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Numerical modeling for the characteristics study of a focusing ultrashort spatiotemporal optical vortex

Guanghao Rui, Bin Yang, Xinyuan Ying, Bing Gu, Yiping Cui, and Qiwen Zhan

Open Access Open Access

Abstract

Spatiotemporal (ST) wave packet carrying pure transverse orbital angular moment (OAM) with subwavelength spatial size has attracted increasing attentions in recent years, which can be obtained by tightly focusing a linear superposition of ST vortices with different topological charges. In this work, numerical models are proposed to explore the impact of the pulse width of the ST vortex on the characteristics of its focal field. We demonstrate that the rigorous model for calculating the focused ST wave packet is essential for ultrashort optical pulse, while the simplified model has the advantage of high efficiency but can only provide credible results when the pulse width of the illumination is long enough. Specifically, when the pulse width decreases from 100 fs to 5 fs, the accuracy of the simplified model would decrease significantly from 99% to 65.5%. In addition, it is found that the pulse duration would still lead to the collapse of transverse OAM structure near the focus of a high numerical aperture lens, even though the ST astigmatism has already been corrected. To analyze the physical mechanism behind this distortion, Levenberg-Marquardt algorithm is adopted to retrieve the OAM distribution of the focal field. It is shown that the contributions from undesired OAM modes would become nontrivial for short pulse width, leading to the formation of the focal field with hybrid OAM structures. These findings provide insight for the focusing and propagation studies of ultrashort ST wave packets, which could have wide potential applications in microscopy, optical trapping, laser machining, nonlinear light-matter interactions, etc.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The interaction of light with matter at the nanoscale is a topic of rapidly increasing scientific importance and technological relevance. Nanoscale light-matter interaction is essential for the efficient conversion of light into chemical energy in biological light harvesting systems and for the light-to-current conversion in artificial photovoltaic devices. Usually, a high numerical aperture (NA) lens is essential to strongly focus light into small spots, leading to high resolution and concentration of light. Since the properties of the focal field strongly depend on the spatial distribution of the incident light, complex optical fields with spatially structured field distribution in the cross section of fields have attracted increasing attentions in the last decade. For example, various kinds of focal fields with specific pattern in terms of amplitude (e.g. optical cage [1], optical chain [2], optical lattice [3] and optical needle [4]), polarization (e.g. transverse spin [5–7], arbitrary photonic spin [8], three-dimensional (3D) polarization [9]), and phase (e.g. longitudinal vortex structure [10,11]) have been realized by carefully sculpting the incident light. The unique properties of these focal fields lead to numerous applications, including information storage, optical manipulation, microscopic and nanoscopic imaging, remote sensing, and materials micromachining [12–24].

To increase both the temporal and spatial resolution of optical measurements simultaneously, a tightly focused optical wave packet with ultrashort pulse duration is necessary [25]. Utilizing technologies to manipulate the temporal phase, the shape evolution of optical pulses along the propagation direction can be controlled [26]. This so-called spatiotemporal (ST) pulse shaping strategies has been adopted to generate various localized pulses, such as light bullet [27], arbitrarily accelerating ST wave packets [28], etc. Recently, ST wave packet with transverse orbital angular momentum (OAM) has been experimentally demonstrated using 4f pulse shaper, in which the energy flow would circulate around the axis perpendicular to the propagation direction of light [29]. The transverse shifts and time delay of ST optical vortex reflection and refraction at the plane interface is studied [30]. Besides, it has been reported that spatiotemporal vortex structure of Bessel ST optical vortex can be well maintained and confined through much longer propagation in dispersive medium [31]. Interestingly, the ST vortex phase structure of this novel wave packet would collapse after strongly focusing by a high NA objective, since the lens can affect both the spatial and temporal domain of the incident ST wave packet. To generate ST wave packet carrying pure transverse OAM with subwavelength spatial size, the ST astigmatism effect must be eliminated by preprocessing the incident ST wave packet with linear superposition of ST vortices with different topological charges (TC) [32]. Using this method, spin-orbital coupling between the longitudinal spin angular momentum and the transverse OAM carried by a focusing ST optical vortex wave packet has been theoretically predicted [33]. However, the effects of the pulse width of the incident ST wave packet on the focal field are ignored in previous studies.

