Abstract

In this paper, we present a systematic investigation of the characterization of tightly focused vector fields formed by an off-axis parabolic mirror. Based on the Stratton-Chu integral of Green’s theorem, the rigorous diffraction integrals that generate vector fields within and outside the focus arising from a collimated beam incident on an idealized parabolic mirror were derived in detail. In addition, explicit analytical expressions for the far-field vector diffraction electric and magnetic fields suitable for an off-axis parabolic mirror were also obtained. It is shown that there are significant differences in the vector diffraction characterizations between on- and off-axis parabolic mirrors. When the off-axis rate is greater than 4, the longitudinal field is predominant, and the maximum peak intensity ratio between the longitudinal field and the transverse field at the focus is approximately 103. This property is valuable for all applications in which a strong longitudinal field component is desirable. The effective focal length increases with increasing off-axis rate, and the depth of focus and the convergence angle of the focused beam are strongly dependent onf/2ω: smaller values of f/2ω lead to shorter focal depths and larger beam convergence (or tighter focusing).

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

As a focusing device, a parabolic mirror can perfectly collect collimated incident light—the axis of propagation of which is parallel to the axis of revolution of the mirror—into the focal point of the mirror without any chromatic aberration within the accuracy of the Gaussian optics, which is even capable of focusing light nearly within the 4π solid angle [1,2]. Owing to these distinctive features, parabolic mirrors have attracted the attention of increasing numbers of researchers over the past several decades and are widely used in astronomical telescope design [3], telecommunications [4], optical tweezers [5], confocal microscopy [6,7], and high-intensity laser systems [8–10]. When a beam is tightly focused by a parabolic mirror, the vector character of the light becomes crucial to correctly describe such a non-paraxial beam. Furthermore, the expressions of its electromagnetic field must satisfy Maxwell’s four equations beyond the paraxial regime. This analysis of the vector field focusing properties is of particular relevance to the high-intensity laser community, which expends significant effort to achieve the highest possible laser intensities. Due to the evolution of high-intensity laser systems, laser intensities in the focal spot of 1023–1024 W/cm2 have been achieved [10–15]. This highly focused laser intensity can be achieved by tightly focusing the beam in a vacuum using an off-axis parabolic mirror (OAP), which creates many exciting opportunities for particle laser acceleration [16,17], materials processing, and the study of fundamental forces in nature. To model such phenomena, an accurate accounting of the electric and magnetic vector field distributions throughout the region of tight focus is often critical [18].

A rigorous vector diffraction theory of tightly focused light from parabolic mirrors [9,19–28] has been developed over the course of nearly a century, beginning with a treatment in 1920 by Ignatovsky [19]. Ignatovsky first developed vector diffraction for a uniformly polarized collimated beam reflected from a parabolic mirror. Owing to the limitations of the contemporary computing power, he computed the intensity only in the focal plane. Another theoretical approach was provided by Richards and Wolf [29,30], by which strongly focused beams were accurately described, given the field distribution of the collimated input beam at the entrance pupil of the focusing system. In addition, another alternate method for analyzing tightly focused beams is based on the Stratton-Chu formulation of Green’s theorem [31], which involves both the electric and magnetic fields after substitutions from Maxwell’s equations.

To date, most investigations of vector diffraction field properties in the focal region of a parabolic reflector have mainly considered the case of an on-axis parabolic mirror [19–28]. The vector diffraction properties of an OAP have rarely been explored. However, Bahk and colleagues [9] derived the far-field vector diffraction formulae of the focal electromagnetic field formed by an OAP for a circular incident beam based on the Stratton-Chu integral formula [31]. However, they did not provide a detailed investigation of the vectorial diffraction properties of an OAP, likely because they were mainly concerned with the generation of an extreme power density by correcting the characterized aberrations. To the best of our knowledge, a detailed theoretical study of the vectorial field focusing properties of an off-axis parabolic mirror is not available in the literature, whether the incident beam is circular or square.

Seeking solutions to the abovementioned problems was the aim of this study. In Sections 2 and 3, we first outline the rigorous diffraction integrals that generate vector fields within and outside the focus arising from a collimated beam incident on an idealized parabolic mirror based on the Stratton-Chu integral. In Section 4, we derive the expressions of the far-field vector diffraction electromagnetic fields suitable for an OAP. In Section 5, we use the expression derived in Section 4 to investigate the characterization of tightly focused vector fields formed by an OAP, including tightly focused vector field component profiles in the focal plane, the three-dimensional (3D) intensity distribution in the vicinity of the focus, the focal depth and convergence angle of a highly confined optical field, and the effective focal length (EFL) of the OAP, among other characteristics.

2. General formulas of on-axis and off-axis parabolic mirror and total field

We first consider an ideal on-axis rotational parabolic mirror with its axis of revolution symmetry coinciding with the z axis and its focus coinciding with the origin of a Cartesian coordinate systemS(x,y,z), such that its surface intersects the z axis a distance f to the left of the origin, as shown in Fig. 1. The equation of the paraboloidal surface is then given by

zo=xo2+yo24ff,
where xo, yo, and zo are the transverse and longitudinal coordinates of any point O on the paraboloidal surface in the Cartesian system, respectively. fis the parent focal length of the parabolic mirror. In this paper, we assume that the incident wave is reflected only once, i.e., the field scattered off the surface leaves the paraboloid, since a parabolic mirror as a focusing device encounters this circumstance most commonly. The unit inward normal vector to this paraboloidal surface is expressed as
n^(xo,yo)=12f(xox^+yoy^)+z^1+xo2+yo24f2,
wherex^, y^, and z^ are the unit basis vectors of the Cartesian coordinates. For an OAP, the usable area of the paraboloidal surface is specified as
Σ:(xoh)2+yo2R2(foracircularoffaxissurface),
or
Σ:RxohRandRyoR(forasquareoffaxissurface),
where h is the distance from the z axis to the center of the incident beam and 2R is the greatest transverse width of the paraboloid. Equation (3) describes a circular off-axis paraboloidal surface and Eq. (4) describes a square one. We call h the offset. It is noted that the usable area with h=0 is an on-axis paraboloidal surface.

 

Fig. 1 Schematic illustrating reflection of off-axis parabolic mirror. (a) 3D view of paraboloid reflection and Cartesian coordinate systems. (b) Meridional section of off-axis parabolic mirror. Focus of off-axis parabolic mirror coincides with origin of Cartesian coordinate system.

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Next, we consider the total field and the field reflected by the parabolic mirror. Based on the electromagnetic boundary conditions, as discussed in the work of Varga and Tцrцk [25], upon reflection the normal components of the electric field and the tangential components of the magnetic field remain unchanged; that is

Er,n=Ei,n=n^(n^Ei),
Hr,t=Hi,t=HiHi,n,
where Ei and Hi denote the incident electric and magnetic fields, respectively. The normal components of the incident and reflected electric fields are denoted Ei,n and Er,n. Hi,t and Hr,t represent the tangential components of the incident and reflected magnetic fields, respectively. However, the tangential component Er,t of the electric field and the normal component Hr,nof the magnetic field change sign. Consequently, the tangential component of the reflected electric field may be written as
Er,t=Ei,t=(EiEr,n),
and the normal component of the reflected magnetic field may be written as
Hr,n=Hi,n=n^(n^Hi).
Using Eqs. (5)–(7), one obtains the total reflected field is expressed as
Er=2n^(n^Ei)Ei,
Hr=Hi2n^(n^Hi).
Therefore, the total electric field is given as the sum of the incident and reflected fields:
E=Ei+Er=2n^(n^Ei).
A similar procedure yields the following for the total magnetic field:

H=Hi+Hr=2Hi2n^(n^Hi).

