Reflective optical system

Abstract

A reflective optical system (100) comprising at least one reflective aspheric surface (1) of focal length f.sub.0 and optical axis (Z), the surface being configured so that an incident laser beam (2) propagating along an axis (Z′) is focused along the optical axis (Z) with a FWHM ((Full Width at Half Maximum) of the intensity of the reflected beam along the optical axis (Z) being larger, preferably by a factor of at least 10, than the FWHM of the intensity of a focused beam reflected by a parabola having same focal length f.sub.0 and same optical axis (Z), receiving same beam.

Claims

1. A reflective optical system comprising at least one reflective aspheric surface of focal length f.sub.0 and optical axis, the at least one reflective aspheric surface being configured so that a FWHM ((Full Width at Half Maximum) of the intensity of a reflected beam along the optical axis, is larger than the FWHM of the intensity of a focused beam reflected by a parabola having the same focal length f.sub.0 and the same optical axis, and receiving the same incident laser beam propagated along an axis of incidence and focused along the optical axis, wherein the at least one reflective aspheric surface is configured so that rays of the incident laser beam that impinge the at least one reflective aspheric surface at a given distance r from the axis of incidence with r≤R, are focused at a distance f(r) from the intersection O of the optical axis with the at least one reflective aspheric surface given by f(r)=f.sub.0+z.sub.g(r), where z.sub.g(r) is a function depending on rand R is distance of the outer rays from the axis of incidence.

2. A system according to claim 1, wherein a sag function s of the at least one reflective aspheric surface satisfies the relation: $r\frac{ds}{dr}=s\left(r\right)-f\left(r\right)+\sqrt{{\left[s\left(r\right)-f\left(r\right)\right]}^2+r^2}.$

3. A system according to claim 2, wherein for integer n>2 $.Math.\frac{d^{n}s\left(r\right)}{d{r}^n}.Math.>0.$

4. A system according to claim 2, wherein the sag function s of the at least one reflective aspheric surface has no polar symmetry.

5. A system according to claim 1, wherein the axis of incidence coincides with the optical axis.

6. A system according to claim 1, wherein the axis of incidence makes a non-zero angle θ with the optical axis.

7. A system according to claim 1, wherein the at least one reflective aspheric surface has an opening enabling a second laser beam to pass therethrough.

8. A system according to claim 1, wherein the at least one reflective aspheric surface has a non-continuous surface.

9. A system according to claim 1, wherein the at least one reflective aspheric surface is deformable and the system comprises at least one actuator for deforming the surface.

10. A system according to claim 1, wherein the at least one of the reflective aspheric surface has a spiral phase around the optical axis.

11. A method for generating a plasma channel, comprising: generating at least one laser pulse, and focusing the at least one laser pulse in a gas medium to produce the plasma channel, using the reflective optical system of claim 1.

12. A method of guiding a laser pulse, comprising: generating at least one first laser pulse with a first laser source, focusing the at least one first laser pulse into a gas medium using the reflective optical system of claim 1, therefore producing a plasma waveguide, and directing at least one second laser pulse originating from a second laser source along the longitudinal axis of the plasma waveguide channel.

13. A laser-plasma charged particle accelerator, comprising: a first laser source for generating at least one first laser pulse to produce a plasma channel waveguide, the reflective optical system of claim 1, a gas medium in which the at least one first laser pulse is focused by said reflective optical system to produce a plasma channel waveguide, and particles being accelerated by the first laser source and/or an accelerator comprising a second laser source for generating at least one second laser pulse along the axis of the plasma waveguide channel for accelerating particles.

14. A vacuum laser-plasma charged particle accelerator with a high intensity laser pulse, comprising: a laser source for generating at least one high intensity laser beam, an acceleration zone under vacuum, a particle source to inject particles to be accelerated into said acceleration zone, the reflective optical system of claim 1 to focus the laser pulse onto the particles to be accelerated in said acceleration zone.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The accompanying drawings, which are incorporated in and constitute a part of the specification, illustrate presently preferred embodiments of the invention, and together with the description, serve to explain the principles of the invention. In the drawings:

(2) FIG. 1a, 1b, 1c are illustrations of different embodiments of a method according to the invention,

(3) FIG. 2 is an illustration of another embodiment of a method according to the invention,

(4) FIG. 3 is an example of a variation of a laser peak intensity along the focal line,

(5) FIG. 4 is an illustration of an embodiment of a plasma creation according to the invention,

(6) FIG. 5 is an illustration of another embodiment of a method according to the invention,

(7) FIG. 6 is an illustration of focal spots of a laser beam obtained at the inlet of a gas medium and at its outlet with and without a plasma channel waveguide,

(8) FIG. 7 is a schematic of one embodiment of a vacuum charged particles accelerator, and

(9) FIG. 8 is an illustration of a deformable aspheric reflective surface.

