Focusing characteristics of linearly polarized ultrashort pulses at the focal plane

Yali Zheng; Xunming Cai; Xin Zhao

doi:10.1364/OE.421275

1. Introduction

The focusing characteristics of beams have been studied for many years. For example, there have been many studies on the tight focus characteristics of the azimuthally polarized and radially polarized beams [1–6]. Under the condition of being focused by high numerical aperture objective lenses, minimal focus spots with strong longitudinal electric field components are formed at the focal plane by the radially polarized beams and hollow focal spots by the azimuthally polarized beams [7–10]. These unique properties make the vector beams widely used in the field of optical storage [11], particle capture and acceleration [12–15], high-resolution microscopy [2,16–18] and material processing [19].

The RWT is widely used to study the focusing phenomenon of light beams [2,20,21]. The theoretical and experimental researches on the sharper focus of vector beams have been reported [22–24]. These researches reveal some excellent methods to achieve super-resolution spots. Analytical models of beam beyond the paraxial approximation have been developed for many years. The complex point source model and the CSSM are used to study ultrashort tightly focused light pulses [25–32]. Hertz potential method (HPM) combined with the CSSM can be used to study the focusing characteristics of ultrashort tightly focused light pulses [26].

The convergence and divergence focusing phenomena of radially polarized beams are analyzed in the literature [33]. Compared to radially polarized pulses, linearly polarized pulses are more common. The non-centrosymmetric polarization characteristics of linearly polarized pulses bring new focusing characteristics. In this work, the focusing characteristics of linearly polarized ultrashort pulses are studied by both the RWT and CSSM. The differences of the numerical simulations of the two models are analyzed. The dynamic focusing phenomena from the halo to two light lobes to the elliptical focus spot are analyzed.

2. Model and equations

The focusing properties of tightly focused Poisson-like ultrashort pulses with the linearly polarized electric field are studied. Both the RWT and the CSSM are used to simulate the focusing properties of the ultrashort beam. The differences of the two models are studied. The research model is shown in Fig. 1. f is the focal length. $\theta$ is the pitch angle of light and $\alpha = \arcsin (NA/n)$ is the semi-aperture angle (convergence angle) of the objective lens.

Fig. 1. Geometric model of the linearly polarized pulsed beam.

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In Fig. 1, $\overrightarrow {{e_0}}$ represents the electric field direction of the incident beam, and it is along the x-axis. This means that the incident beam is linearly polarized in the x direction. $z = 0$ represents the focal plane. ${\phi _s}$ and ${\rho _s}$ are the polar coordinates in the focal plane. r is the incident beam radius. ${x_0}$, ${y_0}$ are the rectangular coordinates of the incident beam and $\phi$ is the azimuth angle with respect to the $x$-axis. Such the electric field of the incident beam ${E_0}$ be decomposed into the radially polarized component ${E_\rho }$ and the azimuthally polarized component ${E_\phi }$. From Fig. 1(a), ${E_\rho }$ can be written as ${E_\rho } = {E_0} \cdot \cos \phi$ and ${E_\phi }$ can be written as ${E_\phi } = {E_0} \cdot \sin \phi$. Using RWT, The Cartesian components of the electric fields of both the ${E_\rho }$ and the ${E_\phi }$ at the focal plane can be found in the Ref. [2]. Such the electric field expressions of the linearly polarized beam at the focal plane can be written as follows:

(1)$$E(\textrm{x},y,z,t) = \left[ \begin{array}{l} {e_x}\\ {e_y}\\ {e_z} \end{array} \right] ={-} \frac{{ik}}{{2\pi }}\int_0^\alpha {\int_0^{2\pi } {\sin \theta \cdot {{\cos }^{{1 / 2}}}} } \theta \cdot {l_0}(\theta ) \cdot f(t) \cdot \left[ \begin{array}{l} \cos \theta + {\sin^2}\phi (1 - \cos \theta )\\ \cos \phi \sin \phi (\cos \theta - 1)\\ - \sin \theta \cos \phi \end{array} \right]d\theta d\phi, $$

In Eq. (1), $k$ is the wave vector and ${l_0}(\theta )$ is the relative amplitude. we let ${l_0}(\theta )\textrm{ = 2}B$, $B$ is a constant. $f(t)$ is the pulse time-domain function and is written as $1/{(1 - i{\omega _0}t/s)^{s + 1}} \cdot {e^{i{\phi _0}}}$ for the Poisson-like pulse [26], where ${\omega _0}$ is the carrier frequency and s is a real positive parameter. The time t can be written as ${t_0} - (f - {\rho _s}\sin \theta \cos (\phi - {\phi _s}))/c$. We let ${t_1} = {t_0} - f/c$, so t can be written as ${t_1} + {\rho _s}\sin \theta \cos (\phi - {\phi _s})/c$. The expressions of the electric field can be further transformed into the following form