3. Vector diffraction formulas for a parabolic mirror

In this section, we derive an expression for the vector diffraction field from a parabolic mirror, for which we use the Stratton-Chu integral formula [31]:

E(P)=14πSOAP[iωμ(n^×H)G+(n^×E)×G+(n^E)G]dA,
H(P)=14πSOAP[iωε(E×n^)G+(n^×H)×G+(n^H)G]dA,
where εand μ are called the capacitivity and inductivity of the medium, respectively. ω is the angular temporal frequency of the beam. The field component inside this integral is the sum of the incident and reflected fields. It is noted that the contour integral in the original Stratton-Chu integral formula is not included, because it is negligible for the continuous paraboloidal surface calculation. G=exp(ikrOP)/rOP, and the wave number k of the illumination is given by k=ωεμ=2π/λ, where λ is the wavelength of the incident beam. O(xo,yo,zo) is a point on the surface of the paraboloid, and P(xP,yP,zP)is a point on the observation plane. Hence
rOP=|rPrO|=[(xPxo)2+(yPyo)2+(zPzo)2]1/2.
G should be used to calculate the point O(xo,yo,zo), and using Eq. (15), G can then be expressed as
G=ik(11ikrOP)GrOP[(xoxP)x^+(yoyP)y^+(zozP)z^]=ik(11ikrOP)GrOP(Δxx^+Δyy^+Δzz^)=ik(11ikrOP)GrOPrOP
The paraboloid area element dAcan be written as
dA=[1+(zoxo)2+(zoyo)2]1/2dxodyo=(1+xo2+yo24f2)1/2dxodyo.
To calculate the vector diffraction field of monochromatic light from a parabolic mirror, we assume that the incoming beam propagates in the negative z direction, is linearly polarized, and does not have longitudinal components in the electric and magnetic fields, as shown in Fig. 1, such that the assumed incident field is given by
Ei=(Ψ0xx^+Ψ0yy^)exp(iωtikzi),
Hi=1η(Ψ0yx^Ψ0xy^)exp(iωtikzi),
where η=μ/ε is the intrinsic impedance of the medium, zi belongs to the position of the incident plane Ω, and Ψ0x and Ψ0y represent the spatial envelope of the light in the x and y directions, respectively. To find the incident field on the paraboloidal surface, one must multiply by an additional phase ik(zizo) from the plane Ω to the surface assuming that the diffraction effect is negligible. The incident plane Ω must be located close enough to the surface of the paraboloid to satisfy this assumption.

Substituting Eqs. (9)–(11) and (15)–(19) into (13) and (14), we obtain the vector diffraction field from a parabolic mirror:

E(P)=ikexp(iωt)2πSOAPexp(ikzo)dxodyo×{[(1(11ikrOP)xo2fΔxrOP)Ψ0xG(11ikrOP)yo2fΔxrOPΨ0yG]x^+[(1(11ikrOP)yo2fΔyrOP)Ψ0yG(11ikrOP)xo2fΔyrOPΨ0xG]y^+(xo2fΨ0x+yo2fΨ0y)[1(11ikrOP)ΔzrOP]Gz^}
H(P)=ikexp(iωt)2πSOAPexp(ikzo)η(11ikrOP)GrOPdxodyo×{[(Δyxo2f)Ψ0x+(xo2+yo24ffΔyyo2f)Ψ0y]x^+[(xo2+yo24ffΔxxo2f)Ψ0x+(Δxyo2f)Ψ0y]y^+(ΔyΨ0xΔxΨ0y)z^}
Since Eqs. (20) and (21) match the boundary condition at the mirror surface and obey Maxwell's equations exactly, they may be indisputably evaluated to find the vector diffraction fields anywhere within and outside the focus.

4. Vector diffraction formulas near the focus of an off-axis parabolic mirror

High-intensity laser systems can currently achieve a focused peak intensity of approximately 1022 W/cm2 [8,9] and will reach 1023–1024 W/cm2 in the near future [10–15]. The intensity range can be reached by tightly focusing the beam in a vacuum using an OAP [8–10]. This high focused laser intensity creates many exciting opportunities for laser-matter interactions in the relativistic and even ultra-relativistic regimes [15–18]. To model such interactions, an accurate and rapid accounting of the electric and magnetic vector field distribution throughout a tight focus is often critical.

Therefore, the aim in this section is to determine the vector diffraction formulas near the focus of an OAP. For this purpose, we simplify Eqs. (20) and (21) and make approximations. Considering a point P(xP,yP,zP) on the observation plane lying in the vicinity of focus, due to this focus coinciding with the origin of a Cartesian coordinate system, we make the following approximation: xP0, yP0, and zP0, which results in Δxxo, Δyyo, and Δzzo in Eqs. (16), (20), and (21). The reason for these approximate treatments is that the incident wave on a paraboloid is a plane wave propagating in a direction parallel to the optical axis and the reflected wave will converge toward the focus, which is readily verified by calculating the Poynting vector of the reflected wave:

S=Er×Hr=1η(Ψ0x2+Ψ0y2)exp[2i(ωt+kzo)](xorox^yoroy^zoroz^).
The Poynting vector of the reflected wave from paraboloidal reflection described by the Eq. (22) indeed points toward the focus.

For the second approximation, it is assumed that (ik1/rOP)ik. This approximation is justified if the field is at a distance many wavelengths from the paraboloidal surface.

The third approximation comprises the well-known Debye approximations, which state the following: a) The exponent of the Green’s function G can be approximately expressed as ikrOP=ik|rPrO|ik(rorO·rP/ro). Based on this approximation, spherical wavelets originating from the diffraction paraboloidal surface are replaced by plane wavelets. b) The denominator rOP of Gcan be replaced approximately by ro. On the paraboloidal surface, when zo is replaced by the Eq. (1), we have ro=f+(xo2+yo2)/4f.

Under the Debye and other two above-mentioned approximations, we can reduce Eqs. (20) and (21) to

EF(P)=ikexp(i2kfiωt)2πSOAPexp(ikrO·rP/ro)dxodyo×{[(1roxo22fro2)Ψ0xxoyo2fro2Ψ0y]x^,+[(1royo22fro2)Ψ0yxoyo2fro2Ψ0x]y^+(xoΨ0x+yoΨ0y)1ro2z^}
HF(P)=ikexp(i2kfiωt)2πSOAPexp(ikrO·rP/ro)ηdxodyo×{[(xoyo2fro2)Ψ0x+(xo2yo24fro2fro2)Ψ0y]x^.+[(xo2yo24fro2+fro2)Ψ0x+(xoyo2fro2)Ψ0y]y^+(yoΨ0xxoΨ0y)1ro2z^}
Equations (23) and (24) determine the vector diffraction electromagnetic field distributions near the focus of a parabolic mirror. The phase term in these integrals can be expressed as
ikrO·rP/ro=ik(xoroxPyoroyPzorozP),
where xo/ro, yo/ro, and zo/ro are called direction cosines (with positive directions in the direction of the light propagation) associated with the geometrical optical rays that reach the focus of a parabolic mirror. From Eq. (25), it is found that the observation plane of the numerical diffraction electromagnetic field distributions is either perpendicular (xy plane) or parallel (xz or yzplane) to the symmetrical axis of revolution of the paraboloidal surface, as will be seen below. As far as we know, this is what most of researchers have done [19–28]. Therefore, if one tries to analyze the focusing vector electromagnetic field structure after off-axis paraboloidal reflection in the observation plane, which is perpendicular to the propagation direction of the focused beam, one must build a new Cartesian coordinate system S(x,y,z) so the direction of propagation of the focused beam reflected by the paraboloidal surface coincides with the z axis of the new coordinate system S, as shown in Fig. 1. The new coordinate system S is obtained by rotating the coordinate system S by an angle φaround the y axis. As shown in Fig. 1(b), the rotation angle φ is defined as AFC, where the line CF¯ is an angular bisector such that BFC=CFE. It is noted that the line CF¯ does not coincide with the line DF¯ connected to the center ray of the incoming beam. We chose the line CF¯ as the alignment axis because it gives a symmetric range of direction cosines. Thus, the coordinates (xo,yo,zo)for a point O on the surface of the paraboloid in the S frame are transformed to (xocosφ+zosinφ, yo, xosinφ+zocosφ) in the S frame. Thus, the Eq. (25) can be rewritten as
ikrO·rP/ro=ik(xocosφ+zosinφroxPyoroyPxosinφ+zocosφrozP),
where xP,yP,zP are the coordinates of the point P near the focus of an OAP in the S frame, and
α=(xocosφ+zosinφ)/ro,
β=yo/ro,
γ=(xosinφ+zocosφ)/ro,
which are called direction cosines in the S frame.

