Polycentric spatial focus of time-reversal electromagnetic field in rectangular conductor cavity

Yingming Chen; Bing-Zhong Wang

doi:10.1364/OE.21.026657

Optics Express
Vol. 21,
Issue 22,
pp. 26657-26662
(2013)
•https://doi.org/10.1364/OE.21.026657

Polycentric spatial focus of time-reversal electromagnetic field in rectangular conductor cavity

Yingming Chen and Bing-Zhong Wang

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Yingming Chen and Bing-Zhong Wang, "Polycentric spatial focus of time-reversal electromagnetic field in rectangular conductor cavity," Opt. Express 21, 26657-26662 (2013)

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- Physical Optics
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: July 17, 2013
- Revised Manuscript: October 11, 2013
- Manuscript Accepted: October 21, 2013
- Published: October 29, 2013

Abstract

Polycentric focus effect of time-reversal (TR) electromagnetic field is found in a rectangular resonant cavity. Theoretical deduction shows that the effect is due to the mirror symmetry of the cavity and the maximum number of focus points is 27 including 1 main focus point and 26 secondary focus points. A case of 6 focus points is calculated, in which the numerical results are consistent with the theoretical predictions, and particularly the 5 secondary focus points have directly resulted in inaccurate imaging and pulse signal interception.

1. Introduction

In nearly closed metal cavity structures such as container, armored car, submarine, spacecraft, etc., the multipath effects of pulse propagation are evident. However, many experiments have confirmed that time-reversal (TR) electromagnetic (EM) field can perform temporal and spatial focusing, especially in dense multipath circumstances [1–3]. And such high TR focus gains have attracted extensive attentions from engineers [1–10]. Recently, TR EM field has even been studied to be applied in ultra-wideband communication [4,6,8,9] and high resolution imaging [5,10,11]. It seems that TR EM field should ONLY focus back on its originator. Is it true?

In the phenomenological analysis, TR EM focus arises from the temporal and spatial coherent superposition of EM waves [12–14]. In dense multipath environments, TR operation to reverse impulse response will compensate multipath delays, which results in the temporal coherent superposition (TR temporal focus). If the compensation effect of TR operation will be lost quickly after the transceiver moves a short distance, it will be called the spatial de-correlation [6] (TR spatial focus). In case the distance is far less than Rayleigh limit, super resolution would be reported [1,10]. The theory of monochromatic TR mirrors or equivalently phase conjugate mirrors has been developed for electromagnetic waves [11,12,15], but the uniqueness of TR focus position is still uncertain.

2. Theoretical derivation

The final analysis of the uniqueness can only come from Maxwell equations and boundary conditions. It is possible to get the analytical expression of TR EM fields in some cases of symmetrical boundaries. In a rectangular vacuum resonant cavity ( $L_{a} \times L_{b} \times L_{c}$ ), one pulse is transmitted by the current element $i (t) l$ at the point $R_{T}$ , then the electric field probes at the point $R_{R}$ will receive $E (R_{R}, t)$ . The method of images can be deliberately used to calculate $E (R_{R}, t)$ in the presence of conducting boundaries with mirror symmetry [12,15]

E (R_{R}, t) = \int E (R_{R}, ω) e^{- j ω t} d ω = \int \sum_{n_{x}, n_{y}, n_{z}}^{\infty} \sum_{m = 1}^{8} {\bar{\bar{G}}}_{0} (ω, R_{R}, {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)}) \cdot l_{m} i (ω) e^{- j ω t} d ω

where

i (ω) = F [i (t)]

{\tilde{Τ}}_{n_{x} n_{y} n_{z}}

is a translation operator which periodically extends the cell consisting of 8 current elements.