In this work, we propose a rigorous numerical model to explore the focusing characteristics of ST wave packet, in which the impact of pulse width is thoroughly considered. With the decreased pulse width, it is found that the OAM structure near the focus of a high NA lens would still be distorted, even though the incident ST vortex has already been pre-conditioned. In addition, a simplified model is also proposed to improve the efficiency of the focal field calculation under certain circumstances. To quantitatively evaluate the correlation between these two models, a focusing ST vortex wave packet calculated by both methods is compared by structural similarity index (SSIM) factor, which would decrease from 0.99 to 0.655 when the pulse width of the illumination decreases from 100 fs to 5 fs, indicating that the simplified model is only credible for relatively long pulse width. Furthermore, with the aim of explaining the OAM distortion, Levenberg-Marquardt algorithm has been adopted to measure both the amplitude and phase of the transverse OAM modes of the focal field simultaneously. It is shown that the contribution from undesired OAM modes would become nontrivial for very short pulse width, leading to the formation of focal field with hybrid OAM structures. These findings provide insight for the focusing and propagation studies of ultrashort ST wave packets, which could have wide potential applications in microscopy, optical trapping, laser machining, and nonlinear light-matter interaction.

2. Focused ST wave packet with transverse OAM: rigorous model

To create a ST vortex with TC of -1 in the focal plane of a high NA lens, the required illumination is the linear superposition of two ST waves with TCs of ±1, which can be expressed as [32]:

$${E_{ - 1}}(x,y,t) = (1 + i)(\frac{x}{{{w_p}}} + \frac{t}{{{w_t}}})\textrm{exp} [ - ({x^2} + {y^2})/w_p^2 - \frac{{{t^2}}}{{w_t^2}} - i{\omega _0}t],$$
where (x, y, t) are the spatial and temporal coordinates in the incident plane, wp is the beam width, wt is the pulse width, and ω0 is the central frequency. The 3D ST structure of the incident ST vortices is presented in the inset of Fig. 1(a) near the focal region. Figure 1(b) shows the corresponding phase distribution in the x-t domain, it can be seen that the phase pattern is binarized and the boundary between two areas is a line with −45 degree with respect to t axis. With the assumption that the objective lens satisfies the sine condition of r = f·sinθ, where f is the focal length of the lens and θ is the focusing angle, the apodization function of a single spectral component can be obtained by Fourier transform:
$$\begin{aligned} S(\theta ,\phi ,\omega ) &= \frac{1}{{\sqrt 2 \pi }}\int_{ - \infty }^\infty {{E_\Omega }} (\theta ,\phi ,t^{\prime}){e^{i\omega t}}dt\\ &= \frac{{{w_t}}}{{2\sqrt 2 {w_p}}}[(2 + 2i)r\cos \phi + ( - 1 + i)(\omega - {\omega _0}){w_t}{w_p}]\textrm{exp} [ - {(\frac{r}{{{w_p}}})^2}]\\ & \times \textrm{exp} \{ - {[\frac{{{w_t}(\omega - {\omega _0})}}{2}]^2} - i\omega [(f + {r_f}\sin \theta \cos (\phi - {\phi _r}) + z\cos \theta )]/c\} , \end{aligned}$$
where t’ = t − [f + rfsinθcos(ϕ−ϕf) + zcosθ]/c is the time in the focal region, c is the speed of light in vacuum, rf and ϕf are the polar coordinates in the focal region, respectively. Then, the electric field EΩ after the refraction of the objective lens can be written in spherical coordinates:
$${E_\Omega }(\theta ,\phi ,t^{\prime}) = (1 + i)(\frac{{f\sin \theta \cos \phi }}{{{w_p}}} + \frac{{t^{\prime}}}{{{w_t}}})\textrm{exp} [ - {(\frac{{f\sin \theta }}{{{w_p}}})^2} - {(\frac{{t^{\prime}}}{{{w_t}}})^2} - i{w_0}t^{\prime}].$$