Using Eqs. (27)–(29), we can obtain the inversion formulas from (α,β,γ) to (xo,yo,zo):

xo=2f(αcosφ+γsinφ)(1+αsinφ+γcosφ),
yo=2f(β)(1+αsinφ+γcosφ),
zo=2f(αsinφγcosφ)(1+αsinφ+γcosφ).
Considering a symmetric range of direction cosines, we use the direction cosines as integral variables. As a result, the Jacobian of the transformation can be expressed as
J=4f2γ(1+αsinφ+γcosφ)2.
Collecting our results so far, we may write the following expressions for the vector electromagnetic fields near the focus from the off-axis paraboloidal reflection in the S frame:
E'F(P)=ikexp(i2kfiωt)2πSOAP'Jexp[ik(αx'P+βyP+γz'P)]dαdβ×{[(1roxo22fro2)Ψ0xxoyo2fro2Ψ0y]x^'+[(1royo22fro2)Ψ0yxoyo2fro2Ψ0x]y^',+(xoΨ0x+yoΨ0y)1ro2z^'}
HF(P)=ikexp(i2kfiωt)2πSOAP'Jexp[ik(αxP+βyP+γzP)]ηdαdβ×{[(xoyo2fro2)Ψ0x+(xo2yo24fro2fro2)Ψ0y]x^+[(xo2yo24fro2+fro2)Ψ0x+(xoyo2fro2)Ψ0y]y^,+(yoΨ0xxoΨ0y)1ro2z^}
where x^, y^, and z^ are the unit basis vectors oriented along the new Cartesian axes x,y,z attached to the OAP, the origin of which is positioned at the focus of the mirror. It is necessary to note that, although the envelope functions are expressed in terms of direction cosines, they are not Fourier-transformed quantities.

5. Numerical vector diffraction field calculations

The vector diffraction electromagnetic field distributions of linearly polarized collimated light focused by a parabolic mirror have been computed for different values of the dimensionless parameters that characterize the optical system. The first parameter is defined by α=h/R, which quantifies the amount of the offset from the z axis to the center of the incoming beam. We call this parameter the off-axis rate. When h/R=0, the parabolic mirror is fully on-axis, i.e., the surface of the parabolic mirror under this circumstance is rotationally symmetric about the z axis. When 0<|h/R|<1, the parabolic mirror is partially off-axis. However, for |h/R|1, the parabolic mirror is entirely off-axis. It should be noted that the rotation angle φ of the coordinate system is closely related to this parameter, and is determined by 12[arctan((h+R)/z0|h+R)+arctan((hR)/z0|hR)]. The second dimensionless parameter is defined by f/2ω, where f is the effective focal length (EFL) of the OAP system and 2ω is the waist diameter of the incident beam. For larger values of f/2ω, the ratio is similar to the f-number of the optical system. Note that the EFL f is different from the parent focal length f of the parabolic mirror, as shown in Fig. 1. This parameter is related to the depth of the focus and the convergence angle of the focused beam: smaller values of f/2ω lead to a shorter focal depth and larger beam convergence (or tighter focusing). The EFL of the OAP system is different from that of the lens or on-axis parabolic mirror system, which increases with increasing off-axis rate α, as will be shown below.

We present profiles of the electromagnetic field intensities below, which are defined by IE|E|2=|Ex|2+|Ey|2+|Ez|2 and IH|H|2=|Hx|2+|Hy|2+|Hz|2. Since high-power laser beams generally have square cross-sections [32], we assume that the illuminating beam has a super-Gaussian top-hat beam profile of E0exp{[((xxcenter)/ωx0)2n+(yycenter)/ωy0)2n]}with polarization in the +xdirection, where E0 is constant amplitude and 2ωx0 and 2ωy0 are the beam waists in the x and y directions, respectively. For a square beam profile, ωx0=ωy0. xcenter and ycenter are the transverse coordinates of the center of the incoming beam. The wavelength λ of the incident beam is 1.053 µm. Unless otherwise stated, the size of the square incident beam is 320 mm Ч 320 mm, and the parent focal length of the parabolic mirror is 800 mm. It is beyond the scope of this paper to discuss the optimum values of the beam size and parent focal length of the parabolic mirror.

An animation showing the contour plots of the electromagnetic field intensity distribution of an aberration-free square focused super-Gaussian top-hat beam in the focal plane (the x'-y' plane with zP=0) by an OAP with different off-axis rates is presented in Fig. 2. The top four panels show |Ex|2, |Ey|2, |Ez|2, and |E|2, and the four lower panels show |Hx|2, |Hy|2, |Hz|2, and |H|2 in the x'-y' plane with zP=0. The intensity is indicated by the “heat” of the color, which goes from black (characterizing the minimum intensity) to red, orange, yellow, and finally, white (characterizing the maximum intensity). As the animation progresses, the off-axis rate α is varied from 0 to 22.5, or the offset h is varied from 0 to 3600 mm. This animation reveals the following: 1) for the intensity distribution of the field component|Ex|2, which possesses lobes of relatively weak amplitude surrounding a central high amplitude peak, the width of the central peak of the intensity distribution and the side-lobe distributions shown in the focal plane along the x' and y' directions widen with increasing off-axis rate αor offset h, except for the intensity distribution in the vicinity of α=2f/R(the rotation angle φ=π/2). 2) For the intensity distribution of the field component|Ez|2, which is very similar to that of |Ex|2, the width of the central peak and that of the side-lobe distributions shown in the focal plane along the x' and y' directions also widen as the off-axis rate αor offset h increases, except in the range 0α<0.4. 3) For the intensity distribution of the field component |Ey|2, which exhibits a linear dark center along the x'-axis direction, i.e., a local minimum of intensity will occur along the x' axis with y' = 0, the intensity distribution exhibits bilateral symmetry with respect to the x' and y' axes. Except for the 0α<0.4 region, two peak intensities are symmetrically distributed along the y' axis in the first ring of main lobes around the dark core, and the width of the linear dark center and the width of the side-lobe distributions shown in the focal plane along the y' direction widen with increasing off-axis rate α or offset h. These results can be seen more clearly in Fig. 3. In addition, we can also see the influence of the intensity distribution of magnetic field components on the off-axis rate α. However, at optical frequencies, the magnetic field intensity is negligible compared to the electric field intensity, and the electric field is primarily responsibility for interacting with matter. Hence, in the following section, we mainly focus on the electric field.

 

Fig. 2 Animation showing contour plots of electromagnetic field intensity distribution of focused square super-Gaussian top-hat beam polarized along + x direction by OAP for different off-axis rates (see Visualization 1). Fields are computed in focal plane (x'-y' plane with zP=0). Dashed lines show transverse locations of contours that are plotted in Fig. 3. Intensity is indicated by the “heat” of the color. Numerical computations based on Eqs. (34), (35), and (4).

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Fig. 3 Contour plots of electric field intensity distribution along dashed transverse locations shown in Fig. 2 as function of off-set h. (a) |Ex|2 along x' axis for 0h3600mm, (b) |Ez|2 along x' axis for 0h3600mm, (c) |Ey|2 along y' axis for 0h3600mm, (d) |Ex|2 along x' axis for 1400h1800mm(or in the vicinity of the rotation angle φ=π/2), and (e) |Ez|2 along x' axis for 0h100mm (or 0α0.6). Intensity is indicated by the “heat” of the color. Numerical computations based on Eqs. (34), (35), and (4).