{\begin{cases} {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R = R + 2 (n_{x} L_{a} \hat{x} + n_{y} L_{b} \hat{y} + n_{z} L_{c} \hat{z}) \\ n_{x}, n_{y}, n_{z} \in ℤ \end{cases}

And

{\bar{\bar{G}}}_{0}

is the transmission tensor in free space [12],

{\bar{\bar{G}}}_{0} (ω, R_{R}, {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)}) = (\bar{\bar{I}} + \frac{1}{k_{0}^{2}} \nabla_{(R_{R})} \nabla_{(R_{R})}) \frac{j ω μ_{0} e^{j k_{0} | R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} |}}{4 π | R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} |}

where

k_{0} = ω \sqrt{μ_{0} ε_{0}} = ω / c

c

is the speed of light in free space,

\bar{\bar{I}}

is an identity tensor. The origin is located at a vertex of the cavity, then the positions and polarization directions of excitation current element and 7 image current elements are

{\begin{array}{l} R_{T}^{(1)} = R_{T} = x_{T} \hat{x} + y_{T} \hat{y} + z_{T} \hat{z} = - R_{T}^{(8)} & l_{1} = l = l_{x} \hat{x} + l_{y} \hat{y} + l_{z} \hat{z} = l_{8} \\ R_{T}^{(2)} = - x_{T} \hat{x} + y_{T} \hat{y} + z_{T} \hat{z} = - R_{T}^{(7)} & l_{2} = l_{x} \hat{x} - l_{y} \hat{y} - l_{z} \hat{z} = l_{7} \\ R_{T}^{(3)} = x_{T} \hat{x} - y_{T} \hat{y} + z_{T} \hat{z} = - R_{T}^{(6)} & l_{3} = - l_{x} \hat{x} + l_{y} \hat{y} - l_{z} \hat{z} = l_{6} \\ R_{T}^{(4)} = x_{T} \hat{x} + y_{T} \hat{y} - z_{T} \hat{z} = - R_{T}^{(5)} & l_{4} = - l_{x} \hat{x} - l_{y} \hat{y} + l_{z} \hat{z} = l_{5} \end{array}

After TR operation and retransmission, the probes at the point $R$ will receive TR electric field as follow

E^{T R} (R, t) = \int \sum_{p_{x}, p_{y}, p_{z}}^{\infty} \sum_{q = 1}^{8} {\bar{\bar{G}}}_{0} (ω, R, {\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(q)}) \cdot E_{q}^{*} (R_{R}^{(q)}, ω) e^{- j ω t} d ω

where the positions and polarization directions of 8 current elements are

{\begin{array}{l} R_{R}^{(1)} = R_{R} = x_{R} \hat{x} + y_{R} \hat{y} + z_{R} \hat{z} = - R_{R}^{(8)} & E_{1}^{*} = E_{}^{*} = E_{x}^{*} \hat{x} + E_{y}^{*} \hat{y} + E_{z}^{*} \hat{z} = E_{8}^{*} \\ R_{T}^{(2)} = - x_{R} \hat{x} + y_{R} \hat{y} + z_{R} \hat{z} = - R_{R}^{(7)} & E_{2}^{*} = E_{x}^{*} \hat{x} - E_{y}^{*} \hat{y} - E_{z}^{*} \hat{z} = E_{7}^{*} \\ R_{R}^{(3)} = x_{R} \hat{x} - y_{R} \hat{y} + z_{R} \hat{z} = - R_{R}^{(6)} & E_{3}^{*} = - E_{x}^{*} \hat{x} + E_{y}^{*} \hat{y} - E_{z}^{*} \hat{z} = E_{6}^{*} \\ R_{T}^{(4)} = x_{R} \hat{x} + y_{R} \hat{y} - z_{R} \hat{z} = - R_{R}^{(5)} & E_{4}^{*} = - E_{x}^{*} \hat{x} - E_{y}^{*} \hat{y} + E_{z}^{*} \hat{z} = E_{5}^{*} \end{array}

In the forward propagation process, the multipath delay set

{τ_{n_{x} n_{y} n_{z}}^{(m)}}

{τ_{n_{x} n_{y} n_{z}}^{(m)}} = {| R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} | / c}

In the TR backward propagation process, the multipath delay set

{τ_{p_{x} p_{y} p_{z}}^{(q)}}

{τ_{p_{x} p_{y} p_{z}}^{(q)}} = {| R - {\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(q)} | / c}

R = R_{T}

, from the reciprocity view we will get

{τ_{n_{x} n_{y} n_{z}}^{(m)}} = {τ_{p_{x} p_{y} p_{z}}^{(q)}}

The Eq. (9) is bijective. That is the multipath-delay-compensation effect of TR propagation. Generally, we note the delay-compensation set