(10) Whenever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts.

(11) In accordance with the invention, and as broadly embodied, in FIGS. 1a, 1b and 1c, an optical system 100 is provided.

(12) The optical system 100 includes a reflective aspheric surface 1 with an optical axis Z and a focal line 3.

(13) As shown in FIG. 1a, a laser beam 2 is incident along the axis of incidence Z′ and is deviated towards the optical axis Z.

(14) The reflective surface 1 is configured so that rays of the incident laser beam 2 that impinge the reflective aspheric surface 1 at a given distance r from the optical axis Z are focused on the optical axis Z at a position z=f(r) from the center O of the reflective aspheric surface 1. The distance between the position f(r) and the center O of the reflective aspheric surface 1 depends upon r following the equation z=f(r)=f.sub.0+z.sub.g(r), where z.sub.g(r) is a function depending on r. The focal line 3 corresponds to the zone where the reflected rays cross the optical axis.

(15) The function z.sub.g(r) is related to the peak intensity of the incident laser beam and the peak intensity of the reflected laser beam according to the following relationship:
I.sub.ref(r)dz.sub.g(r)=2πKI.sub.inc(r)rdr,

(16) where K is a parameter depending on the length of the focal line and the area of the transverse section of the reflected beam.

(17) When the peak intensity of the incident laser beam is approximately constant i.e I.sub.inc(r)=I.sub.0, the peak intensity of the reflected beam is also approximately constant over the focal line 3 and the function z.sub.g(r) can be approximated as:
z.sub.g(r)=δ(r/R).sup.2,

(18) where R is distance of the outer rays from the axis of incidence (Z′) and δ is a constant whose absolute value is equal to the length of the focal line 3.

(19) FIG. 1a shows an embodiment of the reflective aspheric surface 1 where the constant δ is positive. In this case, on-axis rays are focused first and f.sub.0 corresponds to the distance from the center O to the start of the focus. The focal line 3 for the reflective aspheric surface 1 is defined by a set of positions z such that f.sub.0≤z≤f(R), which leads to a focal line 3 of length δ.

(20) FIG. 1b shows an alternative embodiment where δ is negative. In this case outer-axis rays are focused first and f.sub.0 corresponds to the distance from the optics to the end of the focus. The focal line 3 for a reflective aspheric surface 1 is defined by a set of positions z such that f (R)≤z≤f.sub.0, which leads to a focal line 3 of length |δ|.

(21) FIGS. 1a and 1b display an exemplary embodiment of the reflective aspheric surface 1 where the axis of incidence Z′ coincides with the optical axis Z. In this case the reflective aspheric surface 1 has a rotational symmetry. Consequently, its sag function s only depends on the radial coordinate r.

(22) A solution for the sag function s in the on-axis case may be obtained analytically as a power series of the radial coordinate r. For example, in case where the peak intensity of the incident beam is constant, the sag function s is given by:

(23) $s\left(r\right)=\frac{1}{4f_0}{r}^2+\frac{\delta }{8f_0^2{R}^2}{r}^4+\frac{\delta \left(8f_0\delta -R^2\right)}{96f_0^4{R}^4}{r}^6+O\left(r^8\right).$

(24) Referring to FIG. 1c, another embodiment of a reflective aspheric surface used to focus the incident laser beam 2 is disclosed. In this embodiment, the axis of incidence Z′ makes an off-axis angle θ with the optical axis Z. The reflective aspheric surface 1 in this case may have no rotational symmetry. Its sag function s may involve two transverse coordinates x and y in a Cartesian coordinate system. The sag function s can be expressed as an expansion of polynomial functions which minimizes a merit function. The coefficients of the polynomial functions may be obtained by minimizing the discrepancy between the location of the reflected laser beam as simulated with some set of coefficients and the targeted location as defined by the formula f(r) given above, through an iterative optimization procedure. To that end, a numerical optimization software such as Zemax may be used.