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Figure 3 shows the dependence of the transverse intensity distribution in the focal plane on the offset h. The intensity distribution of |Ex|2, |Ez|2 and |Ey|2 along the dashed lines shown in Fig. 2 are plotted as functions of the offset h in Figs. 3(a), 3(b), and 3(c), respectively. It is noted that the intensity along the dashed lines shown in Fig. 2 for each offset h is continuously rescaled and normalized to the corresponding maximum amplitude. This scaling allows the relatively weak intensity distribution to be plotted on the same scale as the high-intensity distribution for clarity. As shown in Fig. 3, the size of the focal spot and the width of the side-lobe distributions increase with increasing off-axis rate α or offset h, excluding the special intensity distribution characteristics shown in the above three cases. For the intensity distribution of |Ey|2 in the region 0h<64mm (or 0α<0.4), although it has four peak intensities in the focal plane, the intensity has a very small value along the y' axis (x' = 0). To show the extraordinary features in the other two cases more clearly, we enlarged the intensity distributions for |Ey|2 in the region 1400mmh1800mm (in the vicinity of α=2f/R) shown in Fig. 3(a) and for |Ez|2 in the region 0h100mm (or 0α0.6) shown in Fig. 3(b), which are shown in Figs. 3(d) and 3(e), respectively. From the result shown in Fig. 3(d), it can be seen that the electric field intensity |Ex|2 in the center decreases and the dark center gradually appears with increasing h, until h = 1608 mm (the rotation angleφ=π/2). As the value of h increases, the electric field intensity is enhanced, and the dark center gradually disappears (also see Fig. 4). However, from the result shown in Fig. 3(e), it can be seen that the intensity distribution of |Ez|2 in the center is small compared to its maximum value, providing an intensity distribution with a dark center; however, as the value of h increases, the electric field intensity is enhanced (also see Fig. 4) and the dark center gradually disappears.

 

Fig. 4 Peak intensity of electric field components of focused square super-Gaussian top-hat beam by OAP as function of (a) off-axis rate α and (b) rotation angle φ. Dependence of effective focal length (EFL) f of OAP system on rotation angle is also shown in (b). Fields are computed in focal plane (x'-y' plane with zP=0). A logarithmic scale for intensity of field components (|Ex|2,|Ey|2, and|Ez|2) is used in (a), but an absolute value scale for intensity of field components is used in (b). Cyan dashed line shows |Ez|2=|Ex|2 case, magenta dash-dotted line location of the maximum peak intensity of |Ez|2 and |Ey|2, and olive dash-dot-dotted line location of the maximum dark center of |Ex|2.

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Figure 4 shows the peak intensity of the electric field components|Ex|2, |Ey|2, and |Ez|2for the focused square super-Gaussian top-hat beam polarized along the + x direction by the OAP as a function of the off-axis rate α [see Fig. 4(a)] and the rotation angle φ [see Fig. 4(b)]. The dependence of the EFL f of the OAP system on the rotation angle is also presented in the Fig. 4(b). It is noted that a logarithmic scale for the intensity of field components (|Ex|2, |Ey|2, and|Ez|2) is used in Fig. 4(a), but an absolute value scale for the intensity of field components is used in Fig. 4(b). From the results shown in Fig. 4, it is easily seen that, based on the overall variation trend, the peak intensity of |Ex|2 decreases with increasing off-axis rate α or rotation angle φ, of course, excluding the region in the vicinity of α=2f/R(in this circumstance, the rotation angle φ=π/2). For a rotation angle of approximatelyπ/2, there is a process in which the peak intensity of |Ex|2 first decreases and then increases as α or φ increases. This observation can be understood by the fact that a central main lobe is split into two lobes first, and then the two main lobes are combined into one with increasing off-axis rate α or rotation angle φ, which is verified by Figs. 2, 3(a), and 3(d). These figures also show that when α>6 (orφ>1.07rad), the peak intensities of |Ey|2 and |Ez|2decrease with increasing α or φ. However, for α<6 (or φ<1.07rad), their peak intensities increase. Furthermore, the OAP with α4(in this circumstance, the rotation angle φπ/4) produces |Ex|2 and |Ez|2 with comparable peak intensities. For α<4 (or φ<π/4), the field component |Ex|2 is predominant. However, when α>4 (or φ>π/4), the longitudinal field |Ez|2 is predominant, and the maximum peak intensity ratio between the longitudinal field and transverse fields at the focus is approximately 103. This property is valuable for all applications in which a strong longitudinal field component is desirable, e.g., (1) for generation of plasma channels [33], or (2) for particle acceleration [16,17]. In addition, we can see that the EFL f increases with increasing rotation angle φ(or the off-axis rate α).

To comprehensively understand the vectorial diffraction properties of the electric field near the focus, the 3D intensity distribution of a focused square super-Gaussian top-hat beam by an OAP for α=1.844 (in this circumstance the rotation angle φ=0.36rad) is shown in Fig. 5. To clearly show the focused field 3D intensity distributions along the propagating direction, we plot the intensity distributions in the yzplane with x=0 and cross-sections (xyplanes) with different values of z before and after the focal plane (z=0). Figure 5(a) presents the intensity profile of the total diffraction electric field Eon the y'-z' plane (x=0) and the transverse intensity cross-sections at z' = 40λ,20λ,0λ,20λ, and 40λ. The intensity distribution exhibits a central square main spot and bilateral symmetric off-axis rectangular side lobes. Clearly, the central main spot is formed by constructive interference on the optical axis z and the appearance of off-axis side lobes is caused by the diffraction effect. This figure clearly reveals the diffraction characteristics of the electric field in the vicinity of the focus. In addition, the focused electromagnetic fields indeed propagate along the z axis and not the z axis [also see Fig. 5(b)]. In the focal plane, the intensity distribution is very sharp, but it becomes increasingly more blurred far from the focal plane.

 

Fig. 5 3D intensity distribution of total diffraction electric field of focused square super-Gaussian top-hat beam by OAP with α=1.844 (rotation angleφ=0.36rad). (a) |E|2in coordinate system S(x,y,z). (b) |E|2 in coordinate system S(x,y,z). Intensity is indicated by the “heat” of the color. Dashed line in (b) represents propagation direction of focused field.

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To clearly reveal the differences in the vectorial diffraction characterizations of the electric field in the vicinity of the focus in the coordinate system S(x,y,z) and the new coordinate system S(x,y,z), the 3D intensity distributions in the coordinate system S(x,y,z)for the focused square super-Gaussian top-hat beam by an OAP with α=1.844 are shown in Fig. 5(b). This numerical computation is based on Eqs. (23), (24), and (4), and was obtained using the same parameters used in Fig. 5(a), apart from the coordinate system. Here, the electric field intensities are defined by IE|E|2=|Ex|2+|Ey|2+|Ez|2. Discrepancies in the electric field intensity distributions between the coordinate systems S(x,y,z) and S(x,y,z) are apparent. The first difference is that the intensity distribution in the coordinate system S has lost bilateral symmetry. An extended tail develops along the +xaxis before the focal plane. However, after the focal plane, this behavior is the opposite, i.e., the extended tail develops along the xaxis. This result can be explained by the fact that the electric energy converges to the focus on the z axis from the +x direction, while after the focal plane it diverges from the focus to the x direction deviating from the z axis. The second difference is that the focused electromagnetic fields propagate along the yellow short dashed line shown in Fig. 5(b), not the zaxis. Thus, the propagation of the electromagnetic field in the coordinate system S is non-paraxial. It should be noted that the angle between the dashed line and the z axis is the rotation angle φ of the coordinate previously defined, such that analytical expressions (34) and (35) are more suitable for characterizing the vectorial diffraction fields focused by an OAP than Eqs. (23) and (24) are.

