Spatio-temporal modification of femtosecond focal spot under tight focusing condition

Tae Moon Jeong; Stefan Weber; Bruno Le Garrec; Daniele Margarone; Tomas Mocek; Georg Korn

doi:10.1364/OE.23.011641

1. Introduction

The advance in the laser technology enables one to produce a laser pulse having a peak power of over 1 PW [1–4]. Charged particle (electron, proton, and ions) acceleration [5–7], high-energy photon (x-ray and γ-ray) generation [8, 9], and the study of exotic science [10] are of primary interest utilizing such high peak power lasers. One particular interest is to tightly focus such a high-peak-power laser pulse to reach an unprecedented intensity level. The highest laser intensity of ~10²² W/cm² was reported by focusing a 45-TW laser pulse to a spot size of wavelength (0.8 μm) using an f/0.6 parabolic mirror [11, 12]. Recently, a tight imaging scheme employing a combination of a parabolic mirror and an ellipsoidal mirror has been devised to demonstrate a tiny spot size of ~λ (~0.9 μm) [13]. With such an intense and tiny intensity distribution, a precise description on focused electromagnetic (EM) fields becomes important to accurately trace the motion of charged particles under a high-intensity, tightly-focused EM field. The paraxial approximation, which is commonly used in calculating focal spots under high f-number conditions, becomes invalid under tight focusing conditions decreasing to an f-number of 1 or below.

Dating back to late 1930s [14], the tight focusing is still an important research topic in many disciplines [15, 16] including ultra-intense laser-matter interaction. Under a tight focusing condition (f-number ≤ 1), the focused EM fields are affected by the polarization of an incident field. The scalar approximation becomes invalid in predicting EM fields in this regime, and vectorial features for EM fields should be considered to improve the accuracy of the description of EM fields. Typical aspects occurred under tight focusing conditions are the increase in the intensity (or strength) of a longitudinal EM field and the elongation of a focal spot along the polarization direction for ideal (monochromatic, spatially uniform, and wavefront-aberration-free) EM fields. Several approaches, using propagating models such as the Maxwell’s equation with a longitudinal field correction in a few cycle regime [17] and the vector potential with Gaussian beam profile [18, 19], have been developed to include these aspects when simulating laser-matter interactions in the high-intensity regime. However, as a 10 PW and even higher power lasers are becoming available [4], a general demand for precisely describing focused EM fields under tight focusing conditions is greatly increasing for a realistic (broadband and aberrated) EM field to be reached at an intensity of 10²³ to 10²⁴ W/cm².

In this paper, we propose a method to calculate vectorial EM fields for a femtosecond laser pulse under a tight focusing condition in the focal plane and its vicinity. The method contains two major procedures: One is to calculate vector diffraction integrals for transverse and longitudinal EM fields for a monochromatic EM wave, and the other is to perform the coherent superposition with calculated transverse and longitudinal EM fields to form a femtosecond focal spot near the focal plane. The integral expressions specified by Varga and Török [20,21], based on vector diffraction integrals developed by Stratton and Chu [14], are employed with modification to calculate focused EM fields for an aberrated incident EM field. Then, the focusing properties for an aberrated EM field are discussed from numerical results calculated under various focusing conditions such as the tight focusing with a parabolic mirror and the tight imaging with a parabolic-ellipsoidal mirror combination that will be an important example in near future. Finally, a focal spot of a femtosecond laser pulse (so-called femtosecond focal spot) obtained by coherently superposing calculated monochromatic EM fields with a given amount of spectral power and phase relation over an entire wavelength range, and its spatio-temporal modification is investigated as the f-number of a focusing optic decreases.

The method described in this paper can be used to accurately assess the peak intensity of a high-power femtosecond laser pulse focused under a tight focusing condition. The accurate assessment of the peak power and the practical information on EM field distribution for a realistic high-power femtosecond laser pulse will be beneficial in simulating and predicting the motion of charged particles under a super-strong EM field which will be provided by a tight focusing scheme with femtosecond high-power laser facilities in the near future.

2. Mathematical forms for describing EM fields focused with low f-number optics

2.1 Modeling of tight focusing scheme with parabolic surface

In order to calculate electric field distributions formed by a parabolic surface, let us consider a linearly-polarized (polarized in the x-axis), free-propagating, monochromatic EM field ( $E_{i n c} (θ_{S}, ϕ_{S})$ ) incident on a parabolic surface with a focal length of $f$ from the right (see Fig. 1). The subscripts, s and p, mean source and observation points. Thus, $E_{i n c} (θ_{S}, ϕ_{S})$ means the incident electric field on the paraboloidal surface position, $(θ_{S}, ϕ_{S})$ . According to [20], theelectric fields dominantly responsible for intensity distributions can be re-expressed as follows (A constant phase factor is omitted in equations for convenience.):

\begin{matrix} E_{x} (x_{P}, y_{P}, z_{P}) \sim \int_{θ m}^{π} \int_{0}^{2 π} d θ_{S} d ϕ_{S} E_{i n c} (θ_{S}, ϕ_{S}) \exp {- i k φ (x_{P}, y_{P}, z_{P}, θ_{S}, ϕ_{S})} \frac{2 f \sin θ_{S}}{(1 - \cos θ_{S})}, \\ \times {1 - \frac{\sin θ_{S} \cos ϕ_{S}}{1 - \cos θ_{S}} (1 - \frac{1 - \cos θ_{S}}{i 2 k f}) \frac{2 f \sin θ_{S} \cos ϕ_{S} - x_{P} (1 - \cos θ_{S})}{2 f}} \end{matrix}

