Propagation of intense electromagnetic pulse with a small conical phase shift induced by Axicon optics

Tae Moon Jeong; Sergei V. Bulanov; Sergei V. Bulanov; Pavel Sasorov; Prokopis Hadjisolomou

doi:10.1364/OE.484968

1. Introduction

The non-diffracting feature of a light beam is one of the interesting optical phenomena since its first report in [1]. Such a non-diffracting beam can be realized by forming Bessel [1,2] or Bessel-Gaussian [3] beam profile with an axicon optical element (hereafter, axicon) [4–6]. The axicon introduces a conical phase shift to the beam and the optical phase delay over the beam becomes proportional to the beam radius, $\rho$. The optical phase delay introduced by the axicon makes the electromagnetic wave located at a different radial position to be focused at a different axial point, resulting in a long focal depth [7]. The formation of a non-diffracting higher-order Bessel beam is also possible by using the spiral diffraction grating [8] or by using the Laguerre-Gaussian (LG) beam as input [9]. Due to the long focal depth, the non-diffracting beam is widely used in micromachining [10], optical manipulation [11], and biomedical engineering [12]. Recently, its applications to the laser-plasma interaction study are also rapidly growing [13–15].

For the axicon with a large amount of conical phase shift ($k d_{max} \gg \sim 50 \times 2 \pi$), the focused electric field is well described by the Bessel function as in [16]. The $k$ and $d_{max}$ stand for the wavenumber and the maximum optical path length at the periphery of the axicon, respectively. In this case, the focused peak intensity is limited by the long focal depth [13] and/or by the annular-shaped intensity distribution [14]. This fact might limit the direct application of the focused field by the axicon to the laser-matter interaction in the relativistic regime [17]. On the other hand, for the axicon with a small amount of conical phase shift ($k d_{max } \ll 2 \pi$), the conical phase shift is considered as a small wavefront error and can produce a relatively higher peak intensity with freedom to produce a desired (or customized) laser spot shape required for specific applications. Recent discussions on the focal spot shaping can be found in [18–20]. In practice, a thin waveplate imposing a small amount of conical phase shift known as the axicon waveplate (AWP) [21] can be considered for this purpose. However, the exact mathematical form describing the focused field configuration with the small conical phase shift is not yet known, so it is of fundamental interest to examine the field configuration focused with a small amount of conical phase shift.

In this paper, we derive mathematical formula describing the focused field distribution for a general LG light wave with a small amount of conical phase shift. The LG light wave is widely encountered in recent applications [22,23] since it not only generates the non-diffracting Bessel beam but delivers the orbital angular momentum (OAM) [24]. In general, the focused field distribution for a monochromatic and weakly-aberrated light wave can be calculated by using the diffraction integral with the Seidel or Zernike decomposition [25] of the wavefront. The spatial field distributions near focus for several low-order aberration modes are well described in [26]. Therefore, the wavefront by the conical phase shift can be decomposed of many higher-order radial Zernike or Seidel aberration modes. However, for mathematical simplicity, we directly calculate the diffraction integral, without decomposing the conical phase shift into Seidel or Zernike modes, as in [16] under the paraxial approximation. The focused fields are calculated for continuous wave (CW) and femtosecond LG beams. It has been found that the focused field distribution with a small conical phase is described by the product of Laguerre, Gaussian, and Bessel functions, and the axial symmetry along the light propagation direction is broken as the spherical aberration in Zernike mode breaks the axial symmetry in the field distribution near focus [26]. Furthermore, depending on the axial position near focus and the amount of conical phase shift, the conical phase shift by the axicon shows a focal spot-shaping capability by controlling the central intensity (i.e., higher than/almost equal to/lower than the surrounding intensity).

The focal spot-shaping capability can be applied to laser-plasma interaction experiments using the high-intensity laser pulse. For example, although the relativistic flying mirror (RFM), which is a plasma optic upshifting the angular frequency of the reflected electromagnetic (EM) wave, can form a very strong field strength close to the Schwinger field limit [27], the low reflectivity of the RFM limits the attainable field strength. A double-sided RFM forced by the front side of a high-intensity pulse has been proposed to enhance the reflectivity [28]. By applying the focal spot-shaping capability, the concavity of the double-sided relativistic flying mirror can be controlled, and high field strength can be obtained by focusing the reflected EM wave or high harmonics. The other application can be found in laser-driven hadron therapy using energetic proton/ion beams [29]. In this case, it is desirable to produce spatially uniform and energetic proton/ion distribution. A relatively uniform intensity distribution formed with a conical phase shift might be beneficial for generating spatially uniform and energetic proton/ion beams. The manuscript is organized as follows. In Sec. 2, the field expression focused by a thin lens with an AWP is derived for the CW case. Section 3 describes the focused laser pulse with a small amount of conical phase shift and its propagation property near focus. The conclusion is given in Sec. 4.