In the following section, we consider the main features in the behavior of the depth of focus and the size of the central focal spot of the focused beam as a function of f/2ω. Figure 6 presents the contour plots of the electric field intensity distribution along the z axis (x=0,y=0,100λz100λ) of a focused square super-Gaussian top-hat beam with a fixed ω by an OAP as function of off-set h. The contour of the electric field intensity distribution |Ex|2 along the zaxis is plotted in Fig. 6(a) and for |Ez|2 in Fig. 6(b). Each profile is on the z axis (y=0), such that |Ey|2=0(see Figs. 2, 3, and 5). Note that the intensity along the z axis for each offset h is continuously rescaled and normalized to the corresponding maximum amplitudes. Although Fig. 6 shows only the dependence of the intensity distribution along the z axis on offset h, it indirectly manifests the behavior of the depth of focus as a function of f/2ω. This is because the intensity distribution along the z axis characterizes the depth of focus, and the EFL f is closely related to offset h. As shown in Fig. 4, a larger value of h leads to a longer EFLf. Thus, from these figures, we readily find that the depth of focus increases with increasing f/2ω (ωis fixed), i.e., smaller values of f/2ω lead to a shorter focal depth. It is necessary to note that a dark center appears in the intensity distribution of |Ex|2 for h = 1608 mm (the rotation angle φ=π/2), as shown in Fig. 6(a), which means that focal depth does not exist due to the appearance of the dark center.

 

Fig. 6 Contour plots of electric field intensity distribution along z' axis (or depth of focus) of focused square super-Gaussian top-hat beam by OAP as function of h. (a) |Ex|2 for 0h3600mm(or 2.5f/2ω15.2) andx=0,y=0 and (b) |Ez|2 for 0h3600mm(or 2.5f/2ω15.2) andx=0,y=0. Intensity is indicated by the “heat” of the color.

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To further demonstrate the dependences of the depth of focus and the size of the central focal spot of the focused beam on f/2ω, Fig. 7 presents contour plots of the electric field intensity distributions |Ex|2 and |Ez|2 along the x and z axes of a focused square super-Gaussian top-hat beam by an OAP with fixed EFL f=1088mm(corrodingh=960mm) as a function of f/2ω. For clarity, the intensity is continuously rescaled and normalized to the corresponding maximum amplitudes, which is indicated by the “heat” of the color. Figures 7(a) and 7(b) show contour plots of the electric field intensity distributions |Ex|2 and |Ez|2 along the x axis (y=0) of the focused beam as function of f/2ω, respectively. These electric field intensity distributions are computed in the focal plane z=0. These figures clearly reveal that the size of the central focal spot of the focused beam decreases with decreasing f/2ω, until it reaches the diffraction limit; that is, smaller values of f/2ω lead to tighter focus. Figures 7(c) and 7(d) show contour plots of the electric field intensity distributions |Ex|2 and |Ez|2 along the z axis (x=0,y=0) of the focused beam as a function of f/2ω, respectively. Therefore, the depth of focus increases with increasing f/2ω.

 

Fig. 7 Contour plots of focused electric field intensity distribution in focal plane along x' and z' axes as function off/2ω (fis fixed). (a) |Ex|2 along x' axis for 0.42f/2ω50 and y=0,z=0, (b) |Ez|2 along x' axis for 0.42f/2ω50 and y=0,z=0, (c) |Ex|2 along z' axis for 0.42f/2ω50andx=0,y=0, and (d) |Ez|2 along z' axis for 0.42f/2ω50 andx=0,y=0. Intensity is indicated by the “heat” of the color.

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Finally, to provide a comparison of the vectorial diffraction field properties formed by an OAP for square and circular incident beams, a movie showing the contour plots of the electromagnetic field intensity distributions near the focus of an aberration-free circularly focused super-Gaussian top-hat beam in the focal plane (the x'-y' plane with zP=0) by an OAP with different off-axis rates is presented as Fig. 8. The top four cells plot |Ex|2, |Ey|2, |Ez|2, and |E|2, and the four lower cells plot |Hx|2, |Hy|2, |Hz|2, and |H|2 in the x'-y' plane with zP=0. The intensity is indicated by the “heat” of the color. Numerical computations are based on Eqs. (34), (35), and (3). The vector diffraction field properties formed by an OAP for a circular incident beam have been investigated and discussed by many researchers [19–28]. However, only in the work of Bahk and associates was the focal vector field formed by an OAP for a circular incident beam considered. Nonetheless, these researchers did not provide a detailed investigation of the intensity distribution of a focused circular beam by an OAP for different off-axis rates [8]. We present this analysis here for direct comparison with the intensity distributions in the focal region for a square incident beam. From the results shown in Fig. 8, one can see the following: 1) For the intensity distribution of the field component |Ex|2, which possesses a circular main spot and concentric ring-shaped side-lobes, the size of the central circular main spot and width of the side-lobe distributions progressively increase with increasing αor h, except for the intensity distribution in the vicinity of α=2f/R. 2) For the intensity distribution of the field component |Ez|2, which is similar to that of |Ex|2, the width of the central circular main spot and width of the side-lobe distributions also widen as αor h increases, except in the range 0α<0.6. 3) For the intensity distribution of the field component |Ey|2, which exhibits a linear dark center along the x'-axis direction, the intensity distribution exhibits bilateral symmetry with respect to the x' and y' axes. Except for the 0α<0.4range, two peak intensities are symmetrically distributed along the y' axis in the first ring side-lobes around the dark core, and the width of the linear dark center and that of the side-lobe distributions widen with increasing αor h. 4) The intensity distribution changes in the focused vector field components |Ex|2 before and after α=2f/R(or φ=π/2), |Ez|2for 0α<0.6, and |Ey|2 for 0α<0.4 exhibit behaviors similar to those of the vector-focused field components for a square incident beam. Moreover, we can also see the influence of the intensity distribution of the magnetic field components on α. By comparing the vectorial electromagnetic field intensity distribution properties for circular and square incident beams, it can be determined that the influence of α on the circular field intensity distribution properties is the same as in the case with a square incident beam, other than a different focal spot pattern. In addition, we observed that the numerical intensity distribution of the vector field components formed by an on-axis parabolic mirror based on Eqs. (34), (35), and (3) is identical to that based on other rigorous vector diffraction theories [19–28].

 

Fig. 8 Movie showing contour plots of electromagnetic field intensity distribution of focused circular super-Gaussian top-hat beam polarized along + x direction by OAP for different off-axis rates (see Visualization 2). Fields are computed in focal plane (x'-y' plane with zP=0). Intensity is indicated by the “heat” of the color. Numerical computations based on Eqs. (34), (35), and (3).

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6. Conclusions

In this paper, we presented an exact analytical approach for the description of the vector electromagnetic fields formed by an on- and off-axis paraboloidal mirror for circular and square incident beams based on the Stratton-Chu integral. General analytical expressions for the spatial distributions of electromagnetic field components anywhere within and outside of the focus were derived in detail. Using several approximations, including the well-known Debye approximations, the far-field vector diffraction formulas suitable for an OAP were obtained from the general analytical expressions derived herein.

According to our rigorous vector diffraction theory, the characterizations of the tightly focused vector fields formed by off- and on-axis parabolic mirrors for square and circular incident beams with a super-Gaussian top-hat profile polarized along the positive x direction were analyzed and discussed in detail. The results show that there are significant differences in vector diffraction characterizations between on- and off-axis parabolic mirrors. For the intensity distribution of the field component |Ex|2, which possesses lobes of relatively weak amplitude surrounding a central high-amplitude peak, the width of the central peak of the intensity distribution and that of the side-lobe distributions in the focal plane along the x' and y' directions widen with an increasing off-axis rate α. However, before and after α=2f/R (orφ=π/2), the intensity of |Ex|2 in the center decreases, and the dark center gradually appears with increasing α, until α=2f/R. As the value of αfurther increases, the field intensity is enhanced, and the dark center gradually disappears. For the intensity distribution of |Ez|2, which also possesses lobes of relatively weak amplitude surrounding a central high-amplitude peak, the width of the central peak and that of the side-lobe distributions shown in the focal plane also widen as α increases overall. However, for 0α<0.6, the intensity distribution of |Ez|2has a dark center, and as the value of α increases, the center intensity is enhanced and the dark center gradually disappears. For the intensity distribution of |Ey|2, which exhibits a linear dark center along the x'-axis direction, the intensity distribution exhibits bilateral symmetry with respect to the x' and y' axes. Apart from 0α<0.4, two peak intensities are symmetrically distributed along the y' axis in the first-ring main lobes around the dark core, and the width of the linear dark center and that of the side-lobe distributions shown in the focal plane along the y' direction widen with increasing α. Although the intensity distribution of |Ey|2 has four peak intensities in the focal plane for 0α<0.4, the intensity has a very small value along the y' axis (x' = 0). In addition, the influence of the intensity distribution of the magnetic field components on αwas clearly shown.

