Abstract

This study proposes and successfully demonstrates an imaging technique to visualize in-plane electric current vector distributions using a plate-shape magneto-optical (MO) sensor. The technique is based on the method proposed by Roth et al. [J. Appl. Phys. 65, 361 (1989) [CrossRef]  ] but configured with the original algorithm for the use of a planar sensor and for future prompt display. The division in the Fourier domain is avoided for the suppression of unnecessary noise enhancement. The signal-to-noise ratio and dynamic range are evaluated, and the effects of the MO characteristics are discussed.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Demands for advanced systems in power electronics have been greatly expanded, especially in the fields of automobiles and renewable energy networks. High-electric-current-density circuits and energy-dense storage batteries are representative elements in such systems, where capacity enlargement is always required with downsizing and lightening. To overcome obstacles caused by unexpected behaviors of the electric currents lurking therein, their invisible spatial behaviors should be grasped, where techniques to promptly visualize electric current paths are highly attractive.

Roth et al. proposed an algorithm to image a two-dimensional (2D) electric current distribution using a magnetometer [1]. Based on the algorithm, quite a few imaging techniques for electric current trajectories have been experimentally demonstrated with the following scanning probes: a scanning superconducting quantum interference device [2], a fiber-edge-type magneto-optical (MO) probe [3], a tunneling magnetic resistance sensor [4], an optical probe combined with nitrogen vacancy color centers in diamond [5], a single electron transistor-based sensor [6], etc. However, these methods have relied on the spatial scan of a magnetometer for the acquisition of a magnetic field distribution, which is generally inferior in the prompt display of electric current images. By contrast, the real-time video display functionality for electric field distribution imaging has been successfully demonstrated through the elimination of mechanical scanning [711], where the ultra-parallel property of photonics [12] has been utilized using planar sensors [13,14]. Although the real-time visualization of electric field distributions is an example, the real-time image display of invisible phenomena tends generally to accelerate and extend related studies and developments, as experienced in many previous technological innovations.

In this paper, a non-scanning method to image in-plane electric current vector distributions using an MO sensor in the form of a plate is proposed theoretically and demonstrated experimentally. The existing effectiveness and limitations of this method are analyzed, suggesting a possible real-time electric current vector imaging. In addition, its signal-to-noise ratio (SNR) and dynamic range are analyzed experimentally.

2. Algorithm

The algorithm of the present method is based on the modification of the method proposed by Roth et al. [1]. Here, the partial differential in the real space is utilized instead of the division in the Fourier space. The avoidance of the division leads to the advantageous suppression of divergence and noise enhancement (see the Eq. (12) and relevant description in [1]). A fast Fourier transform (FFT) filter is used to eliminate the effect of the magnetic domain structures of the planar MO sensor for accurate discussions although the necessity of the FFT filter in realistic applications of the present method depends upon required spatial resolutions, and an alternative means to FFT filter usage is optical defocusing.

As shown in Figs. 1(a) and 1(b), the in-plane electric current element i = (ix, iy, 0) at position A (xP, yP, 0) in the z = 0 plane P, a part of electric current paths, generates a magnetic field vector H (x, y, z). Here, we notice the magnetic field at position B (x, y, h) in the z = h plane Q. Although i in the y direction is depicted in the figures where ix = 0 and iy ≠ 0, the following conditions are effective in any in-plane direction. The Biot–Savart law gives

$${\textbf {H}}({x,y,h} )= \int\!\!\!\int d{x_\textrm{P}}d{y_\textrm{P}}\frac{{{\textbf {i}} \times {\textbf {R}}}}{{4\mathrm{\pi }{R^3}}}, $$
where the distance between positions A and B is given by R = |R|. Here, R = (xxP, yyP, h) and the range of integrals are from −∞ to ∞. Regarding the z-component Hz of Eq. (1), its partial differential by x is expressed as
$$\frac{\partial }{{\partial x}}{H_z}({x,y,h} )= \frac{1}{{4\mathrm{\pi }{h^3}}}\int\!\!\!\int d{x_\textrm{P}}d{y_\textrm{P}}\frac{{ - {i_y} + {i_y}\left[ {2{{\left( {\frac{{x - {x_\textrm{P}}}}{h}} \right)}^2} - {{\left( {\frac{{y - {y_\textrm{P}}}}{h}} \right)}^2}} \right] - 3{i_x}\left( {\frac{{x - {x_\textrm{P}}}}{h}} \right)\left( {\frac{{y - {y_\textrm{P}}}}{h}} \right)}}{{{{({R/h} )}^5}}}. $$

 

Fig. 1. (a) In-plane electric current element i = (ix, iy, 0) at a point A in the z = 0 plane P is indicated by an arrow. Here, a case of ix = 0 is depicted as an example. (b) Side view of i with the magnetic field H generated by i, a planar magneto-optical (MO) sensor placed in the z = h plane Q, and polarization optics. The broken lines indicate the oblique directions of 45° from i. The polarization of the illuminating light is rotated by the MO sensor in accordance with the z component of the magnetic field Hz. DBR: distributed Bragg reflector. (c) Normalized integrant in the right-hand side of Eq. (2) plotted in the x or y directions. The solid line shows the first term, and the broken and dotted-dashed lines are the summation of all the terms in the x and y directions, respectively. (d) The same normalized integrant in Eq. (2) is plotted in the u or v directions. The solid line is the same as in (c), and the broken and dotted-dashed lines shows the third term.