\begin{matrix} E_{y} (x_{P}, y_{P}, z_{P}) \sim \int_{θ m}^{π} \int_{0}^{2 π} d θ_{S} d ϕ_{S} E_{i n c} (θ_{S}, ϕ_{S}) \exp {- i φ (x_{P}, y_{P}, z_{P}, θ_{S}, ϕ_{S})} \frac{2 f \sin θ_{S}}{{(1 - \cos θ_{s})}^{2}}, \\ \times {\sin θ_{S} \cos ϕ_{S} (1 - \frac{1 - \cos θ_{S}}{i 2 k f}) \frac{2 f \sin θ_{S} \sin ϕ_{S} - y_{P} (1 - \cos θ_{S})}{2 f}} \end{matrix}

\begin{matrix} E_{z} (x_{P}, y_{P}, z_{P}) \sim \int_{θ m}^{π} \int_{0}^{2 π} d θ_{S} d ϕ_{S} E_{i n c} (θ_{S}, ϕ_{S}) \exp {- i φ (x_{P}, y_{P}, z_{P}, θ_{S}, ϕ_{S})} \frac{2 f \sin θ_{S}}{{(1 - \cos θ_{S})}^{2}}, \\ \times \sin θ_{S} \cos ϕ_{S} {1 - (1 - \frac{1 - \cos θ_{S}}{i 2 k f}) \frac{2 f \cos θ_{S} - z_{P} (1 - \cos θ_{S})}{2 f}} \end{matrix}

and

φ (x_{P}, y_{P}, z_{P}, θ_{S}, ϕ_{S}) = k (z_{P} \cos θ_{S} + x_{P} \sin θ_{S} \cos ϕ_{S} + y_{P} \sin θ_{S} \sin ϕ_{S}) .

Here,

x_{P}

,

y_{P}

, and

z_{P}

represent positions at an observation point.

θ_{S}

is the polar angle measured from the positive z-axis and

ϕ_{S}

is the rotational angle measured from the positive x-axis. In the paper, an on-axis focusing condition is assumed and the EM field propagates along the negative z-direction. The minimum angle,

θ_{\min}

, determines the f-number of a parabolic mirror. The slowly varying amplitude (SVA) approximation was used, thus a distance between the source (s) and observation (p) points is expressed as

2 f / (1 - \cos θ_{S})

for the field amplitude and as

2 f / (1 - \cos θ_{S}) - ρ_{P} {\cos θ_{S} \cos θ_{P} + \sin θ_{S} \sin θ_{P} \cos (ϕ_{S} - ϕ_{P})}

for the phase of the EM field.

Fig. 1 On-axis focusing scheme for an aberrated laser beam with a low f-number parabolic mirror.

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2.2 Implementation of wavefront aberration

The wavefront aberration of an incident EM wave is one of the major factors that determine the electric field distribution in the focal plane. The wavefront aberration is the phase delay function across the laser beam, and can be included in the incident EM field as follows:

E_{i n c} (θ_{S}, ϕ_{S}) = E_{0} (θ_{S}, ϕ_{S}) \exp {i k W_{i n c} (θ_{n}, ϕ_{S})} .

Here,

θ_{n}

can be interpreted as a normalized radius, defined by

(π - θ_{S}) / (π - θ_{m})

. The wavefront aberration is expanded by Zernike polynomials as

W_{i n c} (θ_{n}, ϕ_{S}) = \sum_{u, v} c_{u}^{v} Z_{u}^{v} (θ_{n}, ϕ_{S})

. In this case,

c_{u}^{v}

means the Zernike coefficient, and

Z_{u}^{v} (θ_{n}, ϕ_{S})

the Zernike polynomial for u-th radial and v-th azimuthal orders, respectively. Then, the entire phase function including the wavefront aberration is modified as

[z_{P} \cos θ_{S} + x_{P} \sin θ_{S} \cos ϕ_{S} + y_{P} \sin θ_{S} \sin ϕ_{S}] + W_{i n c} (θ_{n}, ϕ_{S})

on the reflecting surface. For a high f-number case,

θ_{S}

and

θ_{n}

are almost the same, thus the wavefront,

W_{S} (θ_{n}, ϕ_{S})

, on the surface is almost equivalent to

W_{i n c} (θ_{S}, ϕ_{S})

. However, as the f-number of a parabolic surface decreases, the wavefront on the surface becomes different from the wavefront of an incident wave. The modification appears as a different value for the polar angle (

θ

) in expressing the location on the parabolic surface. After a straightforward mathematical derivation with the help of Fig. 1, the modified normalized radius,

θ_{n}

, on the surface is given by

θ_{n} = \frac{\cos θ_{S}}{1 - \cos θ_{S}} \frac{1 - \cos θ_{m}}{\cos θ_{m}} \frac{\tan (π - θ_{S})}{\tan (π - θ_{m})} .

The second correction originates from the contribution of the change in wavefront aberration due to the polarization rotation after the reflection from a surface, which is dependent on the location ( $θ_{S}$ , $ϕ_{S}$ ) on the surface. To investigate this effect, let us define a normal vector perpendicular to the wavefront as $\vec{S} = \hat{z} W_{S} (θ_{n}, ϕ_{S})$ , with the assumption of a small amount of wavefront aberration. Then, after reflection on a parabolic surface, the normal vector to the wavefront surface is expressed by $2 \hat{n} (\vec{S} \cdot \hat{n}) - \vec{S}$ as follows:

2 \hat{n} (\vec{S} \cdot \hat{n}) - \vec{S} = (\hat{x} \sin θ_{S} \cos ϕ_{S} + \hat{y} \sin θ_{S} \sin ϕ_{S} - \hat{z} \cos θ_{S}) W_{S} (θ_{n}, ϕ_{S})

with the help of expressions for normal vectors on a parabolic surface given by²⁰

n_{x} = \frac{\sin θ_{S} \cos ϕ_{S}}{{[2 (1 - \cos θ_{S})]}^{1 / 2}}, n_{y} = \frac{\sin θ_{S} \sin ϕ_{S}}{{[2 (1 - \cos θ_{S})]}^{1 / 2}}, and n_{z} = {(\frac{1 - \cos θ_{S}}{2})}^{1 / 2} .