2. Focused electromagnetic wave with a small conical phase shift: CW case

Let us consider an incident LG light wave expressed by,

(1a)$$\begin{aligned}E(\rho;\omega) = E_0 \left( \frac{\rho}{w_0} \right)^l e^{ \Psi_{\rho} + \Psi_{\omega} + il \varphi }, \end{aligned}$$

(1b)$$\begin{aligned}\Psi_{\rho} ={-} \frac{\rho^2}{w_0^2}, \quad \textrm{and} \quad \Psi_{\omega} ={-}\frac{(\omega -\omega_c )^2}{\Delta \omega^2} , \end{aligned}$$

where $\rho$ is the radial distance from the beam center, $\varphi$ the azimuthal angle, $w_0$ the Gaussian beam radius, $\omega _c$ the center frequency, $\Delta \omega$ the spectral bandwidth, $l$ the topological charge (TC) for OAM [30], respectively. In this section, we consider a specific frequency component, $\omega$, in the Gaussian frequency spectrum, so the incident LG light wave at frequency $\omega$ can be considered a monochromatic and continuous wave. The electric field passes through an optical element consisting of a focusing optic and an AWP (See Fig. 1). The optical phase, $\Phi _a$, introduced by the optical element is given by

(2)$$\Phi_a ={-}i \alpha k \rho - i \frac{k}{2f} \rho^2.$$

The first term on the right-hand side in Eq. (2) means the conical phase shift. In Eq. (2), $\alpha$ is a proportional constant amounting to the conical phase shift to the beam and $k$ is the wavenumber. The $\alpha$ can be determined by equating $\alpha k w_0$ to the phase shift, $k d_{w_0}$, at the Gaussian radius of $w_0$ as,

(3)$$\alpha = \frac{d_{w_0}}{w_0},$$

where $d_{w_0}$ is the optical path length induced by the conical phase shift. Thus, the conical phase shift implicitly contains information on the refractive index of the axicon material and the thickness of the axicon. The second term in Eq. (2) refers to the quadratic phase shift by a focusing optic with a focal length of $f$. Then, the electric field after the optical element is given by,

(4)$$\begin{aligned} E_T (\rho;\omega) & = T (\rho;\omega) E (\rho;\omega)\\ & = E_0 \left( \frac{\rho}{w_0} \right)^l e^{\Psi_{\rho} + \Psi_{\omega} -i \alpha k \rho - i \frac{k}{2f} \rho^2 + il \varphi} . \end{aligned}$$

Here, $T(\rho ;\omega )$ is the transmission function for the optical element. The unit transmittance is assumed so that $|T(\rho ;\omega )| = 1$.

Fig. 1. (a) Focusing scheme using a conventional axicon lens (AL). (b) Modified focusing scheme using a thin lens (TL) and an axicon waveplate (AWP). Compared to the AL, the AWP introduces a small amount of conical phase shift. Depending on the sign of conical phase shift, the resultant phase profile becomes oe-31-13-21614-i001 -shaped ($\alpha > 0$) or oe-31-13-21614-i002 -shaped ($\alpha < 0$). LG beam stands for Laguerre-Gaussian beam.

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The focused electric field can be calculated from the Huygens diffraction integral [31] as,

(5)$$dE_f (r;\omega) = \frac{i \omega}{2 \pi c} E_0 \left( \frac{\rho}{w_0} \right)^l e^{ \Psi_{\rho} + \Psi_{\omega} -i \alpha k \rho - i \frac{k}{2f} \rho^2 + il \varphi} \frac{ e^{i \vec{k} \cdot \left( \vec{r} - \vec{\rho} \right)} }{ | \vec{r} - \vec{\rho} |} dA,$$

where, $\vec {\rho }$ is the vector to a source point from the origin (focus), $(\rho \cos \varphi, \rho \sin \varphi, -f)$, and $\vec {r}$ the vector to an observation point near focus from the origin, $(r \cos \phi, r \sin \phi, z)$. The $\vec {k}$ and $dA$ are the wavevector and the infinitesimal area on the exit plane of the focusing optic element, respectively. The optical path-length difference (OPD) between the source point and the observation point is given by

(6)$$\begin{aligned} | \vec{r} - \vec{\rho} | &= \sqrt{ \left( r \cos \phi - \rho \cos \varphi \right)^2 + \left( r \sin \phi - \rho \sin \varphi \right)^2 + \left( z + f \right)^2}\\ &\approx z+f - \frac{r \rho \cos \left( \phi - \varphi \right)}{z+f} + \frac{r^2 + \rho^2}{2 \left( z + f \right)} . \end{aligned}$$

Here, $\phi$ and $\varphi$ are the azimuthal angles in observation and source planes. Under the paraxial approximation in which the focal length is long enough compared to the incident beam diameter, so $\rho \approx f \vartheta$ through

(7)$$\rho = f \sin \left( \pi - \theta \right) \quad \textrm{and} \quad \sin \left( \pi - \theta \right) \approx \left( \pi - \theta \right) \equiv \vartheta.$$