(2)$$\begin{aligned} e_x^{(s)} &={-} \frac{{\textrm{i}B}}{\pi }\int_0^{2\pi } {\int_0^\alpha {{{\cos }^{{\textrm{1} / \textrm{2}}}}} } \theta \cdot \sin \theta [{\cos \theta + {{\sin }^2}\phi \cdot (1 - \cos \theta )} ]\\&\quad\cdot {\textrm{(1} - \textrm{i}{\omega _0} \cdot \textrm{(}{\textrm{t}_1}\textrm{ + }{\rho _s}\textrm{sin}\theta \textrm{cos(}\phi \textrm{ - }{\phi _s}\textrm{)/c)/s)}^{ - (s\textrm{ + 1)}}} \cdot {\textrm{e}^{i{\phi _0}}}d\theta d\phi \\ e_y^{(s)} &={-} \frac{{\textrm{i}B}}{\pi }\int_0^{2\pi } {\int_0^\alpha {{{\cos }^{{\textrm{1} / \textrm{2}}}}} } \theta \cdot \sin \theta [{\cos \phi \cdot \sin \phi \cdot (\cos \theta - 1)} ]\\&\quad\cdot {\textrm{(1} - \textrm{i}{\omega _0} \cdot \textrm{(}{\textrm{t}_1}\textrm{ + }{\rho _s}\textrm{sin}\theta \textrm{cos(}\phi \textrm{ - }{\phi _s}\textrm{)/c)/s)}^{ - (s\textrm{ + 1)}}} \cdot {\textrm{e}^{i{\phi _0}}}d\theta d\phi \\ e_z^{(s)} &= \frac{{\textrm{i}B}}{\pi }\int_0^{2\pi } {\int_0^\alpha {{{\cos }^{{\textrm{1} / \textrm{2}}}}} } \theta \cdot \sin {\theta ^2}\cos \phi \cdot {\textrm{(1} - \textrm{i}{\omega _0} \cdot \textrm{(}{\textrm{t}_1}\textrm{ + }{\rho _s}\textrm{sin}\theta \textrm{cos(}\phi \textrm{ - }{\phi _s}\textrm{)/c)/s)}^{ - (s\textrm{ + 1)}}} \cdot {\textrm{e}^{i{\phi _0}}}d\theta d\phi \end{aligned}. $$

The total intensity distributions of light are given by $I\textrm{ = e}_x^{(s)} \cdot \textrm{e}_x^{(s)\ast } + \textrm{e}_y^{(s)} \cdot \textrm{e}_y^{(s)\ast } + \textrm{e}_z^{(s)} \cdot \textrm{e}_z^{(s)\ast }$

Analytical expressions of the focused pulsed beam are used to study the physical properties of ultrashort pulses and the interaction of pulses and materials. Using both the sink-source method and the dipole moment model [26,32], the analytical equations can be derived and used to study the focusing phenomena at the focal plane of the linearly polarized pulses. The dipole moment can be defined as

(3)$$P(\textrm{r},t) = {p_0}f(t)\delta ({\mathop{r}^\rightharpoonup}), $$

where $f\textrm{(t) = 1/(1} - i{\omega _\textrm{0}}t\textrm{/s}{\textrm{)}^{\textrm{s + 1}}} \cdot {e^{i{\phi _0}}}$. The electric Hertz vector and the magnetic Hertz vector of the x-linearly polarized field can be written as

(4)$$\overrightarrow {{\Pi _e}} = P(r,t)\overrightarrow {{e_x}},\;\;\;\overrightarrow {{\Pi _m}} = P(r,t)\eta _0^{ - 1}\overrightarrow {{e_y}}$$

where ${\eta _0} = \sqrt {{{{\mu _0}} / {{\varepsilon _0}}}}$. The spatiotemporal translation is introduced for the pulse propagating in the z direction as

(5)$$z \to {z^{\prime}} = z + ia.\;\;\;t \to {t^{\prime}} = {t_0} + i\frac{a}{c}$$

where $a = 1\sqrt {{{(1 + {k^2}w_0^2/2)}^2} - 1} /k$ . ${w_0}$ is the waist spot size parameter. The complex distance is ${R^{\prime}} = \sqrt {{x^2} + {y^2} + {{(z + ia)}^2}}$. The complex retarded times are ${\tau ^{\prime}} = t^{\prime} - R^{\prime}/c$ and ${\tau ^{^{\prime\prime}}} = t^{\prime} + R^{\prime}/c$. The expressions of the electric Hertz vector and the magnetic Hertz vector in the CSSM can be obtained by folding the source with the $\delta$ function,