Finally, the wavefront component that propagates to the ρ-direction contributes to the formation of an intensity distribution near the focal plane and is given by

[2 \hat{n} (\vec{S} \cdot \hat{n}) - \vec{S}] \cdot {\vec{k}}_{ρ} = \frac{2 π}{λ} W_{S} (θ_{n}, ϕ_{S}) {\sin^{2} θ_{S} \cos 2 ϕ_{S} + \cos^{2} θ_{S}}

with

\hat{ρ} = \hat{x} \sin θ_{S} \cos ϕ_{S} - \hat{y} \sin θ_{S} \sin ϕ_{S} - \hat{z} \cos θ_{S}

. But, as expected in Eq. (6), the contribution by the

{\cdot}

term is not apparent when

\sin^{2} θ_{S} \cos 2 ϕ_{S}

<<

\cos^{2} θ_{S}

.

2.3 Diffraction integrals for tight imaging with parabolic and ellipsoidal surfaces

A tight imaging scheme using a combination of parabolic and ellipsoidal surfaces was recently proposed and experimentally demonstrated to attain a focal spot having a sub-μm spot size [13]. Thus, for the modeling purpose, it is useful to have mathematical expressions for describing EM fields under such a situation. In this section, an approach to model a tight imaging scheme with a combination of parabolic and ellipsoidal surfaces is described, based on the vector diffraction theory. In the modeling, an incident EM field is assumed to be focused in the first focal plane by a parabolic mirror located in the right side (see Fig. 2). The focal spot formed by a parabolic mirror is imaged onto the second focal plane by an ellipsoidal surface. In the first step, electric and magnetic fields, formed by a parabolic mirror, of an incident EM field are calculated in the first focal plane. Then, in the second step, the electric and magnetic fields on the surface of an ellipsoidal mirror are calculated with electric and magnetic fields in the first focal plane as inputs. The resultant fields in the second focal plane are calculated based on electric and magnetic field distributions on the surface of an ellipsoidal surface.

Fig. 2 Expressions for electric and magnetic fields in the tight imaging scheme with ellipsoidal mirror.

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In the modeling, the ellipsoidal surface is assumed to have one major axis ( $b$ ) in the z-direction and two minor axes ( $a$ ) in the x- and y-directions. Then, the eccentricity ( $e$ ) of the ellipsoid is defined by $\sqrt{1 - {(a / b)}^{2}}$ . The second focal point is chosen as an origin in thecoordinate system, thus, the first focal point is located at $z_{0} = 2 f = 2 \sqrt{b^{2} - a^{2}}$ from the origin (see Fig. 2). The electric field distributions ( $E_{x}^{S}$ , $E_{y}^{S}$ , $E_{z}^{S}$ ) on the ellipsoidal surface are calculated from electric and magnetic fields ( $E_{x}$ , $E_{y}$ , $E_{z}$ , $H_{x}$ , $H_{y}$ , and $H_{z}$ ) at $(x_{P}, y_{P}, z_{0})$ from the following integral formulas below:

\begin{array}{l} E_{x}^{S} (θ'_{S}, ϕ'_{S}) ~ \\ \int_{Σ} d x_{P} d y_{P} {H_{y} (x_{P}, y_{P}, z_{0}) - (1 - \frac{1}{i k R_{P}}) (\begin{array}{l} - E_{x} (x_{P}, y_{P}, z_{0}) \cos θ'_{S} \\ + E_{z} (x_{P}, y_{P}, z_{0}) \frac{x_{P} - R_{P} \sin θ'_{S} \cos ϕ'_{S}}{R_{P}} \end{array})} \frac{\exp [- i φ']}{R_{P}}, \end{array}

\begin{array}{l} E_{y}^{S} (θ'_{S}, ϕ'_{S}) ~ \\ \int_{Σ} d x_{P} d y_{P} {H_{x} (x_{P}, y_{P}, z_{0}) - (1 - \frac{1}{i k R_{P}}) (\begin{array}{l} - E_{y} (x_{P}, y_{P}, z_{0}) \cos θ'_{S} \\ + E_{z} (x_{P}, y_{P}, z_{0}) \frac{y_{P} - R_{P} \sin θ'_{S} \sin ϕ'_{S}}{R_{P}} \end{array})} \frac{\exp [- i φ']}{R_{P}}, \end{array}

\begin{array}{l} E_{z}^{S} (θ'_{S}, ϕ'_{S}) ~ \\ \int_{Σ} d x_{S} d y_{S} (1 - \frac{1}{i k R_{P}}) (\begin{array}{l} E_{x} (x_{P}, y_{P}, z_{0}) \frac{x_{P} - R_{P} \sin θ'_{S} \cos ϕ'_{S}}{R_{P}} + \\ E_{y} (x_{P}, y_{P}, z_{0}) \frac{y_{P} - R_{P} \sin θ'_{S} \sin ϕ'_{S}}{R_{P}} + E_{z} (x_{P}, y_{P}, z_{0}) \cos θ'_{S} \end{array}) \frac{\exp [- i φ']}{R_{P}}, \end{array}

and,

φ' (x_{P}, y_{P}, z_{P}, θ_{S}, ϕ_{S}) = k (x_{P} \sin θ'_{S} \cos ϕ'_{S} - y_{P} \sin θ'_{S} \sin ϕ'_{S}) .