D

and the delay-compensation coefficient

η

as follows

\begin{matrix} D = {τ_{n_{x} n_{y} n_{z}}^{(m)}} \cap {τ_{p_{x} p_{y} p_{z}}^{(q)}} & η = \frac{C a r d : D}{C a r d : {τ_{n_{x} n_{y} n_{z}}^{(m)}}} \end{matrix}

In the set

D

, we always have

{| R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} |}_{D} = {| R - {\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(q)} |}_{D}

R = R_{f o c u s}

makes

η = η_{f o c u s}

(here

η_{f o c u s} \in (0, 1]

), we will call

R_{f o c u s}

focus point. When

R_{f o c u s} = R_{T}

(i.e.

η = 1

), we call

R_{T}

main focus point (MFP). From Eq. (11), the general solution of

R_{f o c u s}

will be

{R_{f o c u s}} = {{\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(q)} \pm (R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)})}

Because

R_{f o c u s}

must be in the cavity, the number of focus points

C a r d : {R_{f o c u s}}

is finite, at most 27 including 1 MFP and 26 secondary focus points (SFP). Equation (12) will give theoretical positions of all possible focus points if cavity size and transmit-receive positions are given.

The reason of 27 is $3 \times 3 \times 3$ , where each dimension contributes three possibilities, namely, twice mirror and self. However, observable focus points must be positioned in the cavity, so 27 is the most. And the 27 possible points depend on the size of the cavity and the positions of the transceivers.

3. Numerical verification

The vacuum cavity sizes in Fig. 1 are $L_{a} = 40 c m$ , $L_{b} = 50 c m$ , $L_{c} = 60 c m$ , and the transmit-receive positions are $R_{T} : (20, 38, 49) c m$ , $R_{R} : (10, 19, 20) c m$ , and the band width of the pulse excitation is $(3.1, 10.7) GHz$ , then Eq. (11) predicts 6 possible focus points exhibited in Table 1. To verify the theoretical predictions, we will use Microwave CST studio to numerically simulate TR focus process in accordance with 5 steps tagged in Fig. 1.

Fig. 1 Diagram of polycentric time-reversal (TR) focus process in rectangular resonant cavity, showing main focus point (MFP) and secondary focus points (SFP).

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Table 1. Theoretical predictions and numerical simulation results about the coordinates of 6 time-reversal (TR) focus points in the cavity of Fig. 1.

View Table

In Table 1, the numerical simulation results derive from TR imaging in the plane $x = 20 c m$ at the focus moment, which are consistent with the theoretical predictions. Furthermore, if the theoretical focus positions are brought back into Eq. (5), we can also calculate TR gains of all focus points theoretically.

In Fig. 2, the polycentric TR focus effect has directly affected TR imaging quality because the accurate image should be only one focus point. To improve the quality or to inhibit the polycentric effect, we need to further investigate the relationship between the spatial coherent superposition of EM waves and the spatial symmetry of EM boundaries.

Fig. 2 Inaccurate imaging due to polycentric time-reversal (TR) focus effect. There are 1 main focus point (MFP) and 5 secondary focus points (SFP) in the plane $x = 20 c m$ at the focus moment.

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Figure 3 shows another consequence of the polycentric TR focus effect, i.e. signal interception. Although TR focus gain at MFP is highest, TR focus gains at SFP are high enough to intercept the pulse signal. Thus, TR post-filter scheme [9] seems to be a better scheme for anti interception than TR pre-filter scheme (The complexity at the receiving end will be higher in post-filter scheme than in pre-filter scheme.).

Fig. 3 Secondary focus points (SFP) intercepted the pulse signal transmitted back to main focus point (MFP) because of polycentric TR focus effect.

