

_{2}parallel to A

_{1}B (NA

_{1}normal to A

_{1}B) intercepting the apex A

_{2}of prism 2. This ray is deviated by prism 2 along the path, A

_{2}A

_{3}. As A

_{1}B and NA

_{2}are parallel incident rays, the light paths, Λ

_{opt}= A

_{1}BCD and NA

_{2}A

_{3}, are identical. Let l be the distance between the apices of the two prisms and ρ the angle, (NA

_{2}A

_{1}) (Figure 5), the optical pass, Λ

_{opt}, can be written as:

_{1}A

_{2}O), between A

_{1}A

_{2}and A

_{1}O, one obtains the following relation:

_{4}is the external refraction angle at the exit of prism 1 in air, a the slant distance, A

_{1}O, between the output face of prism 1 and input face of prism 2, b the distance, OA

_{2}, and A

_{2}A

_{3}the distance between the apex of the second prism and a reflecting mirror (used in a two pass configuration to remove spatial chirp). A

_{2}A

_{3}being a constant, we can remove it, and the spectral phase can thus be written:

_{g}, the CEP shift, , and second order dispersion, , have, respectively, the following expression:

_{0}(CEP shift without induced group delay), corresponds to:

_{20}and θ

_{40}are respectively the values of θ

_{2}(internal refraction angle in first prism) and θ

_{4}(external refraction angle in air at the exit of first prism) for E = 0 (Figure 5) and β is the apex angle of the prisms. Similarly, generation of a group delay without variation of the CEP leads to the condition:

#### 3.2.2. Minimal Deviation Configuration

_{0}. This condition, which gives the highest angular dispersion, can be written as:

_{2}and d

_{3}, corresponding, respectively, to the distances covered between prism 1 and 2 in air and to the total path inside the prisms for the ray at the carrier frequency (Figure 6). The spectral phase takes the form (Appendix III):

_{0}, corresponding to the central angular frequency, , and using λ instead of ω. Equation 21 becomes (Appendix IV):

_{0}the variation of the spectral phase with respect to the applied electric field does not depend on the distance, d

_{2}, nor on the apex angle, β.

_{3}and d

_{2}. Analytical expression of the group dispersion with respect to the electric field can also be derived, but, as in the general configuration, is not given here for conciseness.

_{2}/d

_{3}, at for isochronous CEP shifter and PGD generator configuration are, respectively, given below:

_{2}/d

_{3}, for which the system introduces no dispersion (i.e., ) at :

#### 3.3. RTP Prism Pair CEP Shifter

_{2}/d

_{3}, versus the apex angle, β, of the prisms when the isochronous ( , thick red), the PGD ( , thick blue) and the iso-dispersive conditions ( , thin green) are imposed. This gives a means of comparing the performances of each configuration. The thick brown line corresponds to a zero dispersion condition when no electric field is applied ( ) to the set-up.

**Figure 8.**Ratio, d

_{2}/d

_{3}, versus apex angle for different configurations and corresponding CEP shift

_{3}= 40 mm and E = 1 kV/cm) on the same graph in the iso-dispersive configuration (dotted green line). Finally, the dotted red line curve (which corresponds to the isochronous CEP shift as a function of β) illustrates the fact that, in the isochronous configuration, the CEP shift (at central frequency ω

_{0}is independent of the apex angle of the prism—as can be seen from Equation 22.

_{2}corresponding to each configuration, Figure 9 plots the induced CEP shift (red line), the induced group delay (i.e., —blue line) and the induced GDD (brown line) at central frequency, ω

_{0}, as a function of d

_{2}. The group delay dispersion, (GDD) for E = 0 is also plotted (dotted brown line). These results are obtained in the case of an apex angle of the prisms, , a total path length, d

_{3}= 40 mm, at nm and a static electric field, E = 1 kV/cm.

**Figure 9.**CEP shift, Δτ

_{g}ω

_{0}phase, and introduced by a RTP prism pair CEP shifter versus d

_{2}for the following parameters (apex angle , d

_{3}= 40 mm, E = 1 kV/cm; a value E = 0 kV/cm is assumed for the calculation).

_{2}and that the CEP phase shifter is not efficient in the iso-dispersive configuration. One can also conclude that at fixed CEP shift and, for a given pair of prisms, increasing d

_{2}(i.e., the distance between the prisms) can increase the CEP shift range or reduce the electric field. This, however, cannot be done while maintaining the isochronous condition. In any case, d

_{2}is limited by the size of the beam on the second crystal, which depends on its spectral extent.

#### 3.4. Comparison between Different CEP Shifters

_{g}, and the induced GDD of each system and added, when significant, specific data, such as the mechanical displacement for the grating compressor and the electric field for the EO devices. Table 1 gives these parameters, when required, for all the systems, for a CEP phase shift of π radians at a wavelength of 800 nm. In the case of the EO CEP shifters, the single pass configuration and the double pass two-wedged crystal isochronous configuration are selected for comparison and a single pass propagation length, L = 40mm, chosen in RTP for the central wavelength rays. This corresponds to RTP crystal lengths, which are commercially available and that lead to relevant CEP shift at moderate values of the electric field. 1,200 grooves/mm gratings were considered at 37° incidence for the grating compressor. Concerning the glass wedge system, the basic configuration described in [10] is considered, with a displacement of two silica wedges perpendicular to the beam axis to vary the thickness of silica. We also give an estimate of the correction bandwidth, defined as the inverse of the time needed to change from one CEP value to another.