The input magnetic fields are obtained from relations of

H_{x} = {E_{y} - (k_{y} / k_{z}) E_{z}} ~ E_{y}

,

H_{y} = {(k_{x} / k_{z}) E_{z} - E_{x}} ~ - E_{x}

, and

H_{z} = {(k_{y} / k_{z}) E_{x} - (k_{x} / k_{z}) E_{y}} ~ 0

, by assuming

k_{x} / k_{z} ≪ 1

and

k_{y} / k_{z} ≪ 1

. The surface normal vectors (

n_{x}' = 0

,

n_{y}' = 0

, and

n_{z}' = - 1

) in the first focal plane are used to derive the above expressions because the EM field is assumed to propagate along the negative z-direction. The distance,

u

, between the source point in the first focal plane and the observation point on the ellipsoidal surface is given by

u ≃ R_{P} - (x_{P} \sin θ'_{S} \cos ϕ'_{S} - y_{P} \sin θ'_{S} \sin ϕ'_{S})

, and

u ≃ R_{P}

was assumed for the field amplitude with the SVA, where

R_{P} = R_{P} (θ'_{S}) = b (1 - e^{2}) / (1 + e \cos θ'_{S})

is the distance from a focus in the first focal plane to a location on an ellipsoidal surface. The integral range of +/− 10λ was taken for the x- and y-directions in case of f/3 focusing condition.

Finally, electric fields in the second focal plane and its vicinity formed by an ellipsoidal surface are calculated from the following formulas with electric and magnetic fields on the surface.

\begin{array}{l} E_{x}^{E M} (x, y, z) ~ \\ \int_{Σ} d θ d ϕ \frac{ρ \sin θ \exp (i k u)}{\sqrt{{(a / ρ)}^{2} - \sin^{2} θ}} {\begin{cases} {- H_{z}^{S} (θ, ϕ) \frac{\sin θ \sin ϕ}{\sqrt{1 - e^{2}}} - H_{y}^{S} (θ, ϕ) \sqrt{{(a / ρ)}^{2} - \sin^{2} θ}} + \\ (1 - \frac{1}{i k ρ}) \frac{Δ x}{ρ} {\begin{cases} - E_{x}^{S} (θ, ϕ) \frac{\sin θ \cos ϕ}{\sqrt{1 - e^{2}}} - E_{y}^{S} (θ, ϕ) \frac{\sin θ \sin ϕ}{\sqrt{1 - e^{2}}} \\ + E_{z}^{S} (θ, ϕ) \sqrt{{(a / ρ)}^{2} - \sin^{2} θ} \end{cases}} \end{cases}}, \end{array}

\begin{array}{l} E_{y}^{E M} (x, y, z) ~ \\ \int_{Σ} d θ d ϕ \frac{ρ \sin θ \exp (i k u)}{\sqrt{{(a / ρ)}^{2} - \sin^{2} θ}} {\begin{cases} {H_{x}^{S} (θ, ϕ) \sqrt{{(a / ρ)}^{2} - \sin^{2} θ} + H_{z}^{S} (θ, ϕ) \frac{\sin θ \cos ϕ}{\sqrt{1 - e^{2}}}} + \\ (1 - \frac{1}{i k ρ}) \frac{Δ y}{ρ} {\begin{cases} - E_{x}^{S} (θ, ϕ) \frac{\sin θ \cos ϕ}{\sqrt{1 - e^{2}}} - E_{y}^{S} (θ, ϕ) \frac{\sin θ \sin ϕ}{\sqrt{1 - e^{2}}} \\ + E_{z}^{S} (θ, ϕ) \sqrt{{(a / ρ)}^{2} - \sin^{2} θ} \end{cases}} \end{cases}}, \end{array}

and

\begin{array}{l} E_{z}^{E M} (x, y, z) ~ \\ \int_{Σ} d θ d ϕ \frac{ρ \sin θ \exp (i k u)}{\sqrt{{(a / ρ)}^{2} - \sin^{2} θ}} {\begin{cases} {- H_{y}^{S} (θ, ϕ) \frac{\sin θ \cos ϕ}{\sqrt{1 - e^{2}}} + H_{x}^{S} (θ, ϕ) \frac{si n θ \sin ϕ}{\sqrt{1 - e^{2}}}} + \\ (1 - \frac{1}{i k ρ}) \frac{Δ z}{ρ} {\begin{cases} - E_{x}^{S} (θ, ϕ) \frac{\sin θ \cos ϕ}{\sqrt{1 - e^{2}}} - E_{y}^{S} (θ, ϕ) \frac{\sin θ \sin ϕ}{\sqrt{1 - e^{2}}} \\ + E_{z}^{S} (θ, ϕ) \sqrt{{(a / ρ)}^{2} - \sin^{2} θ} \end{cases}} \end{cases}} . \end{array}

Here,

Δ x

,

Δ y

,

Δ z

, and

u

are expressed by

ρ \sin θ \cos ϕ - x

,

ρ \sin θ \sin ϕ - y

,

ρ \cos θ - z

, and

ρ - (x \sin θ \cos ϕ + y \sin θ \sin ϕ + z \cos θ)

, respectively. In this case, the distance from a location on an ellipsoidal surface to the focus in the second focal plane is

ρ (θ) = b (1 - e^{2}) / 1 - e \cos θ

with the definition of

θ = \tan^{- 1} [{R_{P} (θ'_{S}) \cos (π - θ'_{S}) \tan (θ'_{S})} / {R_{P} (θ'_{S}) \cos (π - θ'_{S}) - 2 f}]

. The surface normal vectors for the ellipsoidal surface (

n_{x}^{S} = - b ρ \sin θ \cos ϕ / \sqrt{f^{2} ρ^{2} \sin^{2} θ + a^{4}}

,

n_{y}^{S} = - b ρ \sin θ \sin ϕ / \sqrt{f^{2} ρ^{2} \sin^{2} θ + a^{4}}

, and

n_{z}^{S} = a \sqrt{a^{2} - ρ^{2} \sin^{2} θ} / \sqrt{f^{2} ρ^{2} \sin^{2} θ + a^{4}}

) are used to obtain integral forms with the relation of

n_{x}^{2} + n_{y}^{2} + n_{z}^{2} = 1

. By the definition of ellipsoid,

R_{P} (θ'_{S}) + ρ (θ)

remains constant for all

θ_{S}

. Again, the relations of

H_{x} ~ E_{y}

,

H_{y} ~ - E_{x}

, and

H_{z} ~ 0

are used to obtain information on magnetic fields (

H_{x}^{S}

,

H_{y}^{S}

, and

H_{z}^{S}

) in integrals. The nominal magnification is given by

(π - θ'_{S, \min}) / (π - θ_{\min})

.