We found from our calculations that, for α<4 (or φ<π/4), the field component |Ex|2 is predominant. However, when α>4 (or φ>π/4), the longitudinal field |Ez|2 is predominant, and the maximum peak intensity ratio between the longitudinal field and transverse fields at the focus is approximately 103. This property is valuable for all applications in which a strong longitudinal field component is desirable.

In addition, we explored the focal depth and convergence angle of a highly confined optical field as well as the EFL of the OAP. We found that the EFL increases with increasing off-axis rate α, and the depth of focus and convergence angle of the focused beam are strongly dependent on f/2ω: smaller values of f/2ω lead to shorter focal depths and larger beams convergence (or tighter focusing).

Funding

Fujian Provincial Natural Science Foundation of China (2013J05095, 2017J01560); Science Foundation for the Youth Scholars of Minjiang University (Mj9n201602); National Science and Technology Major Project of the Ministry of Science and Technology of China.

References

1. N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008). [CrossRef]  

2. A. April and M. Piché, “4π Focusing of TM(01) beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010). [CrossRef]   [PubMed]  

3. R. N. Wilson, Reflecting Telescope Optics (Springer, 2004).

4. U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006). [CrossRef]  

5. F. Merenda, J. Rohner, J. M. Fournier, and R. P. Salathé, “Miniaturized high-NA focusing-mirror multiple optical tweezers,” Opt. Express 15(10), 6075–6086 (2007). [CrossRef]   [PubMed]  

6. A. Drechsler, M. Lieb, C. Debus, A. Meixner, and G. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9(12), 637–644 (2001). [CrossRef]   [PubMed]  

7. M. Lieb and A. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef]   [PubMed]  

8. S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004). [PubMed]  

9. S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

10. T. M. Jeong, S. Weber, B. Le Garrec, D. Margarone, T. Mocek, and G. Korn, “Spatio-temporal modification of femtosecond focal spot under tight focusing condition,” Opt. Express 23(9), 11641–11656 (2015). [CrossRef]   [PubMed]  

11. G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006). [CrossRef]  

12. G. Mourou and T. Tajima, “Physics. More intense, shorter pulses,” Science 331(6013), 41–42 (2011). [CrossRef]   [PubMed]  

13. A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011). [CrossRef]  

14. T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526(3-4), 157–172 (2014). [CrossRef]  

15. N. V. Zamfir, “Nuclear Physics with 10PW laser beams at extreme light infrastructure–nuclear physics (ELI-NP),” Eur. Phys. J. Spec. Top. 223(6), 1221–1227 (2014). [CrossRef]  

16. K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15(1), 013108 (2008). [CrossRef]  

17. W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006). [CrossRef]  

18. G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016). [CrossRef]  

19. V.S.Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans.Opt.Inst. Petrograd 1, paper 5 (1920).

20. C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977). [CrossRef]  

21. R. Barakat, “Diffracted electromagnetic fields in the neighborhood of the focus of a paraboloidal mirror having a central obscuration,” Appl. Opt. 26(18), 3790–3795 (1987). [CrossRef]   [PubMed]  

22. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004). [CrossRef]   [PubMed]  

23. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008). [CrossRef]   [PubMed]  

24. A. April, P. Bilodeau, and M. Piché, “Focusing a TM(01) beam with a slightly tilted parabolic mirror,” Opt. Express 19(10), 9201–9212 (2011). [CrossRef]   [PubMed]  

25. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000). [CrossRef]   [PubMed]  

26. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A 17(11), 2090–2095 (2000). [CrossRef]   [PubMed]  

27. J. Peatross, M. Berrondo, D. Smith, and M. Ware, “Vector fields in a tight laser focus: comparison of models,” Opt. Express 25(13), 13990–14007 (2017). [CrossRef]   [PubMed]  

28. A. Couairon, O. G. Kosareva, N. A. Panov, D. E. Shipilo, V. A. Andreeva, V. Jukna, and F. Nesa, “Propagation equation for tight-focusing by a parabolic mirror,” Opt. Express 23(24), 31240–31252 (2015). [CrossRef]   [PubMed]  

29. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959). [CrossRef]  

30. B. Richards and E. Wolf, “Electromagnetic di_raction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]  

31. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939). [CrossRef]  

32. K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007). [CrossRef]  

33. N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008). [CrossRef]  

References

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  1. N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008).
    [Crossref]
  2. A. April and M. Piché, “4π Focusing of TM(01) beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010).
    [Crossref] [PubMed]
  3. R. N. Wilson, Reflecting Telescope Optics (Springer, 2004).
  4. U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
    [Crossref]
  5. F. Merenda, J. Rohner, J. M. Fournier, and R. P. Salathé, “Miniaturized high-NA focusing-mirror multiple optical tweezers,” Opt. Express 15(10), 6075–6086 (2007).
    [Crossref] [PubMed]
  6. A. Drechsler, M. Lieb, C. Debus, A. Meixner, and G. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9(12), 637–644 (2001).
    [Crossref] [PubMed]
  7. M. Lieb and A. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001).
    [Crossref] [PubMed]
  8. S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
    [PubMed]
  9. S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
  10. T. M. Jeong, S. Weber, B. Le Garrec, D. Margarone, T. Mocek, and G. Korn, “Spatio-temporal modification of femtosecond focal spot under tight focusing condition,” Opt. Express 23(9), 11641–11656 (2015).
    [Crossref] [PubMed]
  11. G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
    [Crossref]
  12. G. Mourou and T. Tajima, “Physics. More intense, shorter pulses,” Science 331(6013), 41–42 (2011).
    [Crossref] [PubMed]
  13. A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
    [Crossref]
  14. T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526(3-4), 157–172 (2014).
    [Crossref]
  15. N. V. Zamfir, “Nuclear Physics with 10PW laser beams at extreme light infrastructure–nuclear physics (ELI-NP),” Eur. Phys. J. Spec. Top. 223(6), 1221–1227 (2014).
    [Crossref]
  16. K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15(1), 013108 (2008).
    [Crossref]
  17. W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
    [Crossref]
  18. G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
    [Crossref]
  19. V.S.Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans.Opt.Inst. Petrograd 1, paper 5 (1920).
  20. C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977).
    [Crossref]
  21. R. Barakat, “Diffracted electromagnetic fields in the neighborhood of the focus of a paraboloidal mirror having a central obscuration,” Appl. Opt. 26(18), 3790–3795 (1987).
    [Crossref] [PubMed]
  22. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004).
    [Crossref] [PubMed]
  23. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
    [Crossref] [PubMed]
  24. A. April, P. Bilodeau, and M. Piché, “Focusing a TM(01) beam with a slightly tilted parabolic mirror,” Opt. Express 19(10), 9201–9212 (2011).
    [Crossref] [PubMed]
  25. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000).
    [Crossref] [PubMed]
  26. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A 17(11), 2090–2095 (2000).
    [Crossref] [PubMed]
  27. J. Peatross, M. Berrondo, D. Smith, and M. Ware, “Vector fields in a tight laser focus: comparison of models,” Opt. Express 25(13), 13990–14007 (2017).
    [Crossref] [PubMed]
  28. A. Couairon, O. G. Kosareva, N. A. Panov, D. E. Shipilo, V. A. Andreeva, V. Jukna, and F. Nesa, “Propagation equation for tight-focusing by a parabolic mirror,” Opt. Express 23(24), 31240–31252 (2015).
    [Crossref] [PubMed]
  29. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
    [Crossref]
  30. B. Richards and E. Wolf, “Electromagnetic di_raction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
    [Crossref]
  31. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
    [Crossref]
  32. K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
    [Crossref]
  33. N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
    [Crossref]