The angle, $\theta$, is the polar angle to the source point. Since the field strength is a slowly varying function, $|\vec {r} - \vec {\rho }| \approx z+f$ and $dA \approx f^2 \sin \vartheta d \vartheta d \varphi \approx f^2 \vartheta d \vartheta d \varphi$. Then, Eq. (5) can be written as

(8)$$\begin{aligned} dE_f = \; & \frac{i \omega f^2 E_0}{2 \pi c \left( z+f \right)} \left( \frac{f}{w_0} \right)^l e^{- \frac{\left( \omega - \omega_c \right)^2}{ \Delta \omega^2} } e^{ {-}i \omega t +i k (z+f) + i \frac{kr^2}{2f} }\\ & \times e^{-\frac{f^2}{w_0^2} \vartheta^2 - i \frac{k r f \vartheta \cos \left( \phi - \varphi \right)}{z+f} -i \alpha k f \vartheta - i \frac{kz}{2} \vartheta^2 + i l \varphi} \vartheta^{l+1} d \vartheta d \varphi . \end{aligned}$$

By using the Jacobi-Anger identity of $\int _0^{2 \pi } e^{ip \phi } e^{ix \cos (\phi - \phi ')} d \phi = 2 \pi i^p e^{ip \phi '} J_p (x)$, the integration over $\varphi$ yields

(9)$$\begin{aligned} dE_f \approx \; & i^{l+1} \frac{\omega f^2 E_0}{c (z+f)} \left( \frac{f}{w_0} \right)^l e^{-\frac{(\omega - \omega_c)^2}{\Delta \omega^2}} e^{{-}i \omega t + i k (z+f) + i \frac{kr^2}{2f} +il \phi} e^{- a \vartheta^2 -i \alpha k f \vartheta } J_l \left( b \vartheta \right) \vartheta^{l+1} d \vartheta\\ = \; & i^{l+1} \frac{\omega f^2 E_0}{c (z+f)} \left( \frac{f}{w_0} \right)^l e^{-\frac{(\omega - \omega_c)^2}{\Delta \omega^2}} e^{{-}i \omega t + i k (z+f) + i \frac{kr^2}{2f} +il \phi}\\ & \times \sum_{m=0}^\infty \frac{ ({-}i \alpha kf)^m}{m!} e^{{-}a \vartheta^2} J_l \left( b \vartheta \right) \vartheta^{l+m+1} d \vartheta, \end{aligned}$$

with definitions of $a = \frac {f^2}{w_0^2} + i \frac {kz}{2}$ and $b = \frac {krf}{z+f} \approx kr$. In the following, a small amount of conical phase shift induced by the axicon, which linearly depends on the radial coordinate, is expanded through the Taylor series in order to analytically calculate the diffraction integration. It is shown that the result for the intensity distribution is expressed in terms of the product of Laguerre, Gaussian, and Bessel functions. In Eq. (9), the integration domain for $\vartheta$ is [0, $\pi - \theta _{min}$], but by assuming the beam size is sufficiently smaller than the optic size the integration domain can be extended from 0 to $\infty$. Then, by using the integral identity [32] of

(10)$$\int_0^\infty x^\mu e^{{-}ax^2} J_\nu \left( bx \right) dx = \frac{\Gamma \left( \nu/2 + \mu/2 +1/2 \right)}{b a^{\mu/2} \Gamma\left( \nu +1 \right)} e^{-\frac{b^2}{4a}} \left( \frac{b^2}{4a} \right)^{\nu/2 + 1/2} M \left( \frac{\nu}{2} - \frac{\mu}{2} + \frac{1}{2}, 1+\nu, \frac{b^2}{4a} \right),$$

the resultant focused electric field for a given angular frequency, $\omega$, is given by

(11)$$\begin{aligned} E_f^{cw} \approx \; & i^{l+1} E_0 \frac{f^2}{z+f} \frac{k}{2a} \left( \frac{f}{w_0} \right)^l e^{-\frac{(\omega - \omega_c)^2}{\Delta \omega^2}} e^{{-}i \omega t + i k (z+f) + i \frac{kr^2}{2f} +il \phi} e^{-\frac{b^2}{4a}} \left( \frac{b}{2a} \right)^l\\ & \times \sum_{m=0}^\infty \frac{\Gamma (l+m/2 +1)}{m! \Gamma (l+1)} \left({-}i \frac{\alpha k f}{\sqrt{a}} \right)^m M \left( -\frac{m}{2},1+l,\frac{b^2}{4a} \right). \end{aligned}$$

Here, $M(p,q,x)$ is the Kummer function of the first kind given by [33]

(12)$$M \left( p,q,x \right) = {_1F_1} \left( p,q,x \right) = \sum_{m=0}^{\infty} \frac{p^{(m)}}{q^{(m)}} \frac{x^m}{m!}.$$

Here, $_1F_1 (p,q,z)$ is the confluent hypergeometric function, and $p^{(m)}$ and $q^{(m)}$ are Pochhammer symbols defined as $p^{(0)} \equiv 1$ and $p^{(m)} \equiv p (p+1) \cdots (p+m-1)$. Equation 11 describes the monochromatic field distribution focused by the optical element consisting of a focusing optic and an AWP.