Using the schematic illustrated in Fig. 1(a), the electric field of a single spectral component of S (θ, ϕ, ω) in the focal region can be calculated with Richard-Wolf diffraction theory [34–36]:

$$\begin{aligned} {\overrightarrow E _{rig}}({r_f},\theta ,\phi ,z,\omega ) &= \left( {\begin{array}{{c}} {{{\overrightarrow E }_x}}\\ {{{\overrightarrow E }_y}}\\ {{{\overrightarrow E }_z}} \end{array}} \right) = \frac{{ - ikf{l_0}}}{{2\pi }}\int_0^\alpha {\int_0^{2\pi } {S(\theta ,\phi ,\omega )} } \left( {\begin{array}{c} {\cos \theta + \frac{1}{2}(1 - \cos 2\phi )(1 - \cos \theta )}\\ {\frac{1}{2}(\cos \theta - 1)\sin 2\phi }\\ { - \sin \theta \cos \phi } \end{array}} \right)\\ & \times \sqrt {\cos \theta } \sin \theta \textrm{exp} [ik{r_f}\sin \theta \cos (\phi - {\phi _f})]d\theta d\phi . \end{aligned}$$

The electric field of the focusing ultrashort ST wave packet in the vicinity of the focus can be obtained by the coherent superposition of each spectral component using inverse Fourier transformation:

$${\overrightarrow E _{rig,j}}({r_f},\theta ,\phi ,z,t) = \int_0^\infty {{{\overrightarrow E }_{rig,j}}({r_f},\theta ,\phi ,z,\omega ){e^{ - i\omega t}}d\omega \textrm{ }} (j = x,y,z).$$

Then, the intensity of the electric field near the focus is expressed as:

$${I_{rig}}({r_f},\theta ,\phi ,z,t) = \sum\limits_{j = x,y,z} {{I_{rig,j}}({r_f},\theta ,\phi ,z,t)} = {\sum\limits_{j = x,y,z} {|{{{\overrightarrow E }_{rig,j}}({r_f},\theta ,\phi ,z,t)} |} ^2}.$$

As an example, considering an incident ST wave packet (wp = 0.5 mm, wt = 100 fs, ω0 =2π/1064 nm) depicted by Eq. (1) is focused by a lens (f = 1 mm, NA = 0.95), the intensity of the focal field in the x-t plane (shown in the inset of Fig. 1(a) around the focus of the lens) has doughnut shape with a dark center, and the corresponding phase distribution of Ex in the kx -ω plane varies clockwise in the range of [-π, π], indicating the generation of focused ST optical vortex with transverse OAM along y axis and the TC is −1 [30]. However, it takes a lot of time to solve the coupled equations involving triple integrals. Next, we will discuss the strategy to improve the calculation efficiency by restricting the pulse width of the ST wave packet.

 figure: Fig. 1.

Fig. 1. (a) Schematic diagram of the tight focusing for ST wave packet. The inset around the lens shows the 3D ST structure of the incident ST vortex with pulse width of 100 fs. The inset around the focus of the lens shows the corresponding focused ST vortex in the x-t plane, indicating a transverse OAM along y axis. Phase distribution of the (b) incident and (c) Ex of focused ST vortex presented in the insets of Fig. 1(a). Spectrum of the pupil apodization function for ST wave packet with pulse width of (d) 5 fs and (e) 100 fs.

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3. Focused ST wave packet with transverse OAM: simplified model

To speed up the calculation speed, the approach we used is to simplify the apodization function term S (θ, ϕ, ω) in Eq. (4). Figures 1(d) and 1(e) compare the spectrum of the pupil apodization function for two kinds of ST wave packet. It can be seen that the short pulse width of 5 fs leads to a broadband spectrum (shown in Fig. 1(d)), thus the contributions from a wide range of spectral components must be considered. With the increase of pulse width up to 100 fs, the spectrum of the pupil apodization function gradually evolves into a delta function centered at ω0. Consequently, if the pulse width of the illumination is long enough, the apodization function in Eq. (4) can be reduced to a constant value determined by the central frequency of the wave packet. In this case, the electric field in the focal region can be written as:

$$\begin{aligned} {\overrightarrow E _{sim}}({r_f},\theta ,\phi ,z,t) &={-} \frac{{ikf{l_0}}}{{2\pi }}\int_0^\alpha {\int_0^{2\pi } {{E_\Omega }(\theta ,\phi ,t^{\prime})\left( {\begin{array}{{c}} {\cos \theta + \frac{1}{2}(1 - \cos 2\phi )(1 - \cos \theta )}\\ {\frac{1}{2}(\cos \theta - 1)\sin 2\phi }\\ { - \sin \theta \cos \phi } \end{array}} \right)} } \\ & \times \sqrt {\cos \theta } \sin \theta \textrm{exp} [ik{r_f}\sin \theta \cos (\phi - {\phi _f})]d\theta d\phi . \end{aligned}$$

Clearly, the calculation of focal field is simplified from triple to double integral, leading to the significantly decreased computation time. However, one may be curious about the applicable range of this simplified model. Here, we will study the impact of the pulse width on the accuracy of model. Considering an incident ST vortex with pulse width wt = 100 fs, Figs. 2(a)–2(c) illustrate the normalized intensity distributions of the focal field in the longitudinal (x-z), transverse (x-y), and ST (x-t) plane, respectively, which are calculated with the simplified model given by Eq. (7). As a comparison, rigorous model is also applied to calculate the focal field (shown in Figs. 2(d)–2(f)). Clearly, the results from these two models agree well, and a subwavelength ST wave packet with OAM oriented along y axis is obtained. However, as the spectrum of the pupil apodization function shown in Figs. 1(d) and 1(e) suggests, the simplified model is only suitable for ST wave packet with long pulse width. For example, when the pulse width decreases to 5 fs, there are obvious differences between the focused ST vortex calculated by both models (shown in Figs. 2(g)–2(l)). In addition, it should be noticed that the symmetry of the focal field is broken for incident ST wave packet with short pulse width. It is crucial to introduce the time correction term t’ during the calculation, otherwise the decreased pulse width would not lead to unsymmetrical intensity pattern even with the simplified model.

 figure: Fig. 2.

Fig. 2. Comparison of the intensity distribution of the focused ST wave packet calculated with (a)-(c), (g)-(i) simplified model and (d)-(f), (j)-(l) rigorous model. The first, second and third rows are the intensity patterns of the focused wave packet in the x-z plane (y = t = 0), x-y plane (z = t = 0), and x-t plane (y = z = 0), respectively. The first two and last two columns are the intensity patterns of the focused wave packet with pulse width of 100 fs and 5 fs, respectively.

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4. Impact of the pulse width on the accuracy of the simplified model

To quantitatively explore the impact of the pulse width on the accuracy of the simplified model, the correlation between the proposed two models is evaluated by SSIM factor [37]:

$$SSIM(sim,rig) = \frac{{(2{\mu _{sim}}{\mu _{rig}} + {C_1})(2{\sigma _{sr}} + {C_2})}}{{(\mu _{sim}^2 + \mu _{rig}^2 + {C_1})(\sigma _{sim}^2 + \sigma _{rig}^2 + {C_2})}},$$
where µsim, µrig, σsim, σrig and σsr are the local means, standard deviations, and cross-covariance for the focal field calculated with simplified model and rigorous model, respectively. Generally, these local sample statistics are computed within overlapping windows, and weighted within each window, e.g., by a Gaussian-like profile. Constants C1 and C2 would stabilize the computations of Eq. (8) when the denominator become small. Note that the closer the value of SSIM factor is to 1, the higher the similarity between two models is. Here, we compare the similarity between Ex, Ez and total intensity of the focusing ST optical vortex, which are indicated by Sx, Sz and St respectively, and the relations between SSIM and pulse width of the illumination in the x-z plane, x-t plane and x-y-z space are presented in Figs. 3(a)–3(c). It should be noted that the similarity between Ey is not considered separately since this field component is relatively weak for the focal field with transverse OAM along y axis. In general, SSIM factor would decrease with the decreased pulse width. For pulse with of 100 fs, (Sx, Sz, St) = (0.9968,0.9996,0.9928) is achieved for focused wave packet in 3D space, indicating that the accuracy of the simplified model is very high under this circumstance. In addition, the curves of SSIM factor are relatively flat for pulse width in the range of 20-100 fs, and S3D would slowly decrease to 0.92. However, the SSIM factor dramatically decreases when the pulse width is shorter than 20 fs. For pulse width of 5 fs, (Sx, Sz, St) = (0.6998,0.9296,0.6548) is obtained in 3D space, demonstrating that the accuracy of simplified model is significantly decreased and the results from this model are not reliable for ST wave packet with ultrashort pulse width.

 figure: Fig. 3.