2.4 Femtosecond focal spot: from monochromatic to coherent polychromatic spectrum

In previous sections, the mathematical expressions for calculating field distributions in the focal plane have been discussed for a monochromatic EM field. However, a femtosecond laser pulse typically has a broad spectrum of several tens of nm, so the effect of broad spectrum of a femtosecond laser pulse on the focal spot should be considered in order to describe the focal spot of a femtosecond laser pulse (femtosecond focal spot). The temporal profile of a femtosecond laser pulse is, in general, given by the superposition of EM waves within a broad spectrum. In our research, unlike previous studies using the propagation method [17–19], we adapted the superposition concept to calculate a femtosecond focal spot. A broadband femtosecond laser pulse is considered as a coherent polychromatic EM wave having a spectral power and a spectral phase. And, electric fields ( $E_{x} (λ : x, y, z)$ , $E_{y} (λ : x, y, z)$ , and $E_{z} (λ : x, y, z)$ ) are calculated in the focal plane and its vicinity by diffraction integrals (Eqs. 1(a) to 1(c)) for a monochromatic EM wave at a wavelength ( $λ$ ). This process is repeated to have all electric fields for every wavelength in the spectrum. Then, electric fields are coherently superposed together in space ( $x_{P}, y_{P}, z_{P}$ ) with a given spectral amplitude ( $W_{λ} = \sqrt{I_{λ} / I_{λ, \max}}$ ) and phase ( $α_{λ}$ ), by assuming no change in spectral power and spectral phase after reflection from a focusing optic. Computationally, the resultant electric fields for a femtosecond focal spot are expressed with spectral amplitude and phase as below:

\begin{matrix} E_{x, y, z} (x_{P}, y_{P}, z_{P}) = W_{λ 1} \exp (i α_{λ 1}) E_{x, y, z} (λ_{1} : x_{P}, y_{P}, z_{P}) + W_{λ 2} \exp (i α_{λ 2}) E_{x, y, z} (λ_{2} : x_{P}, y_{P}, z_{P}) + . \\ \dots + W_{λ n} \exp (i α_{λ n}) E_{x, y, z} (λ_{n} : x_{P}, y_{P}, z_{P}) \end{matrix}

In the expression, the spectrum was sliced into n components. The subscripts ( $x, y, z$ ) mean the polarization directions of the EM field, and the monochromatic electric field ( $E_{x, y, z} (λ_{n} : x_{P}, y_{P}, z_{P})$ ) is given by $E_{x, y, z} (x_{P}, y_{P}, z_{P}) / λ_{n}$ . The resultant electric fields, $E_{x, y, z} (x_{P}, y_{P}, z_{P})$ , provide the intensity distributions of a coherent polychromatic EM wave in three dimensional space. Contrary to the monochromatic case, a different field oscillation period at a different wavelength induces a phase mismatch among waves at different wavelengths, and reduces the intensity quickly as the observation position moves away from the origin of focal plane [22]. The intensity distribution of a femtosecond focal spot along the propagation direction is interpreted as a temporal intensity profile of the spot. The spatial spread of a laser pulse in the propagation direction is considered as a temporal profile in the single-shot autocorrelation measurement. Thus, the spatial spread in the propagation direction represents a temporal profile of the focal spot at a moment.

3. Numerical results for focal spots with low f-number optics

The calculations were performed with home-made simulation codes written in Python. A circular, uniform, and x-polarized electric field with a wavefront aberration is used as an incident field. However, an arbitrary electric field based on the measurement can be used as an input with a proper grid size as well. The Python built-in function (simps) using the Simpson’s rule was used to perform the finite integration with sampled data. Electric fields in x-, y-, and z-polarizations were calculated in the focal plane and its vicinity. Finally, ${| E_{x} (x, y, z) |}^{2}$ , ${| E_{y} (x, y, z) |}^{2}$ , ${| E_{z} (x, y, z) |}^{2}$ , and ${| E_{x} (x, y, z) |}^{2} + {| E_{y} (x, y, z) |}^{2} + {| E_{z} (x, y, z) |}^{2}$ are plotted as intensity distributions for x-, y-, z-polarized, and entire electric fields, respectively. The intensity ( ${| E_{x} (x, y, z) |}^{2} + {| E_{y} (x, y, z) |}^{2} + {| E_{z} (x, y, z) |}^{2}$ ) for the entire electric field can be regarded as an intensity distribution measurable by an image sensing device such as a charge-coupled device (CCD).

3.1 Accuracy and validity of integral method

In order to examine the accuracy of vector diffraction integrals, a focal spot for a flat-top, monochromatic (800 nm) and circular EM field was calculated under high f-number (f/10) condition by assuming the far-field approximation, and the calculated focal spot was compared to the Airy disk and an intensity distribution obtained with the fast Fourier-transform (FT) (see the left-top sub-figure in Fig. 3). The first, second, third minima of the Airy disk should be observed at radii of 9.8 μm, 17.9 μm, and 25.9 μm, respectively. When a focal spot was calculated by the fast FT with a step of 0.25 μm, the first, second, and third minima were observed at 9.75 μm, 18.0 μm, and 26.0 μm from the origin, respectively. With diffraction integrals, the first, second, and third minima were shown at 9.75 μm, 18.0 μm, and 26.0 μm as well. The slight difference between values originated from the digitization of the focal plane because of the grid size (0.25 μm) chosen in the calculation. As the f-number decreases to 1, due to the polarization effect, the minima calculated with diffraction integrals started to deviate from values predicted by the Airy disk and an intensity distribution with the fast FT (see the first row of Fig. 3). For example, for the f/1 case, the minima with diffraction integral were observed at 1.05 μm, 1.90 μm, and 2.75 μm. Instead, the minima were predicted at 0.98 μm, 1.79 μm, and 2.58 μm by the Airy disk, and at 0.98 μm, 1.80 μm, and 2.60 μm by the fast FT. Thus, it can be inferred that the prediction of the spot size based on the scalar diffraction theory becomes more invalid as the f-number becomes lower than 1.