In order to observe the connection to the Bessel beam, it is convenient to express the Kummer function, $M\left ( -\frac {m}{2} ,1+l, \frac {b^2}{4a} \right )$, via Bessel functions as [34],

(13)$$M \left( -\frac{m}{2}, 1+l, \frac{b^2}{4a} \right) = \Gamma (1+l) \sum_{n=0}^\infty \left(- \frac{1}{\alpha} \right)^n L_n^{\frac{m}{2}-n} (\alpha) \left( \frac{\sqrt{\alpha} b}{2 \sqrt{a}} \right)^{n-l} J_{l+n} \left( \frac{\sqrt{\alpha} b}{\sqrt{a}} \right).$$

Here, $L_n^{\frac {m}{2} - n} (\cdot )$ is the associated Laguerre polynomial, $J_{l+n} (\cdot )$ the Bessel function of the first kind. By inserting the above expression into Eq. (11), we obtain

(14)$$\begin{aligned} E_f^{cw} \approx \; & i^{l+1} E_0 \frac{kf}{2a} \left( \frac{f}{w_0} \right)^l \left( \frac{b}{2a} \right)^l e^{-\frac{(\omega - \omega_c)^2}{\Delta \omega^2}} e^{{-}i \omega t + i k (z+f) + i \frac{kr^2}{2f} +il \phi} e^{-\frac{b^2}{4a}}\\ & \times \sum_{m=0}^\infty \frac{\Gamma (l+ m/2 +1)}{m!} ({-}i \alpha \beta )^m \sum_{n=0}^\infty \left( - \frac{1}{\alpha} \right)^n L_n^{\frac{m}{2} - n} (\alpha) ( \sqrt{\alpha} \gamma )^{n-l} J_{l+n} ( 2 \sqrt{\alpha} \gamma ). \end{aligned}$$

Here, $\beta = \frac {k f}{\sqrt {a}} = \frac {k w_0}{\sqrt {1+z^2/z_R^2}} \sqrt {1 - i z/z_R}$ and $\gamma = \frac {b}{2 \sqrt {a}} = \frac {r}{w(z)} \sqrt {1 - i z/z_R}$ with the definition of the Rayleigh range given by $z_R \equiv \frac {2f^2}{k w_0^2}$. The first line in Eq. (14) represents the LG beam mode (LG$_0^l$) and the second line modifies the intensity distribution with the conical phase shift expressed by $\alpha$. By the second line of Eq. (14), the general solution for the focused field distribution turns into the product of Laguerre-Gaussian and Bessel functions.

The convergence of the first series in the second line of Eq. (14) can be investigated from Fig. 2. In Fig. 2, $\mathcal {A}_m (l)$ is the element of a series defined as $\frac {\Gamma (l+ m/2 +1)}{m!} (\alpha k w_0)^m$, which is greater than or equal to $\frac {\Gamma (l+ m/2 +1)}{m!} | \alpha \beta |^m$. Thus, the element, $\mathcal {A}_m (l)$, represents the maximum value of $\frac {\Gamma (l+ m/2 +1)}{m!} | \alpha \beta |^m$ appearing in the focal plane, i.e., $z=0$. In the figure, the element is further normalized to the maximum element value on the index $m$. To ensure the accuracy in the field distribution calculation, the first series in Eq. (14) should be summed up to a certain minimum index of $m_{min}$. For example, according to Fig. 2, the series should be summed up to $m=7$ for $\alpha = 2 \times 10^{-6}$ and $l=0$. For a higher $l$, more elements should be summed up to ensure accuracy in the field distribution calculation [see Fig. 2(a)]. Since the series slowly converges as $\alpha$ increases, the minimum index of $m$ quickly increases for a larger $\alpha$ as shown in Fig. 2(b).

Fig. 2. (a) The normalized value of an element, $\mathcal {A}_m (l)$, for different $l$s. The parameter, $\alpha$, is 2 $\times$ 10$^{-6}$ in this case. For low $l$s (0 or 1), the series, $\sum _{m=0}^\infty \frac {\Gamma (l+ m/2 +1)}{m!} ( -i \alpha \beta )^m$, should be summed up to $m=7$ for the reasonable accuracy of the calculation. (b) The normalized value of $\mathcal {A}_m (l)$ for different $\alpha$s. The TC is 1 in this case. The minimum index for $m$ quickly increases with a larger value of $\alpha$.