(6)$$D({R^{\prime}},{t^{\prime}}) = \frac{{{c^2}{\mu _0}}}{{4\pi }}\frac{{\delta ({t^{\prime}} - {R^{\prime}}/c) - \delta ({t^{\prime}} + {R^{\prime}}/c)}}{{{R^{\prime}}}}$$

The time domain pulse functions $f(t)$ can be written as ${f_1}({\tau ^{\prime}})$ and ${f_2}({\tau ^{^{\prime\prime}}})$. Such the electric Hertz vector and the magnetic Hertz vector in the CSSM can be written as

(7)$$\overrightarrow {{\Pi _e}} = \frac{{{c^2}{\mu _0}{p_0}}}{{4\pi {R^{\prime}}}}({f_1}({\tau ^{\prime}}) - {f_2}({\tau ^{^{\prime\prime}}}))\overrightarrow {{e_x}},\;\;\;\overrightarrow {{\Pi _m}} = \frac{{c{\mu _0}{p_0}}}{{4\pi {R^{\prime}}}}({f_1}({\tau ^{\prime}}) - {f_2}({\tau ^{^{\prime\prime}}}))\overrightarrow {{e_y}}$$

as the derivation process in [26], the following expressions of field can be obtained:

\begin{aligned}&\mathop {{E_x}}\limits^ \to = \frac{{{c^2}{\mu _0}{p_0}}}{{4\pi }}\\ &\cdot real\left\{ \frac{{{x^2}}}{{{R^{\prime}}^2}} \cdot (\frac{{{{\mathop f\limits^{..} }_1} - {{\mathop f\limits^{..} }_2}}}{{{c^2}{R^{\prime}}}} + \frac{{3({{\mathop f\limits^. }_1} + {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}^2}} + \frac{{3({{\mathop f\limits^{} }_1} - {{\mathop f\limits^{} }_2})}}{{{R^{\prime}}^3}}) - (\frac{{{{\mathop f\limits^{..} }_1} - {{\mathop f\limits^{..} }_2}}}{{{c^2}{R^{\prime}}}} + \frac{{({{\mathop f\limits^. }_1} + {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}^2}} + \frac{{({{\mathop f\limits^{} }_1} - {{\mathop f\limits^{} }_2})}}{{{R^{\prime}}^3}}) \right.\\ & \left.- \frac{{{z^{\prime}}}}{{{R^{\prime}}^2}} \cdot (\frac{{{{\mathop f\limits^{..} }_1} + {{\mathop f\limits^{..} }_2}}}{{{c^2}}} + \frac{{({{\mathop f\limits^. }_1} - {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}}}) \right\} \quad\mathop {{e_x}}\limits^ \to \end{aligned}

\mathop {{E_y}}\limits^ \to = \frac{{{c^2}{\mu _0}{p_0}}}{{4\pi }} \cdot real\left\{ {\frac{{xy}}{{{R^{\prime}}^2}} \cdot \{ \frac{{{{\mathop f\limits^{..} }_1} - {{\mathop f\limits^{..} }_2}}}{{{c^2}{R^{\prime}}}} + \frac{{3({{\mathop f\limits^. }_1} + {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}^2}} + \frac{{3({{\mathop f\limits^{} }_1} - {{\mathop f\limits^{} }_2})}}{{{R^{\prime}}^3}}\} } \right\} \cdot \mathop {{e_y}}\limits^ \to ,

(8)$$\mathop {{E_z}}\limits^ \to = \frac{{{c^2}{\mu _0}{p_0}}}{{4\pi }} \cdot real\left\{ {\frac{{x{z^{\prime}}}}{{{R^{\prime}}^2}} \cdot (\frac{{{{\mathop f\limits^{..} }_1} - {{\mathop f\limits^{..} }_2}}}{{{c^2}{R^{\prime}}}} + \frac{{3({{\mathop f\limits^. }_1} + {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}^2}} + \frac{{3({{\mathop f\limits^{} }_1} - {{\mathop f\limits^{} }_2})}}{{{R^{\prime}}^3}}) + \frac{x}{{{R^{\prime}}^2}} \cdot (\frac{{{{\mathop f\limits^{..} }_1} + {{\mathop f\limits^{..} }_2}}}{{{c^2}}} + \frac{{({{\mathop f\limits^. }_1} - {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}}})} \right\}\mathop {{e_z}}\limits^ \to . $$