Fig. 3 Line plots of intensity distributions calculated with fast FT (red line) and vectorial diffraction integral (blue line). An x-polarized EM field is used as an input and ${| E_{x} (x, y, z) |}^{2}$ are plotted in the positive x-axis.

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Next, the focal spots for an aberrated EM field were examined at different defocus values expressed in the Zernike coefficient. Wavefront aberration for three different amounts of defocus values (i.e., $c_{2}^{0}$ = 0.1 μm, 0.3 μm, and 0.5 μm) were added to an incident flat-top monochromatic EM field. For the defocus values of 0.1 μm (small) and 0.3 μm (moderate), the focal spots calculated with fast FT and diffraction integral were almost identical under high f-number conditions, validating the thin lens approximation (TLA) used in the fast FT method. The TLA became invalid under conditions of low f-number and a large amount of wavefront aberration, by showing the increase of difference in intensities obtained with the fast FT and the diffraction integral. For the defocus value of 0.5 μm under the high f-numbercondition (f/10), the difference of 8% in intensities obtained with the fast FT and the diffraction integral was observed at the center. The difference increased up to 45% as the f-number decreased to 2.5, and intensity profiles becomes different. On the other hand, the intensity difference decreased from 8% to 5% when increasing the f-number from 10 to 100, by relaxing the invalidity of the TLA broken with a large amount of defocus. This implies that the modification of a focal spot for a highly aberrated beam will be more sensitive to the tight focusing condition.

3.2 Focal spots of ideal and aberrated electromagnetic waves at low f-numbers

In this section, the modification of a focal spot focused with low f-number optics has been further described with two dimensional intensity distributions for ${| E_{x} (x, y, z) |}^{2}$ , ${| E_{y} (x, y, z) |}^{2}$ , and ${| E_{z} (x, y, z) |}^{2}$ . The change of focal spots for electric fields with and without wavefront aberration was traced as the f-number decreases from 5 to 0.25. The flap-top circular fields were used again in calculations. A certain amount of wavefront aberration ( $c_{2}^{- 2}$ = 0.07 μm, $c_{3}^{- 3}$ = 0.05 μm, $c_{3}^{- 1}$ = 0.04 μm, and $c_{3}^{1}$ = 0.02 μm) was introduced to the field because this amount of wavefront aberration can easily exist in a high-power laser either before or after wavefront correction. The calculation results are summarized in Figs. 4(a) and 4(b).

Fig. 4 Intensity distributions for an x-polarized laser beam (a) with and (b) without a wavefront aberration under various focusing conditions. In the figure, the horizontal axis is the x-axis and the vertical axis is the y-axis.

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Figure 4(a) shows the change of a focal spot for an electric field without wavefront aberration as the f-number decreases. As shown in Fig. 4(a), under f/5 condition, the peak intensity of a longitudinal electric field ( $E_{z}$ ) was less than 0.1% of that of x-polarized field component ( $E_{x}$ ). The peak intensity peak of $E_{z}$ increased up to 41% of that of $E_{x}$ under f/0.25 condition. The peak intensity of the y-polarized component was negligible, showing the increase in peak intensity from almost 0% (f/5) to 3% (f/0.25) of $E_{x}$ . Finally, under low f-number conditions below 1, the entire intensity distributions were remarkably affected by the elongation of $E_{x}$ component and the intensity increase in $E_{z}$ component along the polarization direction.

Figure 4(b) shows the change of a focal spot for an aberrated electric field as the f-number decreased. The figure shows how the focal spot of an aberrated electric field becomes influenced by the focusing condition. Under a high f-number condition (f/5), the focal spot of an aberrated field was determined by the characteristics of the electric field (such as wavefront aberration and spatial profile). And, the shape of the focal spot was almost same as that obtained with the fast FT, because the focusing condition and the amount of wavefront aberration did not violate the TLA. Instead, under lower f-number conditions, focal spots were also influenced by the vectorial properties, resulting in the elongation along the polarization direction. With a given amount of wavefront aberration, the peak intensity of $E_{z}$ increased up to 40% of that of $E_{x}$ under f/0.25 condition. Further calculation with a higher amount of wavefront aberration ( $c_{2}^{2}$ = $c_{2}^{- 2}$ = $c_{3}^{- 1}$ = 0.15 μm) showed that intensity distribution under f/0.5 focusing condition was still different from the intensity distribution obtained with the fast FT.

3.3 Tight imaging with a combination of parabolic and ellipsoidal mirrors

In this section, focal spots calculated with the tight imaging scheme were examined under different de-magnification conditions. The use of an f/3 parabolic mirror is assumed, and the length of the major axis was 3.5 cm as indicated in [13]. The de-magnification varied by changing the minor axis from 3.5 mm to 2 mm, which resulted in the change of de-magnification from 1 to 0.2. The calculation results were summarized in Fig. 5.

Fig. 5 Intensity distributions with tight imaging scheme using a parabolic and ellipsoidal mirrors. The first row represents intensities obtained by focusing an aberrated EM field with a f/3 parabolic mirror. The second row represents intensities on the ellipsoidal mirror. The third and fourth rows represent intensities imaged with ellipsoidal mirrors having a different length of the minor axis. In the figure, the horizontal axis is the x-axis and the vertical axis is the y-axis.