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Since it is reasonable to sum up the series to $m=7$ for a small $\alpha = 2 \times 10^{-6}$, which satisfies $d_{w_0} = 0.25 \lambda$ (here, $\lambda$ is the wavelength), and for a lower TC ($l=0$ or $l=1$) that ensures the accuracy, Eq. (14) is approximated as

(15)$$\begin{aligned} E_f^{cw} \approx \; & i^{l+1} E_0 \frac{kf}{2a} \left( \frac{1}{\sqrt{\alpha}} \frac{f}{w_0} \frac{1}{\sqrt{a}} \right)^l e^{-\frac{\left( \omega -\omega_c \right)^2}{\Delta \omega^2}} e^{{-}i\omega t +ik \left( z + f\right) + i \frac{k r^2}{2f} + il \phi} e^{-\frac{b^2}{4a}}\\ & \times \bigg[ J_l - i \Gamma (l+3/2) \alpha \beta J_l - \frac{\Gamma (l+2)}{2 } \alpha^2 \beta^2 \left( J_l - \frac{\gamma^2(\sqrt{\alpha } \gamma )^l}{\Gamma (l+2)} \right)\\ & \quad + i \frac{\Gamma (l+5/2)}{6} \alpha^3 \beta^3 \left( J_l - \frac{3}{2} \frac{\gamma^2(\sqrt{\alpha } \gamma )^l}{\Gamma (l+2)} \right) + HO ( \alpha \beta, \gamma^2) \bigg]. \end{aligned}$$

Here, $J_l$ stands for the Bessel function of the first kind, $J_l (2 \sqrt {\alpha } \gamma )$, and $J_{l+n}$ is approximated as $( \sqrt {\alpha } \gamma )^{l+n} / \Gamma (l+n+1)$ since we consider $\sqrt {\alpha } \gamma \ll 1$. The “ HO ” stands for the higher-order term including from $m=4$ to $m=7$ terms.

For a Gaussian beam ($l=0$) with no conical phase shift ($\alpha =0$), Eq. (15) reduces to the well-known “focused Gaussian field” as

(16)$$E_f^{cw} \approx i E_0 \frac{w_0}{w(z)} e^{-\frac{\left( \omega -\omega_c \right)^2}{\Delta \omega^2}} e^{{-}i\omega t +ik \left( z + f\right) + i \frac{k r^2}{2f} +i \frac{ r^2}{w^2 (z) } \frac{z}{z_R} - i \varphi_g} e^{-\frac{ r^2}{w^2 (z) }},$$

with definitions of Guoy phase given by $\varphi _g \equiv \tan ^{-1} (z/z_R)$ and beam radius by $w(z) = \frac {\lambda f}{\pi w_0} \sqrt {1+z^2/z_R^2} = w_b \sqrt {1+z^2/z_R^2}$. The expression, $\frac {\lambda f}{\pi w_0}$, is replaced by the beam waist radius, $w_b$.

For a small $\alpha$ such that $0< \alpha |\beta | \ll 1$ in which all higher-order terms in $\alpha \beta$ can be ignored, Eq. (15) is approximated as

(17)$$\begin{aligned} E_f^{cw} \approx \; & i^{l+1} E_0 \frac{kf}{2a} \left( \frac{1}{\sqrt{\alpha}} \frac{f}{w_0} \frac{1}{\sqrt{a}} \right)^l e^{-\frac{\left( \omega -\omega_c \right)^2}{\Delta \omega^2}} e^{{-}i\omega t +ik \left( z + f\right) + i \frac{k r^2}{2f} + il \phi} e^{-\frac{b^2}{4a}}\\ & \times J_l (2 \sqrt{\alpha} \gamma) \left[ 1 - i \frac{\Gamma (l+3/2)}{\Gamma(l+1)} \alpha \beta \right]. \end{aligned}$$

In the limit of $f \gg 1$ with a monochromatic continuous wave ($e^{-\left ( \omega -\omega _c \right )^2 / \Delta \omega ^2} \to 1$), $w(z) \to w_0$ and $\left(\frac{1}{\sqrt{\alpha}} \frac{f}{w_0} \frac{1}{\sqrt{a}}\right)^l \approx\left(\frac{1}{\sqrt{\alpha}}\right)^l$. Then, Eq. (17) reduces to the Bessel beam with an OAM property as

(18)$$E_f^{cw} \sim E_0 J_l (2 \sqrt{\alpha} \gamma) \left( 1 - i \frac{\sqrt{\pi}}{2} \alpha k w_0 \right) e^{ il \phi}.$$

As in other publications [9,16], the TC, $l$, determines the order of the Bessel beam. Figure 3 show the intensity distributions (isophotes of intensity) calculated with Eq. (15) for TCs of $l=0$ and $l=1$. The Gaussian and Laguerre-Gaussian, LG$_0^1$, beams with different conical phase shifts were considered as input beam profiles. The focal length of 1 m was assumed with a Gaussian radius, $w_0$, of 0.1 m. Compared to the intensity distributions with no conical phase shift [see Figs. 3(b) and 3(f)], the axial symmetry along the beam propagation direction was broken by the conical phase shift with a wider spot size, which is similar to the focused field with spherical aberration. A wider spot size with the conical phase shift can be understood by the argument, $\sqrt {\alpha } \gamma$, of Bessel function in Eq. (14). When $\alpha = 0$, the minimum spot size is obtained since $J_0 (0) =1$. However, when $\alpha \neq 0$, the contribution from higher order Bessel functions, $J_n (\cdot )$, to the intensity distribution makes the spot size wider since $J_n (\cdot )$ starts to have noticeable value as the radius increases. The peak intensity shift in the z-direction can be more easily understood by the conical phase shift. When $\alpha > 0$, the phase at an outer rim of the beam advances the phase at a central region of the beam, resulting in focusing the beam before the nominal focus, $f$. On the other hand, when $\alpha < 0$, the phase at a central region advances the phase at an outer rim, resulting in focusing the beam after the nominal focus.