\mathop {{B_x}}\limits^ \to = \frac{{c{\mu _0}{p_0}}}{{4\pi }} \cdot real\left\{ {\frac{{xy}}{{{R^{\prime}}^2}} \cdot \{ \frac{{{{\mathop f\limits^{..} }_1} - {{\mathop f\limits^{..} }_2}}}{{{c^2}{R^{\prime}}}} + \frac{{3({{\mathop f\limits^. }_1} + {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}^2}} + \frac{{3({{\mathop f\limits^{} }_1} - {{\mathop f\limits^{} }_2})}}{{{R^{\prime}}^3}}\} } \right\} \cdot \mathop {{e_x}}\limits^ \to ,

\begin{aligned}&\mathop {{B_y}}\limits^ \to = \frac{{c{\mu _0}{p_0}}}{{4\pi }}\\ &\cdot real\left\{ \frac{{{y^2}}}{{{R^{\prime}}^2}} \cdot (\frac{{{{\mathop f\limits^{..} }_1} - {{\mathop f\limits^{..} }_2}}}{{{c^2}{R^{\prime}}}} + \frac{{3({{\mathop f\limits^. }_1} + {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}^2}} + \frac{{3({{\mathop f\limits^{} }_1} - {{\mathop f\limits^{} }_2})}}{{{R^{\prime}}^3}}) - (\frac{{{{\mathop f\limits^{..} }_1} - {{\mathop f\limits^{..} }_2}}}{{{c^2}{R^{\prime}}}} + \frac{{({{\mathop f\limits^. }_1} + {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}^2}} + \frac{{({{\mathop f\limits^{} }_1} - {{\mathop f\limits^{} }_2})}}{{{R^{\prime}}^3}}) \right. \\ & \left.- \frac{{{z^{\prime}}}}{{{R^{\prime}}^2}} \cdot (\frac{{{{\mathop f\limits^{..} }_1} + {{\mathop f\limits^{..} }_2}}}{{{c^2}}} + \frac{{({{\mathop f\limits^. }_1} - {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}}}) \right\} \quad\mathop {{e_y}}\limits^ \to \end{aligned},

\mathop {{B_z}}\limits^ \to = \frac{{c{\mu _0}{p_0}}}{{4\pi }} \cdot real\left\{ {\frac{{y{z^{\prime}}}}{{{R^{\prime}}^2}} \cdot (\frac{{{{\mathop f\limits^{..} }_1} - {{\mathop f\limits^{..} }_2}}}{{{c^2}{R^{\prime}}}} + \frac{{3({{\mathop f\limits^. }_1} + {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}^2}} + \frac{{3({{\mathop f\limits^{} }_1} - {{\mathop f\limits^{} }_2})}}{{{R^{\prime}}^3}}) + \frac{y}{{{R^{\prime}}^2}} \cdot (\frac{{{{\mathop f\limits^{..} }_1} + {{\mathop f\limits^{..} }_2}}}{{{c^2}}} + \frac{{({{\mathop f\limits^. }_1} - {{\mathop f\limits^. }_2})}}{{c{R^{\prime}}}})} \right\}\mathop {{e_z}}\limits^ \to.

The expressions of ${f_1}(\tau ^{\prime})$ and ${f_2}(\tau ^{\prime\prime})$ are $\textrm{1/(1} - i{\omega _\textrm{0}}{\tau ^{\prime}}\textrm{/s}{\textrm{)}^{\textrm{s + 1}}} \cdot {e^{i{\phi _0}}}$ and $\textrm{1/(1} - i{\omega _\textrm{0}}{\tau ^{^{\prime\prime}}}\textrm{/s}{\textrm{)}^{\textrm{s + 1}}} \cdot {e^{i{\phi _0}}}$. ${p_0}$ can be replaced by the peak electric field ${E_0}$ at the focus and is given by ${E_0} = \max \{{|\overrightarrow E |} \}{|_{x,y,z = 0}}$. ${E_0}$ can be written as

(9)$${E_0} = \frac{{{p_0}{c^2}{\mu _0}}}{{4\pi }}\left[ {\frac{\textrm{1}}{{{a^3}}} \cdot (1 - \frac{1}{{{{(1 + \frac{{2{\omega_0}a}}{{c \cdot s}})}^{s + 1}}}}) - \frac{{2(s + 1){\omega_0}}}{{cs{a^2}}} + \frac{{2(s + 1)(s + 2)\omega_0^2}}{{{c^2}{s^2}a}}} \right]$$

3. Results and discussion

The focusing characteristics of the linearly polarized Poisson ultrashort pulses at the focal plane are studied. The value of parameter s is taken as 4.12, 28.4, and 100, corresponding to the full width at half maximum (FWHM) of the pulse is 0.45 cycle, 1.37 cycle and 2.6 cycle. In the RWT, the incident beam is a plane wave and the focal length is $f = 4.6 \times {10^{ - 3}}m$. The numerical aperture of objective lens is 0.96. The wavelength is $\lambda = 433nm$ and the peak electric field is ${E_0} = 5.6 \times {10^{10}}V/m$. Both the RWT and the CSSM are used to study the central focus spot and the convergence and divergence focusing phenomena of linearly polarized ultrashort pulses.