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Compared to the tight focusing scheme using a single parabolic mirror, the tight imaging scheme showed loose polarization-dependent properties. For the case of a relatively long minor axis (a = 2.9 mm) as shown in Fig. 5, the electric field was almost focused within a wavelength, and the peak intensity of $E_{z}$ was only about 1% of that of $E_{x}$ . This tight imaging condition shows a better focusing performance than a parabolic mirror of f/1, in terms of beam size and reduction of $E_{z}$ component. The calculated de-magnifications of the focal spot were 0.3 and 0.25 in x- and y-directions, respectively. When a minor axis decreased to 2.0 mm, the polarization-dependent focusing property became dominant, showing typical intensity properties for all polarization components under low f-number conditions. The peak intensity of $E_{z}$ increased to 8% of that of $E_{x}$ . In this case, the de-magnifications in x- and y-directions were 0.2 and 0.14, respectively, and showed a slightly better focusing performance than a parabolic mirror of f/0.5.

The relaxation of vectorial features of the focal spot calculated with the tight imaging scheme is related to a smaller incident angle to the normal direction on the ellipsoidal surface. Thus, the tight imaging scheme might be beneficial to produce a relatively well-polarized intense electric field, and can be used in ultra-intense laser-matter interaction experiments where a minimized longitudinal electric field component ( $E_{z}$ ) matters in order to avoid complex physical situations. It should be noted that, even though the on-axis imaging properties are mostly investigated in this paper, the off-axis imaging properties can be examined with a proper projection and integration ranges during finite integration processes.

3.4 Femtosecond focal spot in spatio-temporal domain

The spectral power and its spectral phase of a femtosecond laser pulse were shown in Fig. 6(a) and 6(b). The spectral power was taken from an ultrashort laser oscillator in the Advanced Photonics Research Institute (APRI) PW laser [23], and the spectral phase was arbitrarily chosen from a broadband reflection mirror by assuming the development of a <15-fs, 10-PW laser. For the reference, a temporal profile of the femtosecond laser pulse was calculated based on the fast FT with given spectral power and phase. The calculated temporal profile was presented in Fig. 6(c). The FWHM of the femtosecond laser pulse is about 12 fs.

Fig. 6 (a) Spectrum of a femtosecond laser pulse, (b) Spectral phase induced by a broadband mirror, (c) Temporal pulse profile calculated through the fast FT with spectrum and spectral phase.

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We calculated femtosecond focal spots by using diffraction integrals with Eq. (1) and Eq. (9), and Figs. 6(a) and 6(b). In the calculation, the spectral bandwidth and phase were sliced with a step of 1 nm. Figure 7 shows spatial and temporal line plots of femtosecond focal spots under several focusing conditions. In the spatial domain, the intensity for a femtosecond focal spot was reduced in the wing (see Fig. 6(a)) although the spatial profile of a femtosecond focal spot was similar to the shape of the focal spot for a monochromatic field in the central region. The intensity reduction is due to the phase mismatch induced by the difference in wavelength. Although electric fields at different wavelengths interfere constructively at the beam center, those in the wing interfere destructively as the transverse position becomes distant from the beam center.

Fig. 7 (a) Spatial intensity distributions and (b) temporal intensity distributions. In Fig. 7(a), line plots of spatial intensity distributions of focal spots for a monochromatic and a coherent polychromatic (broadband) electric field are shown under f/3 focusing optic. In Fig. 7(b), line plots of temporal intensity distributions of focal spots for a broadband electric field are shown for the cases calculated with fast FT and diffraction integrals under f/3 and f/0.5.

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The coherent superposition of monochromatic fields reproduced the temporal profile along the propagation direction. The destructive interference among different wavelength components induced the phase mismatch again in the longitudinal z-direction, resulting in the formation of a pulse profile in time. The temporal profile calculated with the diffraction integral under f/3 focusing condition was almost identical to that calculated with fast FT (see Fig. 7(b)). However, the shortening in the pulse duration was observed under low f-number conditions. The shortening of the pulse duration can be understood from following explanations. Under the tight focusing condition, the electric fields that contribute the formation of a focal spot come from apparently different angles. As shown in Fig. 8(a), if we take two points (one is from the on-axis position on a parabolic surface and the other is froman off-axis position on the surface), then the phase difference between both electric fields will be $Δ φ = φ_{o f f} - φ_{o n} = k z (1 + \cos θ_{S}) = 2 π z (1 + \cos θ_{S}) / λ$ along the z-axis by the phase function in Eq. (1d). This phase difference induced by angle is related with the Rayleigh range that determines the beam size and the intensity distribution along the z-axis. This can be qualitatively understood as follows. If we define $θ$ as $π - θ_{S}$ , the phase difference is given by $Δ φ = \frac{2 π z}{λ} \sin^{2} \frac{θ}{2}$ . By defining a distance as $z_{R}$ that brings out-of-phase condition ( $Δ φ = π$ ), the distance, $z_{R}$ , is given by $z_{R} \approx \frac{λ}{4} {(\frac{2}{θ})}^{2} \approx λ {(\frac{2 f}{D})}^{2} = \frac{{(2 / 1.22)}^{2}}{λ} {(1.22 \frac{f λ}{D})}^{2} = c o n s t \times \frac{ω_{A i r y}^{2}}{λ}$ . Here, the definition of numerical aperture under the paraxial approximation ( $\sin θ \approx θ$ and $θ \approx D / 2 f$ ) is used. The distance ( $z_{R}$ ) is the same as the definition of Rayleigh range exceptfor the constant factor. The Rayleigh range becomes shorter when the f-number decreases. Under the tight focusing condition (below 1), the Rayleigh range can be compatible or shorterthan the pulse duration, depending on the pulse duration and the focusing condition. Thus, in the vicinity of the focal plane, the intensity becomes weaker when z becomes farther from the focal plane. This phenomenon is shown in Figs. 8(b) and 8(c) by calculating the intensity distribution of 800-nm monochromatic electric field focused under two different conditions (f-number of 3 and f-number of 0.5). For a focused fs laser pulse, the shortened Rayleigh range confines the electric field in a short range that can be interpreted as a temporal range at a moment in a single-shot autocorrelation measurement concept. In the time-bandwidth context, the pulse shortening phenomenon might be understood that increasing the angle, $θ$ , introduces more wavelength components (expressed in $λ / \cos θ$ ) along the z-axis.