Fig. 3. Intensity distributions (isophotes of intensity) in the meridional plane near focus (a)-(d) for the Gaussian beam and (e)-(h) for the Laguerre-Gaussian beam, LG$_0^1$. The conical phase shift are shown on the top for each column. In the figures, the gray dashed lines are the intensity distribution curves at the highest peak location (depicted by the white dotted line). The $w_b$ refers to the beam waist radius and $z_R$ does the Rayleigh range with no conical phase shift.

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A narrower confinement of the LG beam in the radial direction on the AWP makes the focal spot less sensitive to the conical phase shift than the Gaussian beam. Despite the relatively smaller change in the central part of the focal spot, the peak of the LG beam intensity radially shifted from $\sim$0.72$\times w_b$ for $\alpha = 0$ to $\sim$1.44$\times w_b$ for $\alpha = 2 \times 10^{-6}$ (see Fig. 3). This property can be used to optimize the guiding effect of a main laser pulse in the x-ray amplification experiment [35]. In such an experiment, an annular laser beam with a specific radius is required for guiding the main laser pulse. Equation 15 can be used to design an axicon waveplate to produce such an annular laser beam profile in the focal plane.

3. Focused electromagnetic pulse with a small conical phase shift

To calculate the instantaneous field distribution for a femtosecond laser pulse, Eq. (11) has to be integrated over the angular frequency, $\omega$, as,

(19)$$\begin{aligned} E_f^p \approx \; & i^{l+1} E_0 f \left( \frac{f}{w_0} \right)^l \int_{-\infty}^{\infty} d \omega \frac{\omega}{2ac} \left( \frac{b}{2a} \right)^l e^{-\frac{\left( \omega -\omega_c \right)^2}{\Delta \omega^2}} e^{{-}i\omega t +i \omega \frac{z+f}{c} + i \frac{\omega r^2}{2cf} + i l \phi} e^{-\frac{b^2}{4a}}\\ & \times \sum_{m=0}^\infty \frac{\Gamma (l + m/2 +1)}{m! \Gamma (l+1)} \left({-}i \frac{\alpha f}{c \sqrt{a}} \right)^m \omega^m \sum_{n=0}^\infty \frac{({-}m/2)^{(n)}}{(1+l)^{(n)} } \frac{(b^2 / 4a)^n}{n!}, \end{aligned}$$

where $c$ is the speed of light. By defining a dimensionless parameter, $\tilde {\omega }$, as $\frac {\omega - \omega _c}{\omega _c}$, we approximate the Rayleigh range as

(20)$$z_R \approx \frac{2cf^2}{\omega_c w_0^2} (1 - \tilde{\omega}) = z_c (1 - \tilde{\omega}),$$

since $\tilde {\omega } \ll 1$. This is the first order approximation in $\tilde {\omega }$. Here, $z_c \equiv \frac {2cf^2}{\omega _c w_0^2}$ refers to the Rayleigh range at the center frequency, $\omega _c$. From Eq. (20), we obtain the following approximations:

(21)$$\begin{aligned} \frac{z^2}{z_R^2} &\approx \frac{z^2}{z_c^2} (1+ 2 \tilde{\omega}),\\ \frac{1}{a} &\approx \frac{w_0^2}{f^2} \frac{1}{\sqrt{1+z^2 / z_c^2}} \left( 1 - \frac{\tilde{\omega} z^2}{z^2 + z_c^2}\right) e^{{-}i \varphi_c} = \frac{1}{a_c} \left( 1 - \frac{\tilde{\omega} z^2}{z^2 + z_c^2}\right),\\ \textrm{and}\\ \frac{b^2}{4a} &\approx \frac{\omega_c^2 r^2 w_0^2}{4 c^2 f^2 (1+ z^2 / z_c^2)} \left( 1 + \frac{2\tilde{\omega} z^2}{z^2 + z_c^2}\right) \left( 1 - i \frac{z}{z_c}\right) = \frac{b_c^2}{4a_c} \left( 1 + \frac{2\tilde{\omega} z^2}{z^2 + z_c^2}\right), \end{aligned}$$

where $\frac {1}{a_c} = \frac {w_0^2}{f^2} \frac {1}{\sqrt {1+z^2 / z_c^2}} e^{-i \varphi _c}$, $\frac {b_c^2}{4a_c} = \frac {r^2}{w_c^2 (z)} (1 - i z/z_c)$, and $w_c (z) = \frac {2 c f}{w_0 \omega _c} \sqrt {1+z^2/z_c^2}$ with the definition of $\varphi = \tan ^{-1} (z/z_c)$.