3.1. Central focus spots of the pulses

The ${t_1}$ in Eq. (2) and the ${t_0}$ in Eq. (5) are set to 0. Therefore, the central focus spots of the linearly polarized pulses can be simulated by both the RWT and the CSSM. The simulation results are shown in Figs. 2 and 3.

Fig. 2. The intensity profiles of the electric field along the y axis at the focal plane ($z = 0$, $x = 0$). The black solid line shows the result of RWT and the red dotted line shows the result of CSSM. (a) $s = 4.12$. (b) $s = 28.4$.

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Fig. 3. The#intensity profiles of the electric field along the y axis at the focal plane. ${w_0} = 0.002\mu m$ (red dotted line). (a) $s = 4.12$. (b) $s = 28.4$.

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The intensity profiles of the central focus spots along the y axis are shown in Fig. 2. The simulation results in the x direction are similar to the simulation results in the y direction. After considering both of the simulation results in the x direction and y direction, the parameter ${w_0}$ is selected as ${w_0} = 0.16\mu m$ in Fig. 2(a) and ${w_0} = 0.11\mu m$ in Fig. 2(b). Thus, the most consistent results of the RWT model and CSSM model for simulating the central focus spot are shown in Fig. 2. The FWHM is $0.166\mu m$ for the RWT and $0.184\mu m$ for the CSSM in Fig. 2(a). One can see that the simulation results of the RWT and the CSSM are approximately consistent for the sub-cycle pulse. The FWHMs of the RWT and the CSSM are $0.206\mu m$ in Fig. 2(b). Such the simulation results of the two models are well consistent for the single-cycle pulse. In general, the results of the RWT and the CSSM are well consistent when simulating the central focus spots of the linearly polarized ultrashort pulses.

The waist spot size parameter ${w_0}$ of CSSM is $\textrm{0}\textrm{.002}\mu m$ in Fig. 3. It should be pointed out that the waist spot size parameter ${w_0}$ isn’t equal to the FWHM of the focal spot. The FWHM depends on both the beam waist radius parameter and the pulse width in the time domain. For the CSSM, the FWHM is $0.\textrm{086}\mu m$ for the 0.45-cycle pulse in Fig. 3(a) and is $0.\textrm{152}\mu m$ for the 1.37-cycle pulse in Fig. 3(b). Therefore, the super-resolution focus spot can be described by the CSSM. By using the annular diaphragm, the super-resolution focus spot can be obtained by the high numerical aperture optical system [34–36]. The field of the super-resolution focus spot can be calculated numerically by the RWT. But the calculation process is complicated. Using the analytical CSSM, the super-resolution focus spot can be easily simulated.

3.2. Convergence and divergence focusing phenomena at the focal plane

The convergence and divergence focusing phenomena of the linearly polarized ultrashort pulses at the focal plane are studied by both the RWT and the CSSM. We let $x = 0$, which means that the dynamic focusing phenomena along the y axis are studied. In other directions, the dynamic focusing phenomena are similar to that of the y-axis. As shown in Eq. (2) and (8), the electric field components ${E_y}$ and ${E_z}$ are 0 when $x = 0$. So only the electric field component ${E_x}$ needs to be calculated. The calculation results are shown in Fig. 4.

Fig. 4. Amplitude and intensity profiles of the electric field components ${E_x}$ at the focal plane ($z = 0$, $x = 0$).The values of ${t_0}$ are respectively taken as $15fs$, $10fs$, and $5fs$. (a) ${E_x}$ ($s = 4.12$), (b) ${I_x}$ ($s = 4.12$), (c) ${E_x}$ ($s = 28.4$), (d) ${I_x}$ ($s = 28.4$), (e) ${E_x}$ ($s = 100$), (f) ${I_x}$ ($s = 100$).