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The multipath delay set can be grouped into 8 regular delay-compensation subsets due to the mirror symmetry of the cavity. In the case of MFP, the bijective relationship of $D$ is

{\begin{cases} | R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} | = | R_{T} - {\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(q)} | \\ m = q = 1; n_{x}, n_{y}, n_{z} = - p_{x}, - p_{y}, - p_{z} \\ m = q = 2; n_{x}, n_{y}, n_{z} = p_{x}, - p_{y}, - p_{z} \\ m = q = 3; n_{x}, n_{y}, n_{z} = - p_{x}, p_{y}, - p_{z} \\ m = q = 4; n_{x}, n_{y}, n_{z} = - p_{x}, - p_{y}, p_{z} \\ m = q = 5; n_{x}, n_{y}, n_{z} = p_{x}, p_{y}, - p_{z} \\ m = q = 6; n_{x}, n_{y}, n_{z} = p_{x}, - p_{y}, p_{z} \\ m = q = 7; n_{x}, n_{y}, n_{z} = - p_{x}, p_{y}, p_{z} \\ m = q = 8; n_{x}, n_{y}, n_{z} = p_{x}, p_{y}, p_{z} \end{cases}

Any multipath delay in the forward process can be compensated by the only matched one in the TR backward process, which is the detail of the reciprocity in the cavity. In the case of SFP, partial delays can be compensated by the subsets, for example

q = 1

{\begin{cases} R_{f o c u s} = (2 R_{R}^{(1)} - R_{T}^{(m)}) + {\tilde{Τ}}_{(p_{x} - n_{x}) (p_{y} - n_{y}) (p_{z} - n_{z})} 0 \\ | R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} | = | R_{f o c u s} - {\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(1)} | \end{cases}

where up to 8 SFP can appear in the cavity, and

η_{f o c u s} = 1 / 8

4. Conclusion

Both theory and simulation negate the uniqueness of TR focus between two points in rectangular resonant cavity, which suggests that more structures with spatial symmetries need to be further investigated about TR focus patterns. For TR users, actively, TR polycentric focus effect has the potential to help design new ultra-wide band analog signal distributors and multiplex wireless chargers. Passively, it may be also interesting to regulate the polycentric effect by EM polarization, because in the presence of spatial symmetries the effect may also appear in other imaging processes not only TR imaging process.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61071031 and No. 61107018), the Research Fund for the Doctoral Program of Higher Education of China (No. 20100185110021 and No. 20120185130001), the Fundamental Research Funds for the Central Universities (No. ZYGX2012YB020), and the Project ITR1113.

References and links

1. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-Field time reversal,” Science 315(5815), 1120–1122 (2007). [CrossRef] [PubMed]

2. G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, “Time reversal of wideband microwaves,” Appl. Phys. Lett. 88(15), 154101 (2006). [CrossRef]

3. P. Sundaralingam, V. Fusco, D. Zelenchuk, and R. Appleby, “Detection of an object in a reverberant environment using direct and differential time reversal,” 6th European Conference on Antennas and Propagation (2012), pp. 1115–1117. [CrossRef]

4. N. Guo, B. M. Sadler, and R. C. Qiu, “Reduced-complexity UWB time-reversal techniques and experimental results,” IEEE Trans. Wireless Comm. 6(12), 4221–4226 (2007).

5. P. Zhang, X. J. Zhang, and G. Y. Fang, “Comparison of the imaging resolutions of time reversal and back-projection algorithms in EM inverse scattering,” IEEE Geosci. Remote Sci. Lett. 10(2), 357–361 (2013).

6. I. H. Naqvi, S. A. Aleem, O. Usman, S. B. Ali, P. Besnier, and G. El, Zein, “Robustness of a time-reversal ultra-wideband system in non-stationary channel environments,” Wireless Communications and Networking Conference: PHY and Fundamentals (2012), pp. 37–41.

7. B. Wu, W. Cai, M. Alrubaiee, M. Xu, and S. K. Gayen, “Time reversal optical tomography: locating targets in a highly scattering turbid medium,” Opt. Express 19(22), 21956–21976 (2011). [CrossRef] [PubMed]

8. A. Khaleghi, G. El Zein, and I. H. Naqvi, “Demonstration of time-reversal in indoor ultra-wideband communication: time domain measurement,” IEEE ISWCS (2007), pp. 465–468.

9. X. F. Liu, B. Z. Wang, S. Q. Xiao, and S. J. Lai, “Post-time-reversed MIMO ultrawideband transmission scheme,” IEEE Trans. Antennas Propag. 58(5), 1731–1738 (2010).

10. R. Pierrat, C. Vandenbem, M. Fink, and R. Carminati, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna ,” Phys. Rev. A 87, 041801(R) (2013).