2017 (1)

2016 (1)

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

2015 (2)

2014 (2)

T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526(3-4), 157–172 (2014).
[Crossref]

N. V. Zamfir, “Nuclear Physics with 10PW laser beams at extreme light infrastructure–nuclear physics (ELI-NP),” Eur. Phys. J. Spec. Top. 223(6), 1221–1227 (2014).
[Crossref]

2011 (3)

G. Mourou and T. Tajima, “Physics. More intense, shorter pulses,” Science 331(6013), 41–42 (2011).
[Crossref] [PubMed]

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

A. April, P. Bilodeau, and M. Piché, “Focusing a TM(01) beam with a slightly tilted parabolic mirror,” Opt. Express 19(10), 9201–9212 (2011).
[Crossref] [PubMed]

2010 (1)

2008 (4)

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008).
[Crossref]

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15(1), 013108 (2008).
[Crossref]

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[Crossref] [PubMed]

2007 (2)

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

F. Merenda, J. Rohner, J. M. Fournier, and R. P. Salathé, “Miniaturized high-NA focusing-mirror multiple optical tweezers,” Opt. Express 15(10), 6075–6086 (2007).
[Crossref] [PubMed]

2006 (3)

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[Crossref]

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

2005 (1)

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

2004 (2)

2001 (2)

2000 (2)

1987 (1)

1977 (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977).
[Crossref]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic di_raction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

1939 (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Andreeva, V. A.

April, A.

Bahk, S. W.

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Barakat, R.

Berrondo, M.

Bilodeau, P.

Bokor, N.

Borot, A.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Brase, J. M.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Bulanov, S. V.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[Crossref]

Bychenkov, V. Yu.

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15(1), 013108 (2008).
[Crossref]

Choi, W.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977).
[Crossref]

Chu, L. J.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Chvykov, V.

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Combs, R. L.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Couairon, A.

Davidson, N.

Debus, C.

Drechsler, A.

Esarey, E.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Estan, C.

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

Fochs, S. N.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Fournier, J. M.

Freudenreich, M.

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

Gallet, V.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Gannaway, J.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977).
[Crossref]

Geddes, C. G. R.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Gobert, O.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Gonoskov, A. A.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

Gonsalves, A. J.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Hafz, N. A. M.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Hong, K. H.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Hooker, S. M.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Hurd, R. L.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Jeong, T. M.

T. M. Jeong, S. Weber, B. Le Garrec, D. Margarone, T. Mocek, and G. Korn, “Spatio-temporal modification of femtosecond focal spot under tight focusing condition,” Opt. Express 23(9), 11641–11656 (2015).
[Crossref] [PubMed]

T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526(3-4), 157–172 (2014).
[Crossref]

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Jukna, V.

Kalintchenko, G.

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Khazanov, E. A.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

Ko, D. K.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Korn, G.

Korzhimanov, A. V.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

Kosareva, O. G.

Kulagin, V. V.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

LaFortune, K. N.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Le Garrec, B.

Lee, J.

T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526(3-4), 157–172 (2014).
[Crossref]

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Lee, S. K.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Leemans, W. P.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Lieb, M.

Maksimchuk, A.

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Margarone, D.

Meixner, A.

Meixner, A. J.

Merenda, F.

Mescheder, U. M.

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

Mocek, T.

Mourou, G.

G. Mourou and T. Tajima, “Physics. More intense, shorter pulses,” Science 331(6013), 41–42 (2011).
[Crossref] [PubMed]

Mourou, G. A.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[Crossref]

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Nagler, B.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Nakamura, K.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Nesa, F.

Olivier, S. S.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Pae, K. H.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Panov, N. A.

Pariente, G.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Pax, P. H.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Peatross, J.

Piché, M.

Planchon, T. A.

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Popov, K. I.

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15(1), 013108 (2008).
[Crossref]

Quéré, F.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic di_raction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Rohner, J.

Rotter, M. D.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Rousseau, P.

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Rozmus, W.

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15(1), 013108 (2008).
[Crossref]

Salathé, R. P.

Schroeder, C. B.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Sergeev, A. M.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977).
[Crossref]

Shipilo, D. E.

Smith, D.

Somogyi, G.

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

Stadler, J.

Stanciu, C.

Stratton, J. A.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Stupperich, C.

Sung, J. H.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Sydora, R. D.

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15(1), 013108 (2008).
[Crossref]

Tajima, T.

G. Mourou and T. Tajima, “Physics. More intense, shorter pulses,” Science 331(6013), 41–42 (2011).
[Crossref] [PubMed]

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
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Supplementary Material (2)

NameDescription
» Visualization 1       Animation showing the contour plots of the electromagnetic field intensity distribution of a focused square super-Gaussian top-hat beam polarized along the +x direction by an off-axis parabolic mirror for different off-axis rate.
» Visualization 2       Movie showing the contour plots of the electromagnetic field intensity distribution of a focused circular super-Gaussian top-hat beam polarized along the +x direction by an off-axis parabolic mirror for different off-axis rate.

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Figures (8)

Fig. 1
Fig. 1 Schematic illustrating reflection of off-axis parabolic mirror. (a) 3D view of paraboloid reflection and Cartesian coordinate systems. (b) Meridional section of off-axis parabolic mirror. Focus of off-axis parabolic mirror coincides with origin of Cartesian coordinate system.
Fig. 2
Fig. 2 Animation showing contour plots of electromagnetic field intensity distribution of focused square super-Gaussian top-hat beam polarized along + x direction by OAP for different off-axis rates (see Visualization 1). Fields are computed in focal plane (x'-y' plane with z P = 0 ). Dashed lines show transverse locations of contours that are plotted in Fig. 3. Intensity is indicated by the “heat” of the color. Numerical computations based on Eqs. (34), (35), and (4).
Fig. 3
Fig. 3 Contour plots of electric field intensity distribution along dashed transverse locations shown in Fig. 2 as function of off-set h. (a) | E x | 2 along x' axis for 0 h 3600 mm , (b) | E z | 2 along x' axis for 0 h 3600 mm , (c) | E y | 2 along y' axis for 0 h 3600 mm , (d) | E x | 2 along x' axis for 1400 h 1800 mm (or in the vicinity of the rotation angle φ = π / 2 ), and (e) | E z | 2 along x' axis for 0 h 100 mm (or 0 α 0.6 ). Intensity is indicated by the “heat” of the color. Numerical computations based on Eqs. (34), (35), and (4).
Fig. 4
Fig. 4 Peak intensity of electric field components of focused square super-Gaussian top-hat beam by OAP as function of (a) off-axis rate α and (b) rotation angle φ . Dependence of effective focal length (EFL) f of OAP system on rotation angle is also shown in (b). Fields are computed in focal plane (x'-y' plane with z P = 0 ). A logarithmic scale for intensity of field components ( | E x | 2 , | E y | 2 , and | E z | 2 ) is used in (a), but an absolute value scale for intensity of field components is used in (b). Cyan dashed line shows | E z | 2 = | E x | 2 case, magenta dash-dotted line location of the maximum peak intensity of | E z | 2 and | E y | 2 , and olive dash-dot-dotted line location of the maximum dark center of | E x | 2 .
Fig. 5
Fig. 5 3D intensity distribution of total diffraction electric field of focused square super-Gaussian top-hat beam by OAP with α = 1.844 (rotation angle φ = 0.36 rad ). (a) | E | 2 in coordinate system S ( x , y , z ) . (b) | E | 2 in coordinate system S ( x , y , z ) . Intensity is indicated by the “heat” of the color. Dashed line in (b) represents propagation direction of focused field.
Fig. 6
Fig. 6 Contour plots of electric field intensity distribution along z' axis (or depth of focus) of focused square super-Gaussian top-hat beam by OAP as function of h. (a) | E x | 2 for 0 h 3600 mm (or 2.5 f / 2 ω 15.2 ) and x = 0 , y = 0 and (b) | E z | 2 for 0 h 3600 mm (or 2.5 f / 2 ω 15.2 ) and x = 0 , y = 0 . Intensity is indicated by the “heat” of the color.
Fig. 7
Fig. 7 Contour plots of focused electric field intensity distribution in focal plane along x' and z' axes as function of f / 2 ω ( f is fixed). (a) | E x | 2 along x' axis for 0.42 f / 2 ω 50 and y = 0 , z = 0 , (b) | E z | 2 along x' axis for 0.42 f / 2 ω 50 and y = 0 , z = 0 , (c) | E x | 2 along z' axis for 0.42 f / 2 ω 50 and x = 0 , y = 0 , and (d) | E z | 2 along z' axis for 0.42 f / 2 ω 50 and x = 0 , y = 0 . Intensity is indicated by the “heat” of the color.
Fig. 8
Fig. 8 Movie showing contour plots of electromagnetic field intensity distribution of focused circular super-Gaussian top-hat beam polarized along + x direction by OAP for different off-axis rates (see Visualization 2). Fields are computed in focal plane (x'-y' plane with z P = 0 ). Intensity is indicated by the “heat” of the color. Numerical computations based on Eqs. (34), (35), and (3).