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The divergence focusing phenomena along the y axis at the focal plane are simulated at the time ${t_0}$ equals $5fs$, $10fs$ and $15fs$. There are some differences between the simulation results of the RWT and CSSM. The maximum peak powers of the RWT are only a fraction of that of the CSSM at the time ${t_0}$ equals $10fs$ and $15fs$. In other directions, the differences between the simulation results of the RWT and the CSSM model are similar to that of the y-axis direction. This is different from the simulation result of the radially polarized ultrashort pulses case [33]. When the time value ${t_0}$ is negative, the convergent dynamic focusing phenomenon can also be simulated by the RWT and the CSSM. The simulations show that the analytical results of the CSSM and the numerical calculation results of the RWT are not completely consistent for the convergence and divergence focusing phenomena of the linearly polarized ultrashort pulses.

The focusing field of linearly polarized beam in the focal plane is asymmetric and complex. From Fig. 2, one can see that the simulation results by using the RWT and CSSM agree well near the focus in the focal plane. But Fig. 4 shows that the simulation results don’t agree well in the area far away from the focus. The field of CSSM comes from a single electric dipole and a magnetic dipole and can’t well describe the field of linearly polarized beam far away from the focus area. Maybe the model based on the clusters of electric dipole and magnetic dipole can better describe the field of linearly polarized beam. This is a complex problem and deserves further discussion.

3.3. Dynamic two-dimensional focus evolution in the focal plane

The dynamic two-dimensional (2D) field intensity distribution in the focal plane is studied by changing the value of ${t_0}$. Three cases of the 0.45 cycle pulse, 1.37 cycle pulse and 2.6 cycle pulse are analyzed. Both of the simulation results of RWT and CSSM for the 0.45 cycle case are shown in Fig. 5.

Fig. 5. The intensity profiles of the total electric field at the focal plane. $s = 4.12$. The results of RWT are shown in (a)–(c) and the results of CSSM are shown in (e)–(g). (a) ${t_0} = 0fs$, (b) ${t_0} = 1fs$, (c)${t_0} = 3fs$. (e) ${t_0} = 0fs$, (f) ${t_0} = 1.5fs$, (g) ${t_0} = \textrm{4}fs$.

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One can see that the central focus spot is an ellipse at the time ${t_0} = 0fs$. This is also consistent with the results found in [21,37]. From the results of RWT, one can see that the ellipse splits into two focused lobes at the time ${t_0} = 1fs$ and the focused lobes evolve into an outgoing focusing halo at the time ${t_0} = 3fs$. The FWHMs of the central focus ellipse are $166nm$ in the y direction and are $\textrm{234}nm$ in the x direction in Fig. 5(a). Such the ellipse is a super-resolution focus spot in the x direction. The FWHM of the focusing halo is $270nm$ at the time ${t_0} = 3fs$ in Fig. 3(c). For both RWT and CSSM, the dynamic focusing phenomena transitions from the halo to two light lobes to the elliptical focus spot can be observed. The time values in Figs. 5(f) and 5(g) are different from that of Figs. 5(b) and 5(c). The reason is to observe clear focused lobes and the focusing halo in the simulation of CSSM. Figure 5 shows that the simulation results of the RWT and CSSM agree well near the focus and the amplitudes of the results don’t agree well in the area far away from the focus in the focal plane. The reasons are analyzed at the end of Section 3.2. The dynamic focus phenomena of the linearly polarized ultrashort pulses are very important for understanding the interaction between the pulses and matter. The 1.37 cycle case is shown in Fig. 6. The RWT is used to obtain the results in Fig. 6 and Fig. 7.

Fig. 6. The intensity profiles of the total electric field at the focal plane. $s = \textrm{28}\textrm{.4}$. (a) ${t_0} = 0fs$, (b) ${t_0} = \textrm{3}fs$, (c) ${t_0} = \textrm{5}fs$.

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Fig. 7. The intensity profiles of the total electric field at the focal plane. $s = \textrm{100}$. (a) ${t_0} = 0fs$, (b) ${t_0} = \textrm{5}fs$, (c) ${t_0} = \textrm{9}fs$.

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The 2D focus evolution of the 1.37 cycle linearly polarized ultrashort pulses is similar to that of the 0.45 cycle pulse. The FWHMs of the central focus ellipse are $\textrm{206}nm$ in the y direction and are $\textrm{300}nm$ in the x direction at the time ${t_0} = 0fs$, and are $\textrm{294}nm$ in the y direction and are $\textrm{525}nm$ in the x direction at the time ${t_0} = \textrm{3}fs$. The FWHM of the focusing halo is $\textrm{690}nm$ at the time ${t_0} = \textrm{5}fs$. One can see that the FWHMs of the central focus ellipses of the 1.37 cycle pulse are larger than that of the 0.45 cycle pulse. The 2.6 cycle case is shown in Fig. 7.