Then, we re-express Eq. (19) with the dimensionless parameter, $\tilde {\omega }$, as

(22)$$\begin{aligned} E_f^p \approx \; & i^{l+1} f E_0 \left( \frac{f}{w_0} \right)^l \left( \frac{k_c r}{2a_c} \right)^l e^{{-}i \omega_c \left(t - \frac{z+f}{c} - \frac{r^2}{2cf} \right) + i l \phi} e^{- \frac{b_c^2}{4 a_c^2 }} \frac{\omega_c}{2a_c c}\\ & \times \sum_{m=0}^\infty \frac{\Gamma (l+ m/2 +1)}{m! \Gamma (l+1)} \left({-}i \frac{\alpha \omega_c f}{c \sqrt{a_c}} \right)^m \sum_{n=0}^\infty \frac{({-}m/2)^{(n)}}{(1+l)^{(n)}} \frac{ \left( b_c^2 / 4 a_c \right)^n}{n!}\\ & \times \left[ \mathcal{I}_1 + \frac{(m/2) z^2 +(l+m+2n+1) z_c^2 }{1 + z^2 /z_c^2} \mathcal{I}_2 \right] . \end{aligned}$$

Here, $\omega _c/c$ is replaced by $k_c$. The integrals, $\mathcal {I}_1$ and $\mathcal {I}_2$, are defined as $\int _{-\infty }^{\infty } d \tilde {\omega } e^{-A^2 \tilde {\omega }^2 -i B \tilde {\omega }}$ and $\int _{-\infty }^{\infty } d \tilde {\omega } \tilde {\omega } e^{-A^2 \tilde {\omega }^2 -i B \tilde {\omega }}$ with $A^2 = \frac {\omega _c^2}{\Delta \omega ^2}$ and $B = \omega _c \left ( t - \frac {z+f}{c} - \frac {r^2}{2cf} \right ) - i \frac {4 r^2}{w_c^2 (z)} \frac {2}{1+z^2 /z_c^2}$. Now, by applying the integral identity [32] of

(23)$$\int_{\infty}^{\infty} (ix)^\nu e^{-\beta^2 x^2 -iqx} dx = \sqrt{\pi} 2^{-\nu/2} \beta^{-\nu -1} e^{-\frac{q^2}{8\beta^2}} D_\nu \left( \frac{q}{\beta \sqrt{2}} \right),$$

the integrals, $\mathcal {I}_1$ and $\mathcal {I}_2$, become

(24)$$\mathcal{I}_1 = \frac{\sqrt{\pi}}{A} e^{-\frac{B^2}{4A^2}} \quad \text{and} \quad \mathcal{I}_2 ={-}i \frac{\sqrt{\pi}}{2} \frac{B}{A^3} e^{-\frac{B^2}{4A^2}}.$$

In Eq. (23), $D_{\nu } (\cdot )$ is the parabolic cylinder function. Since $|B| / A^2 \ll 1$ in the spatio-temporal domain of interest, $|\mathcal {I}_1| \gg |\mathcal {I}_2|$. And, the integral, $\mathcal {I}_1$, is expressed as

(25)$$\mathcal{I}_1 \approx \sqrt{\pi} \frac{\Delta \omega}{\omega_c} \exp \left[ - \frac{\Delta \omega^2}{4} \left( t - \frac{z+f}{c} - \frac{r^2}{2cf} \right)^2 \right],$$

with $\frac {\Delta \omega ^2}{\omega _c^2} \ll 1$ and $\frac {\Delta \omega }{\omega _c} \ll 1$. The first two lines in Eq. (22) are identical to Eq. (11) but the last line, together with Eq. (25), delivers information on the propagation of a femtosecond light pulse near focus. By introducing new variables, $\beta _c$ and $\gamma _c$, defined at the center frequency as,

(26)$$\beta_c = \frac{\omega_c w_0}{c \sqrt{1+z^2 / z_c^2}} \sqrt{1 - i \frac{z}{z_c}} \quad \textrm{and} \quad \gamma_c = \frac{r}{w_c (z)} \sqrt{ 1 - i \frac{z}{z_c} },$$

we obtain the general field expression for a focused femtosecond laser pulse with a conical phase shift in the form of

(27)$$\begin{aligned} E_f \approx \; & i^{l+1} \sqrt{\pi} E_0 \frac{\Delta\omega}{\omega_c} \frac{w_0}{w_c (z)} \left[ \frac{r}{w_c (z)} \right]^l e^{{-}i \omega_c \left( t - \frac{z+f}{c} - \frac{r^2}{2cf} \right) -i (l+1) \varphi_c + i l \phi} e^{- \frac{r^2}{w_c^2 (z)} - \frac{\Delta \omega^2}{4} \left( t - \frac{z+f}{c} -\frac{r^2}{2cf} \right)^2}\\ & \times \sum_{m=0}^\infty \frac{\Gamma (l+ m/2 +1)}{m!} ({-}i \alpha \beta_c )^m \sum_{n=0}^{\infty} \left( - \frac{1}{\alpha} \right)^n L_n^{\frac{m}{2} -n} ( \alpha ) ( \sqrt{\alpha} \gamma_c )^{n-l} J_{l+n}( 2 \sqrt{\alpha} \gamma_c ) . \end{aligned}$$

The same approach used in the previous section can be applied to explicitly express the Laguerre-Gaussian-Bessel beam at a small conical phase shift.