(25) The reflective aspheric surface 1 creates a correlation between the distance r and the position z along the focal line 3. Therefore, the intensity peak, which is formed from different parts of incident beam, can propagate with superluminal velocity in vacuum. For the case of a uniform intensity along the line, the peak accelerates and deaccelerates for δ>0 and δ<0 respectively.

(26) FIG. 3 compares the peak laser intensity corresponding to a reflected laser beam 4, along the focal line 3 as obtained using the reflective aspheric surface 1 and as obtained using a reflective optical device, for example a parabola mirror, for δ=5 mm, f.sub.0=200 mm, R=21 mm, λ=800 nm and a 10 TW laser source. Both the aspheric surface 1 and the parabola mirror have an aperture in their center with a radius of 8.5 mm.

(27) The peak intensity of the reflected beam in the case the reflective surface (1) is larger than 10.sup.18 Wcm.sup.−2 is over more than 4 mm. Hence the Full Width at the maximum is approximately 4 mm, which corresponds to an increase nearly by a factor of 30 of the FWHM of the peak intensity, when compared to a parabola mirror with same numerical aperture. Hence only about 10 mJ and 100 mJ are required to produce a plasma waveguide over 1 cm and 10 cm, respectively.

(28) Reference is now made to FIG. 4, which is an example of a plasma creation by means of the reflective aspheric surface 1, a gas medium 23 provided by a nozzle 11, and a laser source (not shown) emitting a laser beam 2. The latter is focused into the gas medium 23 by the reflective aspheric surface 1. When the reflected laser beam 4 enters the gas medium 23, the latter is ionized thus forming a plasma 10.

Example of Hydrogen Gas

(29) As shown in FIG. 3, the peak intensity may be larger than 10.sup.18 Wcm.sup.−2, which is higher than the threshold for barrier suppression ionization for Hydrogen, which is on the order of magnitude of 510.sup.14 Wcm.sup.−2. As a result, the reflected laser beam 4 is strong enough to fully ionize an Hydrogen gas and hence generate a plasma channel. When the reflected beam 4 propagates along the gas medium, the energy the electrons in the plasma electrons is raised up to the ponderomotive energy, which is about 200 keV. The peak intensity and the ponderomotive energy are about 4 orders of magnitude larger than those typically obtained with an axicon lens and a similar laser source. Waveguides based on axicons lenses have to rely on collisional ionization, which is inefficient for electronic densities n.sub.e<10.sup.18 cm.sup.−3. In contrast, with the invention, the efficiency of the ionization is independent from the electronic density n.sub.e; thus, the reflective aspheric surface 1 can generate a plasma at arbitrarily low densities. Moreover, because of very different plasma temperatures, plasmas generated with the reflective aspheric surface 1 expand much faster than those produced with axicons lenses. Indeed the expansion velocity of a Hydrogen gas is

(30) 0${\left(k_b\frac{T}{M}\right)}^{\frac{1}{2}}{10}^{-3}{T\left[\mathrm{keV}\right]}^{\frac{1}{2}}c,$
M being the proton mass; a plasma channel of ˜10 μm radius will thus be formed after ˜10 ps with a reflective aspheric surface 1, while ˜1 ns would be required with an axicon lens.

(31) FIG. 5 illustrates a method according to the invention. A first laser source generates at least one first laser pulse directed toward the reflective aspheric surface 1. The laser beam 2 resulting from the at least one laser pulse is then reflected by the reflective surface 1 and the reflected beam 4 is focused over a long focal line. The interaction between the reflected laser beam 4 and the gas medium 23 provided by a nozzle 11 leads to the formation of a plasma waveguide channel 10. The reflective aspheric surface 1 has a hole formed in its center O so as to allow a laser beam 8 to pass therethrough. The laser beam 8 is obtained by reflecting a laser beam 6 generated by the first laser source or a second laser source using a reflective optical device 7 such as a parabola mirror. The laser beam 8 is then guided over the plasma channel 10. A set of two reflective elements 13 and of two focusing elements 15 and neutral density filters 19 may be used to attenuate the laser beam 8 at the exit of the gas medium 23 and measure its peak intensity profile at focus using a suitable measurement tool including a CCD 21 camera.

(32) A vacuum chamber 20 may be provided to host the above-mentioned elements. The vacuum chamber 20 can be set and maintained under vacuum using one or several vacuum pumps. The vacuum chamber 20 can be provided with a window 17 allowing the reflected beam 8 to leave the vacuum chamber 20.