From Fig. 7, we can see that the FWHMs of the central focus ellipse are $\textrm{212}nm$ in the y direction and are $\textrm{310}nm$ in the x direction at the time ${t_0} = 0fs$, and are $\textrm{230}nm$ in the y direction and are $\textrm{348}nm$ in the x direction at the time ${t_0} = \textrm{5}fs$. The FWHM of the focusing halo is $\textrm{1287}nm$ at the time ${t_0} = \textrm{9}fs$. The FWHM of the 2.6 cycle pulse is slightly larger than the FWHM of the 1.37 cycle pulse at the time ${t_0} = 0fs$. But the FWHM of the central elliptical focus spot of the 2.6 cycle pulse increases with time slower than that of the 1.37 cycle pulse.

As shown in Figs. 4–7, the linearly polarized beam amplitude and intensity in the focal plane has a min in the center or a “focusing halo” for certain ${t_0}$. The phenomena can be understood from the Huygens–Fresnel principle [33]. The reason for the min is that the pulse peaks have passed the focus. The reason for the focusing halo is that the pulse peaks emitted by some point sources of the wavefront reach these points in the focal plane.

3.4. Changes of the focus field intensities with the convergence angles

The changes of the maximum intensities of the focusing halos with the convergence angles are shown in the Table 1. Two cases of the 0.45 cycle pulse and the 2.6 cycle pulse are considered. Both the x-linear polarized pulse and the radially polarized pulse are compared in the Table 1. The calculation time is ${t_0} = \textrm{10}fs$ in the Table 1. The RWT is used to study the changes.

Table 1. The focus field intensities vs. the convergence angles.^a

View Table

From Table 1, we can see that the changes of the maximum focus field intensities with the convergence angle are similar for both the x-linear polarized pulse and the radially polarized pulse. But they are different for the 0.45 cycle pulse and the 2.6 cycle pulse. For the 0.45-cycle pulse, the maximum focus field intensities change slowly with the convergence angle if the convergence angle is greater than ${30^\circ }$. The maximum focus field intensities change by four orders of magnitude from ${20^\circ }$ to ${30^\circ }$. But they only change three times from ${30^\circ }$ to ${40^\circ }$. Therefore, ${30^\circ }$ is a particularly important angle for observing the convergence and divergence focusing phenomena at the focal plane. For the 2.6-cycle pulse, the particularly important angle is ${20^\circ }$. Such one can find that the convergence and divergence focusing phenomena can be observed significantly with the smaller convergence angles for the longer duration pulses. The significantly enhanced convergence and divergence focusing phenomena come from the influence of the non-paraxial effect of the beam.

4. Conclusion

The focusing characteristics of linearly polarized ultrashort pulses at the focal plane are studied with both the CSSM and the RWT. The research shows that both the RWT and the CSSM are well consistent when simulating the central focus spots of the linearly polarized ultrashort pulses. The analytical CSSM can easily simulate the super-resolution focus spot. When simulating the convergence and divergence focusing phenomena of the linearly polarized ultrashort pulses, the simulation results of the CSSM and the RWT are not completely consistent. The dynamic 2D field intensity distribution in the focal plane is studied by the RWT. For the convergent focusing process, first there is an inwardly propagating halo, then it evolves into two light lobes, and then the light lobes evolve into a focusing ellipse. The divergent defocusing process is the inverse process of the convergent focusing process. The dynamic focus phenomena are important for understanding the interaction between the pulses and matter. The particularly important convergence angles for observing the significant convergence and divergence focusing phenomena are given for both the sub-cycle pulses and the few cycle pulses.

Funding

National Natural Science Foundation of China (11964007); The Thousand Levels of Innovative Talents of Guizhou Province ([2016]016).

Acknowledgment

We thank Yifu Zhu at department of Physics, Florida International University for helps.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. Q. W. Zhan, “Cylindrical vector beams from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]

2. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef]

3. G. Cincotti, A. Ciattoni, and C. Sapia, “Radially and azimuthally polarized vortices in uniaxial crystals,” Opt. Commun. 220(1-3), 33–40 (2003). [CrossRef]

4. Z. M. Zhang, J. X. Pu, and X. Q. Wang, “Tight focusing of radially and azimuthally polarized vortex beams through a uniaxial birefringent crystal,” Appl. Opt. 47(12), 1963–1967 (2008). [CrossRef]

5. M. Erdélyi and Z. Bor, “Radial and azimuthal polarizers,” J. Opt. A: Pure Appl. Opt. 8(9), 737–742 (2006). [CrossRef]

6. D. Khoptyar, R. Gutbrod, A. Chizhik, J. Enderlein, F. Schleifenbaum, M. Steiner, and A. J. Meixner, “Tight focusing of laser beams in a λ/2-microcavity,” Opt. Express 16(13), 9907–9917 (2008). [CrossRef]