Figure 4 presents snapshot images for femtosecond laser focuses propagating near focus. In the figure, $w_{c0}$ and $z_c$ stand for the beam waist radius and the Rayleigh range at the center wavelength when no conical phase shift is introduced. Figures 4(a) - 4(c) show the propagation of Gaussian femtosecond (46.5 fs) laser pulse focused with different conical phase shift parameters. Again, the focal length of 1 m was assumed with a Gaussian radius, $w_0$, of 0.1 m. In this case, the input pulse duration corresponds to $0.55 \times (z_c /c)$. The propagation property for an OAM $LG_0^1$ laser beam focused with a conical phase shift is shown in Figs. 4(d) and 4(f).

Fig. 4. Propagation of laser focus with different input beam profile and conical phase shift parameter, $\alpha$. (a) Gaussian beam with $\alpha = -2 \times 10^{-6}$, (b) Gaussian beam with $\alpha = 0$, (c) Gaussian beam with $\alpha = 2 \times 10^{-6}$, (d) LG$_0^1$ beam with $\alpha = -2 \times 10^{-6}$, (e) LG$_0^1$ beam with $\alpha = 0$, and (f) LG$_0^1$ beam with $\alpha = 2 \times 10^{-6}$. The time on the left side means the elapsed time, so the peak of laser pulse arrives at the focal plane at t = 0 fs when no conical phase shift is introduced. The $\alpha = 2 \times 10^{-6}$ corresponds to the phase advance of $\lambda /4$ at $r = w_{c0}$, which is the minimum spot size defined as $\frac {2cf}{w_0 \omega _c}$ at $z$ = 0. A pulse duration of 46.5 fs is chosen to represent the pulse duration of 1 PW class Ti:Sapphire laser pulse.

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Another interesting feature related to the generation of a flattened (or concave) intensity profile and the control of convexity of the double-sided mirror is shown in Fig. 5. In this case, the pulse duration of the input laser pulse is about 17 fs, which corresponds to $0.2 \times (z_c /c)$. The solid white line in Fig. 5(a) is the intensity profile along the $y$-direction at the peak. A flattened or slightly concave intensity distribution propagates from -10 fs to 10 fs, corresponding to -3 $\mu$m to 3 $\mu$m along the z-direction. This propagation property might be used for controlling the concavity of the relativistic flying mirror through the double-sided mirror scheme. Since the electrons are pushed by the laser pulse through the ponderomotive force, the concave intensity shape shown in Fig. 5(a) can be projected to the electron density profile, which behaves as a relativistic flying mirror. By changing the conical phase shift parameter, the concavity of the relativistic flying mirror can be adjusted.

Fig. 5. (a) Propagation of the focused Gaussian laser pulse near focus. In this case, the pulse duration is about 17 fs which represents the 4 PW Ti:sapphire laser [36], and $\alpha$ is 2 $\times$ 10$^{-6}$. A flattened or concave front intensity profile is maintained at a certain propagation distance. (b) Intensity distribution in x-y plane at t = 0. All intensity distributions are normalized by the maximum value at z = 0.

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Figure 5(b) presents the focal spot images at different locations in the $z$-direction. These images are understood as time-integrated ones at the position. Since the main difference between Fig. 4 and Fig. 5 is the pulse duration, there is no difference in the footprint images. It is clearly shown that by the conical phase shift, the focal spot image is modified to have a flattened or concave intensity profile. The flattened beam profile shown in Fig. 5(b) uniformly accelerates protons/ions followed by electrons in the spatial domain.

4. Conclusion

The analytical formula describing the focused field distribution for the Laguerre-Gaussian beam with a small amount of conical phase shift has been derived. The intensity distribution near focus is calculated for continuous wave and femtosecond laser pulse. The focused intensity distribution is described by the product of Laguerre-Gaussian and Bessel functions. The axial symmetry in the intensity distribution is broken by the conical phase shift and the conical phase shift modifies the propagation property near focus. Depending on the position and the amount of conical phase shift, the laser focus shows a flattened or concave intensity profile propagating within a certain range without a severe decrease in intensity. The focal spot-shaping capability provided by the conical phase shift might be used to control the concavity of double-sided relativistic flying mirrors or to generate spatially uniform and energetic laser-driven proton/ion beams for hadron therapy.

Funding

Advanced research using high intensity laser produced photons and particles (ADONIS) European Regional Development Fund (CZ.02.1.01/0.0/0.0/16_019/0000789).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Propagation of intense electromagnetic pulse with a small conical phase shift induced by Axicon optics

Abstract

1. Introduction

2. Focused electromagnetic wave with a small conical phase shift: CW case

3. Focused electromagnetic pulse with a small conical phase shift

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (28)

Optics Express