Fig. 8 (a) Phase difference between electric fields coming from the on-axis and an off-axis position on a parabolic mirror. Intensity distributions for a monochromatic 800-nm electric field focused by (b) f/3 optic and (c) f/0.5. The phase difference shorten The Rangleigh range and affects the pulse duration along z-axis at a moment.

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Our calculation showed that the shortening of ~50% in the pulse duration was possible for a specific case of Fig. 6 under f/0.5 focusing condition. This means that a tight focusing scheme can provide an extra benefit for efficiently increasing the laser intensity by reducing not only the spot size but the pulse duration. Thus, a higher intensity of over 10²⁴ W/cm² (or corresponding electric field strength), at which intensity protons become relativistic, might be attained at a relatively lower power than that predicted by the elongation and the growth of $E_{z}$ component. The exact amount of pulse shortening will depend on the spectral bandwidth of a femtosecond laser pulse as well.

Figure 9 shows spatio-temporal intensity distributions of femtosecond focal spots for an aberrated laser pulse (Zernike coefficient used are $c_{2}^{- 2}$ = 0.07 μm, $c_{3}^{- 3}$ = 0.05 μm, $c_{3}^{- 1}$ = 0.04 μm, and $c_{3}^{1}$ = 0.02 μm.) under two different focusing conditions (f/3 and f/0.5). In the figure, the intensity distributions in the X-Y plane provide information on spatial profiles of a femtosecond focal spot, and the intensity distributions in the X-Z plane does information on temporal profiles. By assuming a 12-fs, 10-PW, uniformly-circular, and aberrated laser pulse as an input, peak intensities for $E_{x}$ field component increased up to ~8.8x10²² W/cm² for f/3 and ~2.5x10²⁴ W/cm² for f/0.5, respectively. Under same conditions, the peak intensities for $E_{z}$ component rapidly increased to ~3.1x10²⁰ W/cm² and ~2.4x10²³ W/cm² as in Fig. 9. These intensities along the z-direction should be taken into account to better describe the motion of charged particles under an extremely strong EM field which is formed by tightly focusing a femtosecond high-power laser pulse. Even though a spatio-temporal intensity distribution for a fs focal spot tightly imaged by the combination of paraboloidal and ellipsoidal mirrors was not shown in the paper, it can be straightforwardly computed by the coherent superposition with computational burden.

Fig. 9 Spatio-temporal intensity distributions of a focal spot focused with (a) a f/3 parabolic mirror and (b) a f/0.5 parabolic mirror. The intensity distribution in the XY plane means the spatial intensity distribution of the focal spot and the intensity distribution in the XZ plane means the temporal distribution of the focal spot. +/− 6 μm in the z-axis corresponds to +/− 20 fs in time. Under tight focusing condition, electric fields of a focal spot are compressed in both spatial and temporal domains. The peak intensities are described in figures.

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4. Conclusion

A method for calculating a femtosecond focal spot, which uses vector diffraction integrals and coherent superposition, has been developed to describe an intensity distribution in the spatial and temporal domains under low f-number conditions. The calculation results with the developed method provide information on how transverse and longitudinal electric fields are modified, under low f-number conditions, by the polarization of an incident field in the spatio-temporal domain. In the paper, the modifications in the final intensity distribution were investigated under various conditions using tight focusing and imaging schemes for aberrated monochromatic and/or polychromatic (and having the coherent phase relation) electric fields. The shortening in the pulse duration of a femtosecond focal spot is predicted under low f-number conditions, which should be experimentally confirmed from the measurement. The pulse shortening phenomenon might open a discussion on the definition of pulse duration that should be physically meaningful for a tightly-focused fs laser spot. The precise characterization of a femtosecond focal spot will be crucial in assessing an exact peak intensity of a high-power femtosecond laser pulse and providing more realistic information on extreme intensities formed by ultra-intense laser pulses focused by low f-number optics.

Tae Moon Jeong’s permanent address is Advanced Photonics Research Institute, Gwangju Institute of Science and Technology.

Acknowledgment

This work was supported by the Ministry of Science, ICT and Future Planning of Korea through the Infrastructure for femto-technology program supervised by the National IT Industry Promotion Agency. This work was partially supported by ELI (Project No. CZ.1.05/1.1.00/02.0061) and also by the Academy of Sciences of the Czech Republic (M100101210).

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Spatio-temporal modification of femtosecond focal spot under tight focusing condition

Abstract

1. Introduction

2. Mathematical forms for describing EM fields focused with low f-number optics

2.1 Modeling of tight focusing scheme with parabolic surface

2.2 Implementation of wavefront aberration

2.3 Diffraction integrals for tight imaging with parabolic and ellipsoidal surfaces

2.4 Femtosecond focal spot: from monochromatic to coherent polychromatic spectrum

3. Numerical results for focal spots with low f-number optics

3.1 Accuracy and validity of integral method

3.2 Focal spots of ideal and aberrated electromagnetic waves at low f-numbers

3.3 Tight imaging with a combination of parabolic and ellipsoidal mirrors

3.4 Femtosecond focal spot in spatio-temporal domain

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (9)

Equations (17)

Optics Express