(33) Focal spots obtained at the inlet of the gas medium and at its outlet with and without the plasma channel waveguide 10 are shown in FIGS. 6A-6c. Because of significant aberrations, the focal spot at the entrance of the gas medium 23 has a large part of the energy out of the central spot. Without waveguide, the laser beam 8 diverges in the plasma and its peak intensity is reduced at the plasma outlet (FIG. 6b). In contrast with the plasma channel waveguide 10, a small focal spot is obtained (FIG. 6c). Comparing FIG. 6b and FIG. 6c shows that the laser beam 8 is filtered spatially by the waveguide 10.

(34) FIG. 2 displays an example of focalization relying upon a combination of the optical system 100 and a second optics. This second optics may for example be a deformable mirror, a phase plate or a mirror with a spiral phase. Firstly, one first laser beam 2 is produced by a laser source and directed onto the optics 7. Next, the laser beam 2 which is provided from the optics 7 is directed towards the aspheric reflective surface 1. Finally, the reflected beam 21 is focused by the aspheric reflective surface 1 as explained above in the examples of FIGS. 1a, 1b and 1c.

(35) The reflective aspheric surface 1 or the optics 7 may be deformable, as shown in FIG. 8. The deformable reflective surface shape may be obtained by deforming the reflective surface with actuators 31 or otherwise. In the example of FIG. 8, the reflective aspheric surface 1 may comprise piezoelectric plates which are bonded together and are oppositely polarized (parallel to their axes). An array of electrodes 31 is deposited between the two plates. The front and back surfaces of the plates are connected to ground. The front surface acts as a reflective surface.

(36) When a voltage is applied to an electrode, if affects the shape of the plates.

(37) This variant is of particular interest for focusing a laser beam into a plasma or a gas for acceleration applications. The deformable surface allows to achieve wavefront control through surface deformation and to correct optical aberrations. Hence, one may obtain a laser beam 4 with less or none alignment error.

(38) The combination of the reflective aspheric surface 1 and the optics 7 may also introduce a dephasing in reflection with respect to the case where one reflective surface is used. This dephasing may depend on r, for example the dephasing may be proportional to r.sup.2, r.sup.4 or r.sup.6. By adding a dephasing which is a function of r, it is possible to control the velocity along the focal line. In other words it is possible to tune the two quantities independently. Without this radial dephasing, the focal line and the velocity along the focal line are coupled.

(39) The focusing optics 7 may include a cylindrical reflective optics, a spherical reflective optics or an aspheric reflective optics.

(40) FIG. 7 shows an example of a vacuum charged particles accelerator. It includes a high intense laser source 25, the reflective system 100, a particle source 24 and a vacuum chamber 20. The vacuum chamber 20 contains the laser system and the reflective system 100. In this setup, the laser source generates a laser pulse which is directed toward the optical system 100. A reflected beam 4 is then obtained thanks to the reflective system 100 and subsequently directed to the charged particles to accelerate them.

(41) The particle source 24 is used to introduce at least one particle into the vacuum chamber 20. The at least one particle is then accelerated by the reflected laser beam 4 within the vacuum chamber 20. In some applications, the at least one accelerated particle may be permitted to exit the vacuum chamber 20.

(42) The at least one introduced particle may be an electron or positron.

(43) The particle source 24 may be a plasma mirror or a metal target, wherein at least one laser pulse generated either by laser source 25 or by any other laser source interact with the target to generate a plasma comprising free electrons.

(44) The particle source 24 may be placed inside vacuum chamber 20 or may be placed outside of vacuum chamber 20.

(45) Vacuum chamber 20 may additionally include electronics and other devices for detecting and measuring properties of the at least one accelerated particle including its position, velocity and energy. The electronics and other devices may be disposed inside or outside chamber 20. As an example, chamber 20 may include photographic elements and a phosphorescent or scintillator screen to measure the at least one accelerated particle position. Alternatively, or in addition, chamber 20 may include detection elements or instruments such as a Faraday cup or other detection devices based upon the deflection of charged particles in a magnetic field.

(46) The injection by the particle source 21 into the vacuum chamber 20 allows the charged particles accelerated by the laser beam to reach energies that are many times greater than their initial energies. For example, electrons may be accelerated to energies up to 200 MeV, 600 MeV, 1000 MeV or more. The invention is not limited to the described embodiments, and various variations and modifications may be made without departing from its scope.