Fig. 3. SSIM factor versus pulse width of the incident ST wave packet in the (a) x-z plane (y = t = 0), (b) x-t plane (y = z = 0), and (c) x-y-z space (t = 0). (d) Intensity distribution of the hybrid ST vortex modes in the x-t plane (y = z = 0) and the retrieved (e) amplitude and (f) phase distribution of the transverse OAM mode. The theoretical and simulated transverse OAM distributions are indicated by the orange and blue bars, respectively. (g) Intensity distribution of the focused ST wave packet with pulse width of 5 fs, and the retrieved (h) amplitude and (i) phase distribution of the transverse OAM mode.

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5. Retrieving OAM distribution of the focused ST wave packet

It can be seen from Fig. 2 that the intensity pattern of focused ST optical vortex cannot maintain the doughnut shape in the x-t plane for short pulse width, and its OAM structure would be distorted. That is to say, the field distribution of the focal field can be considered as the linear superposition of ST vortex modes with different amplitudes and phases. Now we assume that the complex amplitudes of the transverse OAM modes are indicated by Al, where l is the TC of corresponding OAM mode, the superposition of the focal field can be expressed as:

$${E_{rig}}\mathrm{\ \propto }\sum {{A_l}{E_{to,l}}} ,$$
where ST vortex traveling with the wave packet is:
$${E_{to,l}}(x,y,t) = {r_t}^{|l |}(x,t)\textrm{exp} [il{\phi _t}(x,t)]\textrm{exp} ( - \frac{{{x^2} + {y^2}}}{{w_p^2}} - \frac{{{t^2}}}{{w_t^2}} - i{w_0}t),$$
where rt = [(x/wp)2+(t/wt)2]1/2 and ϕt = atan[(t/wt)/(x/wp)]. Consequently, to retrieve the complex amplitudes of the hybrid transverse OAM modes, the rest of key issue is to solve the problem in the following:
$$\sigma = \min \{ \sum\limits_{p = 1}^M {\sum\limits_{q = 1}^M {[{{\sum {{{|{{A_l}{E_{to,l}}} |}^2} - |{{E_{rig}}} |} }^2}]} } \} ,$$
where p and q are the pixel values of the detection plane. The non-linear least squares problem can be easily solved by employing the Levenberg-Marquardt algorithm [38].

An example is given to prove the feasibility of the scheme. Without loss of generality, we assume that an ST wave packet contains nine transverse OAM modes whose TCs are integer value from −4 to 4, and the normalized amplitude and phase of modes are set to be (1, 2, 3, 3, 4, 4, 3, 3, 2) and (0, 0.5π, 1π, 1.5π, 1π,0, 0.5π, 0.5π, 0) respectively. Figure 3(d) shows the intensity pattern of this hybrid optical mode in the x-t plane, and the corresponding OAM spectrum is shown by the orange bar in Fig. 3(e) and 3(f). Now, the intensity distribution of the optical field is decomposed by Eq. (11) and then its OAM spectrum can be determined. As the bar plot shown in Figs. 3(e) and 3(f), it can be seen that the simulated (blue bars) distributions of the amplitude and phase of transverse OAM modes consistently match the preset values (orange bars). It proves that the algorithm is accurate enough and the scheme can successfully retrieve the OAM distribution when the optical field contains multiple OAM modes.