11. Y. W. Jin and J. M. F. Moura, “Time-reversal detection using antenna arrays,” IEEE Trans. Signal Proc. 57(4), 1396–1414 (2009).

12. R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32(21), 3107–3109 (2007). [CrossRef] [PubMed]

13. O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Express 17(22), 20605–20614 (2009). [CrossRef] [PubMed]

14. Y. Sivan and J. B. Pendry, “Broadband time-reversal of optical pulses using a switchable photonic-crystal mirror,” Opt. Express 19(15), 14502–14507 (2011). [CrossRef] [PubMed]

15. J. de Rosny, G. Lerosey, and M. Fink, “Theory of electromagnetic time-reversal mirrors,” IEEE Trans. Antennas Propag. 58(10), 3139–3149 (2010).

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Fig. 1 Diagram of polycentric time-reversal (TR) focus process in rectangular resonant cavity, showing main focus point (MFP) and secondary focus points (SFP).

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Fig. 2 Inaccurate imaging due to polycentric time-reversal (TR) focus effect. There are 1 main focus point (MFP) and 5 secondary focus points (SFP) in the plane

x = 20 c m

at the focus moment.

View in Article | Download Full Size | PDF

Fig. 3 Secondary focus points (SFP) intercepted the pulse signal transmitted back to main focus point (MFP) because of polycentric TR focus effect.

View in Article | Download Full Size | PDF

Tables (1)

Table 1 Theoretical predictions and numerical simulation results about the coordinates of 6 time-reversal (TR) focus points in the cavity of Fig. 1.

View Table

Equations (14)

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E (R_{R}, t) = \int E (R_{R}, ω) e^{- j ω t} d ω = \int \sum_{n_{x}, n_{y}, n_{z}}^{\infty} \sum_{m = 1}^{8} {\bar{\bar{G}}}_{0} (ω, R_{R}, {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)}) \cdot l_{m} i (ω) e^{- j ω t} d ω

{\begin{cases} {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R = R + 2 (n_{x} L_{a} \hat{x} + n_{y} L_{b} \hat{y} + n_{z} L_{c} \hat{z}) \\ n_{x}, n_{y}, n_{z} \in ℤ \end{cases}

{\bar{\bar{G}}}_{0} (ω, R_{R}, {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)}) = (\bar{\bar{I}} + \frac{1}{k_{0}^{2}} \nabla_{(R_{R})} \nabla_{(R_{R})}) \frac{j ω μ_{0} e^{j k_{0} | R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} |}}{4 π | R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} |}

{\begin{array}{l} R_{T}^{(1)} = R_{T} = x_{T} \hat{x} + y_{T} \hat{y} + z_{T} \hat{z} = - R_{T}^{(8)} & l_{1} = l = l_{x} \hat{x} + l_{y} \hat{y} + l_{z} \hat{z} = l_{8} \\ R_{T}^{(2)} = - x_{T} \hat{x} + y_{T} \hat{y} + z_{T} \hat{z} = - R_{T}^{(7)} & l_{2} = l_{x} \hat{x} - l_{y} \hat{y} - l_{z} \hat{z} = l_{7} \\ R_{T}^{(3)} = x_{T} \hat{x} - y_{T} \hat{y} + z_{T} \hat{z} = - R_{T}^{(6)} & l_{3} = - l_{x} \hat{x} + l_{y} \hat{y} - l_{z} \hat{z} = l_{6} \\ R_{T}^{(4)} = x_{T} \hat{x} + y_{T} \hat{y} - z_{T} \hat{z} = - R_{T}^{(5)} & l_{4} = - l_{x} \hat{x} - l_{y} \hat{y} + l_{z} \hat{z} = l_{5} \end{array}

E^{T R} (R, t) = \int \sum_{p_{x}, p_{y}, p_{z}}^{\infty} \sum_{q = 1}^{8} {\bar{\bar{G}}}_{0} (ω, R, {\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(q)}) \cdot E_{q}^{*} (R_{R}^{(q)}, ω) e^{- j ω t} d ω