7. Y. J. Zhang, B. F. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010). [CrossRef]

8. B. Tian and J. X. Pu, “Tight focusing of a double-ring-shaped, azimuthally polarized beam,” Opt. Lett. 36(11), 2014–2016 (2011). [CrossRef]

9. K. T. Gahagan and G. A. Swartzlander Jr, “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef]

10. R. M. Herman and T. A. Wiggins, “Hollow beams of simple polarization for trapping and storing atoms,” J. Opt. Soc. Am. A 19(1), 116–121 (2002). [CrossRef]

11. J. P. Bai and Y. J. Zhang, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009). [CrossRef]

12. B. Gu, Y. Pan, J. L. Wu, and Y. P. Cui, “Tight focusing properties of spatial-variant linearly-polarized vector beams,” J. Opt. 43(1), 18–27 (2014). [CrossRef]

13. X. M. Cai, J. Y. Zhao, Q. Lin, H. Tong, and J. T. Liu, “Electron acceleration driven by sub-cycle and single-cycle focused optical pulse with radially polarized electromagnetic field,” Opt. Express 26(23), 30030–30041 (2018). [CrossRef]

14. S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. 32(27), 5222–5229 (1993). [CrossRef]

15. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010). [CrossRef]

16. Y. Kozawa and S. Sato, “Numerical analysis of resolution enhancement in laser scanning microscopy using a radially polarized beam,” Opt. Express 23(3), 2076–2084 (2015). [CrossRef]

17. Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19(17), 15947–15954 (2011). [CrossRef]

18. Y. K. Chen, D. G. Zhang, L. Han, G. H. Rui, X. X. Wang, P. Wang, and H. Ming, “Surface-plasmon-coupled emission microscopy with a polarization converter,” Opt. Lett. 38(5), 736–738 (2013). [CrossRef]

19. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D: Appl. Phys. 32(13), 1455–1461 (1999). [CrossRef]

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

21. B. Richards and E. Wolf, “Electromagnetic Diffraction in optical systems. II. Structure of the image field in an Aplanatic System,” Proc. R. Soc. Lond. A 253(1274), 358–379 (1959). [CrossRef]

22. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000). [CrossRef]

23. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001). [CrossRef]

24. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]

25. Q. Lin, J. Zheng, and W. Becker, “Subcycle pulsed focused vector beams,” Phys. Rev. Lett. 97(25), 253902 (2006). [CrossRef]

26. A. April, Coherence and Ultrashort Pulse Laser Emission, F. J. Duarte, Ed. (InTech, 2010) pp. 355–382.

27. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57(4), 2971–2979 (1998). [CrossRef]

28. E. Heyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4(11), 2081–2091 (1987). [CrossRef]

29. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7(23), 684–685 (1971). [CrossRef]

30. E. Heyman and L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6(6), 806–817 (1989). [CrossRef]

31. Z. Y. Wang, Q. Lin, and Z. Y. Wang, “Single-cycle electromagnetic pulses produced by oscillating electric dipoles,” Phys. Rev. E 67(1), 016503 (2003). [CrossRef]

32. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

33. X. M. Cai, Y. L. Zheng, and Y. F. Zhu, “Convergence and divergence focusing phenomena at the focal plane of ultrashort pulses,” J. Opt. Soc. Am. A 37(6), 969–973 (2020). [CrossRef]

34. L. X. Yang, X. S. Xie, S. C. Wang, and J. Y. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett. 38(8), 1331–1333 (2013). [CrossRef]

35. K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010). [CrossRef]

36. P. Li, S. Liu, G. F. Xie, T. Peng, and J. L. Zhao, “Modulation mechanism of multi-azimuthal masks on the redistributions of focused azimuthally polarized beams,” Opt. Express 23(6), 7131–7139 (2015). [CrossRef]

37. R. Dorn, S. Quabis, and G. Leuchs, “The focus of light-linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50(12), 1917–1926 (2003). [CrossRef]

Focusing characteristics of linearly polarized ultrashort pulses at the focal plane

Abstract

1. Introduction

2. Model and equations

3. Results and discussion

3.1. Central focus spots of the pulses

3.2. Convergence and divergence focusing phenomena at the focal plane

3.3. Dynamic two-dimensional focus evolution in the focal plane

3.4. Changes of the focus field intensities with the convergence angles

4. Conclusion

Funding

Acknowledgment

Disclosures

References

Cited By

Figures (7)

Tables (1)

Equations (14)

Optics Express