Next, we use this method to explore the transverse OAM distribution of the focusing optical vortex with pulse width of 5 fs. By decomposing the intensity pattern (shown in Fig. 3(g)) into transverse OAM modes, a set of complex coefficients are obtained, and the amplitude and phase distributions of the coefficients are illustrated in Figs. 3(h) and 3(i), which are related to the transverse OAM modes. The normalized amplitude and phase distributions of nine transverse OAM modes (TC from −4 to 4) are found to be (0.0536, 0.0241, 0.0082, 0.8431,0.0040, 0.0215, 0.0346, 0.0095, 0.0013) and (0.7854, 0.7854, 3.9270, 3.9270, 3.9270, 0.7854, 0.7854, 3.9270, 0.7854), respectively. Consequently, the ST coupling caused by the short pulse width disturbs the original OAM mode (TC = −1), and the contribution from other undesired modes becomes nontrivial, leading to the hybrid OAM structures.

6. Conclusions

In summary, we have revisited the method of subwavelength focusing of ST wave packet with transverse OAM, and explored the impact of the pulse width of the incident ST optical vortices on the characteristics of the corresponding focal field. With the proposed rigorous numerical model, we have found that the OAM structure near the focus of a high NA lens would still be distorted, even though the ST astigmatism has already been corrected with the use of pre-conditioned ST wave packet. Besides, another simplified model has also been proposed to improve the efficiency of the focal field calculation, which is only reliable for relatively long pulse width, since the spectrum of the pupil apodization function for ST wave packet can be reduced to a delta function. Through evaluating the SSIM factor between focusing ST optical vortex calculated by both methods, we have demonstrated that the accuracy of the simplified model would decrease significantly from 99% to 65.5%. Furthermore, the reason accounting for the collapse of the ST vortex structure can be understood by retrieving the OAM distribution of the focal field. Through decomposing the intensity pattern of the focal field into transverse OAM modes, both the relative amplitude and the relative phase of the different OAM modes can be simultaneously measured with Levenberg-Marquardt algorithm. It is shown that the contributions from undesired OAM modes would become nontrivial for short pulse width, leading to the formation of focal field with hybrid OAM structures. It should be noted that chirped laser pulse can be considered by adding additional time domain phase in Eq. (1), namely exp(iat2/wt2), where a is a constant. However, the impact of the spectral phase profile of the illumination to the OAM spectrum of the focal field is negligible. On the other side, the transverse spatial dimension of the incident wave packet may affect the OAM structure of the focal field severely. If the waist radius of the illumination is much larger than the focal length of the lens, the original doughnut shaped intensity pattern of the focal field in the x-t plane would split even under the large pulse width condition, which is due to the disturbed OAM spectrum. These findings provide insight for the focusing and propagation studies of ultrashort ST wave packets, which could have wide potential applications in microscopy, optical trapping, laser machining, nonlinear light-matter interactions, etc.

Funding

National Natural Science Foundation of China (12274074, 12134013, 12074066, 92050202).

Acknowledgment

G. R. acknowledges the support by the Zhishan Young Scholar Program of Southeast University.

Disclosures

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

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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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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Figures (3)
Fig. 1.
Fig. 1. (a) Schematic diagram of the tight focusing for ST wave packet. The inset around the lens shows the 3D ST structure of the incident ST vortex with pulse width of 100 fs. The inset around the focus of the lens shows the corresponding focused ST vortex in the x-t plane, indicating a transverse OAM along y axis. Phase distribution of the (b) incident and (c) Ex of focused ST vortex presented in the insets of Fig. 1(a). Spectrum of the pupil apodization function for ST wave packet with pulse width of (d) 5 fs and (e) 100 fs.

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Fig. 2.
Fig. 2. Comparison of the intensity distribution of the focused ST wave packet calculated with (a)-(c), (g)-(i) simplified model and (d)-(f), (j)-(l) rigorous model. The first, second and third rows are the intensity patterns of the focused wave packet in the x-z plane (y = t = 0), x-y plane (z = t = 0), and x-t plane (y = z = 0), respectively. The first two and last two columns are the intensity patterns of the focused wave packet with pulse width of 100 fs and 5 fs, respectively.