{\begin{array}{l} R_{R}^{(1)} = R_{R} = x_{R} \hat{x} + y_{R} \hat{y} + z_{R} \hat{z} = - R_{R}^{(8)} & E_{1}^{*} = E_{}^{*} = E_{x}^{*} \hat{x} + E_{y}^{*} \hat{y} + E_{z}^{*} \hat{z} = E_{8}^{*} \\ R_{T}^{(2)} = - x_{R} \hat{x} + y_{R} \hat{y} + z_{R} \hat{z} = - R_{R}^{(7)} & E_{2}^{*} = E_{x}^{*} \hat{x} - E_{y}^{*} \hat{y} - E_{z}^{*} \hat{z} = E_{7}^{*} \\ R_{R}^{(3)} = x_{R} \hat{x} - y_{R} \hat{y} + z_{R} \hat{z} = - R_{R}^{(6)} & E_{3}^{*} = - E_{x}^{*} \hat{x} + E_{y}^{*} \hat{y} - E_{z}^{*} \hat{z} = E_{6}^{*} \\ R_{T}^{(4)} = x_{R} \hat{x} + y_{R} \hat{y} - z_{R} \hat{z} = - R_{R}^{(5)} & E_{4}^{*} = - E_{x}^{*} \hat{x} - E_{y}^{*} \hat{y} + E_{z}^{*} \hat{z} = E_{5}^{*} \end{array}

{τ_{n_{x} n_{y} n_{z}}^{(m)}} = {| R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} | / c}

{τ_{p_{x} p_{y} p_{z}}^{(q)}} = {| R - {\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(q)} | / c}

{τ_{n_{x} n_{y} n_{z}}^{(m)}} = {τ_{p_{x} p_{y} p_{z}}^{(q)}}

\begin{matrix} D = {τ_{n_{x} n_{y} n_{z}}^{(m)}} \cap {τ_{p_{x} p_{y} p_{z}}^{(q)}} & η = \frac{C a r d : D}{C a r d : {τ_{n_{x} n_{y} n_{z}}^{(m)}}} \end{matrix}

{| R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} |}_{D} = {| R - {\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(q)} |}_{D}

{R_{f o c u s}} = {{\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(q)} \pm (R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)})}

{\begin{cases} | R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} | = | R_{T} - {\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(q)} | \\ m = q = 1; n_{x}, n_{y}, n_{z} = - p_{x}, - p_{y}, - p_{z} \\ m = q = 2; n_{x}, n_{y}, n_{z} = p_{x}, - p_{y}, - p_{z} \\ m = q = 3; n_{x}, n_{y}, n_{z} = - p_{x}, p_{y}, - p_{z} \\ m = q = 4; n_{x}, n_{y}, n_{z} = - p_{x}, - p_{y}, p_{z} \\ m = q = 5; n_{x}, n_{y}, n_{z} = p_{x}, p_{y}, - p_{z} \\ m = q = 6; n_{x}, n_{y}, n_{z} = p_{x}, - p_{y}, p_{z} \\ m = q = 7; n_{x}, n_{y}, n_{z} = - p_{x}, p_{y}, p_{z} \\ m = q = 8; n_{x}, n_{y}, n_{z} = p_{x}, p_{y}, p_{z} \end{cases}

{\begin{cases} R_{f o c u s} = (2 R_{R}^{(1)} - R_{T}^{(m)}) + {\tilde{Τ}}_{(p_{x} - n_{x}) (p_{y} - n_{y}) (p_{z} - n_{z})} 0 \\ | R_{R} - {\tilde{Τ}}_{n_{x} n_{y} n_{z}} R_{T}^{(m)} | = | R_{f o c u s} - {\tilde{Τ}}_{p_{x} p_{y} p_{z}} R_{R}^{(1)} | \end{cases}

	Theory	Simulation
MFP	(20,38,49)cm	(20,38.1,48.9)cm
SFP1	(20,38,31)cm	(20,38.1,30.8)cm
SFP2	(20,38,9)cm	(20,37.9,8.7)cm
SFP3	(20,24,49)cm	(20,23.9,49.1)cm
SFP4	(20,24,31)cm	(20,23.9,30.7)cm
SFP5	(20,24,9)cm	(20,24,8.8)cm

Abstract

1. Introduction

2. Theoretical derivation

3. Numerical verification

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Tables (1)

Equations (14)

Optics Express