EO CEP Shifter | Dazzler | Grating compressor | Glass wedges | 4f + LCD | Lens + rotating grating | ||
---|---|---|---|---|---|---|---|

Longitudinal single pass | Prism pair double pass | Double pass | |||||

Δτ_{g} (fs) | −5.78 | 0 | 0 | 7.9 | 43.7 | 0 | 0 |

Δ
Ф^{(2)} (fs^{2}) | -2 | 2.6 | 0 | 1.7 | 1 | 0 | 0 |

E (kV/cm) | 2.86 | −0.43 | - | - | - | - | - |

Displacement (µm) | - | - | - | 0.54 | - | - | - |

Bandwidth | MHz | MHz | <30 kHz | ~10–100 Hz | ~1–10 Hz | ~10–100 Hz | ~1 kHz |

## 4. Conclusions

## Acknowledgments

## Conflict of Interest

## Appendix 1: Derivation of the Isochronous Condition (General Case)—Equation 16

_{1}, on the first prism, calculations are done in the particular case of a ray intercepting the apex, A

_{1}, of prism 1, as shown on Figure 5.

- -
- β apex angle of the prisms;
- -
- θ
_{1}incident angle on prism 1; - -
- θ
_{2}refracted angle in prism 1; - -
- θ
_{3}incident angle on the output face of prism 1; - -
- θ
_{4}refracted angle after prism 1 (in air); - -
- θ
_{5}incident angle on prism 2; - -
- θ
_{6}refracted angle in prism 2; - -
- θ
_{7}= θ_{1}incident angle on output face of prism 2.

_{4}(Figure 5). The variation of the group delay, Δτ

_{g}, when applying the electric field, E, is given by:

_{4}and θ

_{40}are, respectively, the output angle of prism 1, as defined in Figure 5, when an electric field is applied and when it is not. The isochronous condition corresponds to Δτ

_{g}= 0 (no group delay variation due to the application of the electric field).

#### 1.1.Calculation of A:

#### 1.2.Calculation of B:

_{3}is the light pass in prism 2 when the electric field is applied:

#### 1.3.Group Delay

_{3}to the first order in E as a function of x

_{30}. This is obtained using Equation A.13 and the set of Equation A.5. One obtains:

_{30}and a, which does not depend on E. Using Equation A.9 and Equation A.18, one obtains Equation 16.

## Appendix 2: Derivation of the Pure Group Delay Condition (General Case)—Equation (18)

## Appendix 3: Derivation of Equation 21

_{1}and d

_{2}. We immediately see that:

## Appendix 4: Derivation of the Equation 22

_{0}, one obtains immediately Equation 22.

## Appendix 5: Discussion on the Possibility to Stabilize the CEP of a ML Laser Oscillator with an EO System, Outside the Oscillator Cavity

_{0}is the circular carrier wave frequency and Δϕ

_{CE}is the phase slippage (modulo 2π) between two pulses. Applying Fourier transform and Fourier series theory, one obtains the following expression [28]:

_{rep}(or an integer multiple of f

_{rep}), as shown on Figure 11. The electric field at the output of the modulator can then be written:

_{rep}is the angular frequency corresponding to f

_{rep}. Applying the Anger-Jacobi development [30] to the last term of Equation A.29 leads to:

_{k}are Bessel functions. This last expression can be written as:

_{rep}, and, on the other side, a single mode of the comb, whose temporal extension is infinite and which experiences a sinusoidal phase modulation, giving rise to adjacent modes separated by a multiple of ω

_{rep}. This shows that it is not possible to stabilize in this way the CEP of the optical pulse train outside the ML oscillator with our EO device.

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**MDPI and ACS Style**

Gobert, O.; Rovera, D.; Mennerat, G.; Comte, M.
Linear Electro Optic Effect for High Repetition Rate Carrier Envelope Phase Control of Ultra Short Laser Pulses. *Appl. Sci.* **2013**, *3*, 168-188.
https://doi.org/10.3390/app3010168

**AMA Style**

Gobert O, Rovera D, Mennerat G, Comte M.
Linear Electro Optic Effect for High Repetition Rate Carrier Envelope Phase Control of Ultra Short Laser Pulses. *Applied Sciences*. 2013; 3(1):168-188.
https://doi.org/10.3390/app3010168

**Chicago/Turabian Style**

Gobert, Olivier, Daniele Rovera, Gabriel Mennerat, and Michel Comte.
2013. "Linear Electro Optic Effect for High Repetition Rate Carrier Envelope Phase Control of Ultra Short Laser Pulses" *Applied Sciences* 3, no. 1: 168-188.
https://doi.org/10.3390/app3010168