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Fig. 3.
Fig. 3. SSIM factor versus pulse width of the incident ST wave packet in the (a) x-z plane (y = t = 0), (b) x-t plane (y = z = 0), and (c) x-y-z space (t = 0). (d) Intensity distribution of the hybrid ST vortex modes in the x-t plane (y = z = 0) and the retrieved (e) amplitude and (f) phase distribution of the transverse OAM mode. The theoretical and simulated transverse OAM distributions are indicated by the orange and blue bars, respectively. (g) Intensity distribution of the focused ST wave packet with pulse width of 5 fs, and the retrieved (h) amplitude and (i) phase distribution of the transverse OAM mode.

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Equations (11)

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E − 1 ( x , y , t ) = ( 1 + i ) ( x w p + t w t ) exp [ − ( x 2 + y 2 ) / w p 2 − t 2 w t 2 − i ω 0 t ] ,
S ( θ , ϕ , ω ) = 1 2 π ∫ − ∞ ∞ E Ω ( θ , ϕ , t ′ ) e i ω t d t = w t 2 2 w p [ ( 2 + 2 i ) r cos ⁡ ϕ + ( − 1 + i ) ( ω − ω 0 ) w t w p ] exp [ − ( r w p ) 2 ] × exp { − [ w t ( ω − ω 0 ) 2 ] 2 − i ω [ ( f + r f sin ⁡ θ cos ⁡ ( ϕ − ϕ r ) + z cos ⁡ θ ) ] / c } ,
E Ω ( θ , ϕ , t ′ ) = ( 1 + i ) ( f sin ⁡ θ cos ⁡ ϕ w p + t ′ w t ) exp [ − ( f sin ⁡ θ w p ) 2 − ( t ′ w t ) 2 − i w 0 t ′ ] .
E → r i g ( r f , θ , ϕ , z , ω ) = ( E → x E → y E → z ) = − i k f l 0 2 π ∫ 0 α ∫ 0 2 π S ( θ , ϕ , ω ) ( cos ⁡ θ + 1 2 ( 1 − cos ⁡ 2 ϕ ) ( 1 − cos ⁡ θ ) 1 2 ( cos ⁡ θ − 1 ) sin ⁡ 2 ϕ − sin ⁡ θ cos ⁡ ϕ ) × cos ⁡ θ sin ⁡ θ exp [ i k r f sin ⁡ θ cos ⁡ ( ϕ − ϕ f ) ] d θ d ϕ .
E → r i g , j ( r f , θ , ϕ , z , t ) = ∫ 0 ∞ E → r i g , j ( r f , θ , ϕ , z , ω ) e − i ω t d ω   ( j = x , y , z ) .
I r i g ( r f , θ , ϕ , z , t ) = ∑ j = x , y , z I r i g , j ( r f , θ , ϕ , z , t ) = ∑ j = x , y , z | E → r i g , j ( r f , θ , ϕ , z , t ) | 2 .
E → s i m ( r f , θ , ϕ , z , t ) = − i k f l 0 2 π ∫ 0 α ∫ 0 2 π E Ω ( θ , ϕ , t ′ ) ( cos ⁡ θ + 1 2 ( 1 − cos ⁡ 2 ϕ ) ( 1 − cos ⁡ θ ) 1 2 ( cos ⁡ θ − 1 ) sin ⁡ 2 ϕ − sin ⁡ θ cos ⁡ ϕ ) × cos ⁡ θ sin ⁡ θ exp [ i k r f sin ⁡ θ cos ⁡ ( ϕ − ϕ f ) ] d θ d ϕ .
S S I M ( s i m , r i g ) = ( 2 μ s i m μ r i g + C 1 ) ( 2 σ s r + C 2 ) ( μ s i m 2 + μ r i g 2 + C 1 ) ( σ s i m 2 + σ r i g 2 + C 2 ) ,
E r i g   ∝ ∑ A l E t o , l ,
E t o , l ( x , y , t ) = r t | l | ( x , t ) exp [ i l ϕ t ( x , t ) ] exp ( − x 2 + y 2 w p 2 − t 2 w t 2 − i w 0 t ) ,
σ = min { ∑ p = 1 M ∑ q = 1 M [ ∑ | A l E t o , l | 2 − | E r i g | 2 ] } ,
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