Equations (35)

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z o = x o 2 + y o 2 4 f f ,
n ^ ( x o , y o ) = 1 2 f ( x o x ^ + y o y ^ ) + z ^ 1 + x o 2 + y o 2 4 f 2 ,
Σ : ( x o h ) 2 + y o 2 R 2 ( for a circular off axis surface ) ,
Σ : R x o h R and R y o R ( for a square off axis surface ) ,
E r , n = E i , n = n ^ ( n ^ E i ) ,
H r , t = H i , t = H i H i , n ,
E r , t = E i , t = ( E i E r , n ) ,
H r , n = H i , n = n ^ ( n ^ H i ) .
E r = 2 n ^ ( n ^ E i ) E i ,
H r = H i 2 n ^ ( n ^ H i ) .
E = E i + E r = 2 n ^ ( n ^ E i ) .
H = H i + H r = 2 H i 2 n ^ ( n ^ H i ) .
E ( P ) = 1 4 π S OAP [ i ω μ ( n ^ × H ) G + ( n ^ × E ) × G + ( n ^ E ) G ] d A ,
H ( P ) = 1 4 π S OAP [ i ω ε ( E × n ^ ) G + ( n ^ × H ) × G + ( n ^ H ) G ] d A ,
r O P = | r P r O | = [ ( x P x o ) 2 + ( y P y o ) 2 + ( z P z o ) 2 ] 1 / 2 .
G = i k ( 1 1 i k r O P ) G r O P [ ( x o x P ) x ^ + ( y o y P ) y ^ + ( z o z P ) z ^ ] = i k ( 1 1 i k r O P ) G r O P ( Δ x x ^ + Δ y y ^ + Δ z z ^ ) = i k ( 1 1 i k r O P ) G r O P r O P
d A = [ 1 + ( z o x o ) 2 + ( z o y o ) 2 ] 1 / 2 d x o d y o = ( 1 + x o 2 + y o 2 4 f 2 ) 1 / 2 d x o d y o .
E i = ( Ψ 0 x x ^ + Ψ 0 y y ^ ) exp ( i ω t i k z i ) ,
H i = 1 η ( Ψ 0 y x ^ Ψ 0 x y ^ ) exp ( i ω t i k z i ) ,
E ( P ) = i k exp ( i ω t ) 2 π S OAP exp ( i k z o ) d x o d y o × { [ ( 1 ( 1 1 i k r O P ) x o 2 f Δ x r O P ) Ψ 0 x G ( 1 1 i k r O P ) y o 2 f Δ x r O P Ψ 0 y G ] x ^ + [ ( 1 ( 1 1 i k r O P ) y o 2 f Δ y r O P ) Ψ 0 y G ( 1 1 i k r O P ) x o 2 f Δ y r O P Ψ 0 x G ] y ^ + ( x o 2 f Ψ 0 x + y o 2 f Ψ 0 y ) [ 1 ( 1 1 i k r O P ) Δ z r O P ] G z ^ }
H ( P ) = i k exp ( i ω t ) 2 π S OAP exp ( i k z o ) η ( 1 1 i k r O P ) G r O P d x o d y o × { [ ( Δ y x o 2 f ) Ψ 0 x + ( x o 2 + y o 2 4 f f Δ y y o 2 f ) Ψ 0 y ] x ^ + [ ( x o 2 + y o 2 4 f f Δ x x o 2 f ) Ψ 0 x + ( Δ x y o 2 f ) Ψ 0 y ] y ^ + ( Δ y Ψ 0 x Δ x Ψ 0 y ) z ^ }
S = E r × H r = 1 η ( Ψ 0 x 2 + Ψ 0 y 2 ) exp [ 2 i ( ω t + k z o ) ] ( x o r o x ^ y o r o y ^ z o r o z ^ ) .
E F ( P ) = i k exp ( i 2 k f i ω t ) 2 π S OAP exp ( i k r O · r P / r o ) d x o d y o × { [ ( 1 r o x o 2 2 f r o 2 ) Ψ 0 x x o y o 2 f r o 2 Ψ 0 y ] x ^ , + [ ( 1 r o y o 2 2 f r o 2 ) Ψ 0 y x o y o 2 f r o 2 Ψ 0 x ] y ^ + ( x o Ψ 0 x + y o Ψ 0 y ) 1 r o 2 z ^ }
H F ( P ) = i k exp ( i 2 k f i ω t ) 2 π S OAP exp ( i k r O · r P / r o ) η d x o d y o × { [ ( x o y o 2 f r o 2 ) Ψ 0 x + ( x o 2 y o 2 4 f r o 2 f r o 2 ) Ψ 0 y ] x ^ . + [ ( x o 2 y o 2 4 f r o 2 + f r o 2 ) Ψ 0 x + ( x o y o 2 f r o 2 ) Ψ 0 y ] y ^ + ( y o Ψ 0 x x o Ψ 0 y ) 1 r o 2 z ^ }
i k r O · r P / r o = i k ( x o r o x P y o r o y P z o r o z P ) ,
i k r O · r P / r o = i k ( x o cos φ + z o sin φ r o x P y o r o y P x o sin φ + z o cos φ r o z P ) ,
α = ( x o cos φ + z o sin φ ) / r o ,
β = y o / r o ,
γ = ( x o sin φ + z o cos φ ) / r o ,
x o = 2 f ( α cos φ + γ sin φ ) ( 1 + α sin φ + γ cos φ ) ,
y o = 2 f ( β ) ( 1 + α sin φ + γ cos φ ) ,
z o = 2 f ( α sin φ γ cos φ ) ( 1 + α sin φ + γ cos φ ) .
J = 4 f 2 γ ( 1 + α sin φ + γ cos φ ) 2 .
E ' F ( P ) = i k exp ( i 2 k f i ω t ) 2 π S OAP ' J exp [ i k ( α x ' P + β y P + γ z ' P ) ] d α d β × { [ ( 1 r o x o 2 2 f r o 2 ) Ψ 0 x x o y o 2 f r o 2 Ψ 0 y ] x ^ ' + [ ( 1 r o y o 2 2 f r o 2 ) Ψ 0 y x o y o 2 f r o 2 Ψ 0 x ] y ^ ' , + ( x o Ψ 0 x + y o Ψ 0 y ) 1 r o 2 z ^ ' }
H F ( P ) = i k exp ( i 2 k f i ω t ) 2 π S OAP ' J exp [ i k ( α x P + β y P + γ z P ) ] η d α d β × { [ ( x o y o 2 f r o 2 ) Ψ 0 x + ( x o 2 y o 2 4 f r o 2 f r o 2 ) Ψ 0 y ] x ^ + [ ( x o 2 y o 2 4 f r o 2 + f r o 2 ) Ψ 0 x + ( x o y o 2 f r o 2 ) Ψ 0 y ] y ^ , + ( y o Ψ 0 x x o Ψ 0 y ) 1 r o 2 z ^ }

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