Download Full Size | PPT Slide | PDF

Similarly, the partial differential by y is given as

$$\frac{\partial }{{\partial y}}{H_z}({x,y,h} )= \frac{1}{{4\mathrm{\pi }{h^3}}}\int\!\!\!\int d{x_\textrm{P}}d{y_\textrm{P}}\frac{{{i_x} + {i_x}\left[ {{{\left( {\frac{{x - {x_\textrm{P}}}}{h}} \right)}^2} - 2{{\left( {\frac{{y - {y_\textrm{P}}}}{h}} \right)}^2}} \right] + 3{i_y}\left( {\frac{{x - {x_\textrm{P}}}}{h}} \right)\left( {\frac{{y - {y_\textrm{P}}}}{h}} \right)}}{{{{({R/h} )}^5}}}. $$

Figure 1(b), a side view of Fig. 1(a), indicates a planar MO sensor in the plane Q together with the polarization optics for the detection of the Faraday rotation of the illuminating light induced by Hz during a round trip within the MO sensor. The 2D distribution of the Faraday rotation is transferred to an optical intensity distribution through the analyzer.

Here, the integrant in the right-hand side of Eq. (2) is assumed to be a product of iy (xP, yP, 0) and a delta function Cxδ(xxP, yyP), which leads to

$$\frac{\partial }{{\partial x}}{H_z}({x,y,h} )={-} {i_y}({x,y,0} ){C_x}/4\mathrm{\pi }{h^3}. $$

Similarly, the following equation is derived from Eq. (3):

$$\frac{\partial }{{\partial y}}{H_z}({x,y,h} )= {i_x}({x,y,0} ){C_y}/4\mathrm{\pi }{h^3} $$

Here, Cx and Cy are constants. Equations (5) and (4) indicate the partial differentials of a 2D distribution of Hz (x, y, h), i.e., the analyzer output of the MO sensor, by y, and −x gives the corresponding 2D distribution of the respective vector components of i, i.e., ix and iy.

The accuracy of the present imaging method is determined by the accuracy of the above-mentioned delta function assumption. Figure 1(c) and 1(d) shows the calculated terms of the integrant in Eq. (2) divided by “−iy” with horizontal axes normalized by h. The first term and the summation of all the terms in the (xxP) and (yyP) directions are shown in Fig. 1(c), and the first and third terms along the (uuP) and (vvP) directions are independently shown in Fig. 1(d), assuming that ix = iy. Here, u = (x + y)/20.5 and v = (−x + y)/20.5. As shown in these plots, the first term (solid lines) is isotropic, but its width is not zero. The second term is zero at the origin and along the lines of 2(xxP)2 = (yyP)2, and its maxima and minima are on the lines of (yyP) = 0 or (xxP) = 0. The third term is zero along the lines of (yyP) = 0 and (xxP) = 0, and its maxima and minima are on the lines of (vvP) = 0 and (uuP) = 0. Thus, the integrant is like a delta function to some extent but deviates as follows.

The second term in Eq. (2) leads to peripheral depressions in the x direction and additional broadening in the y direction. The spatial resolution in the x direction is h in the full width at the half maximum, and that in the y direction is approximately double. The depth of the x depressions is about a fifth of the summit. The bottom positions of the depressions are approximately ±1.3h, where h can be evaluated from an acquired electric current image. The third term in Eq. (2) is nonzero only for ix ≠ 0 and generates ghost iy images even for iy = 0 or the superpositions of ix on iy images, which appear apart from the origin A(xP, yP, 0) along the u and v directions with features of double peaks or hollows having amplitudes of a third or less of the first term summit.

When electric current paths are complicated and/or convergent, the higher-order terms might degrade the resultant electric current images, although the effect is less for simple electric current paths. More accurate images can be obtained by possible deconvolution processes with the features of the higher terms considered.

3. Experiments

To prove the effectiveness of the above-mentioned method, the following experiments were conducted.

Figure 2(a) shows a ready-made patterned printed circuit board (PCB) commercially available in Akizuki denshi tsusho co., ltd. together with an external voltage source V and a load of parallel light emitting diodes connected in series, which supply a DC electric current I to a part of the PCB electrode pattern. A magnified photograph of the PCB part indicated by a yellow square in Fig. 2(a) is shown on the left-hand side of Fig. 2(b), including a yellow rectangle, which is the area of interest for the electric current vector imaging and is magnified in the right-hand side of the figure. No electric current flows along the vertical line pattern on the left-hand side, and the injected electric current flows along the L-like electrode pattern from the right edge to the upper edge. A planar 8 × 8 mm2 MO sensor shown in Fig. 2(c) was set over the portion shown by the yellow rectangle in Fig. 2(b), together with polarization optics to acquire the Faraday rotation distribution corresponding to the Hz distribution.

 

Fig. 2. (a) Commercially available patterned printed circuit board connected to an external circuit supplying DC electric current I. (b) Left: a magnified picture of the yellow square in (a). Right: a magnified area taken by a stereoscopic microscope corresponding to the yellow rectangle in the left. (c) Commercially available planar magneto-optical (MO) sensor and a schematic of its layer structure. DBR: distributed Bragg reflector. (d) Faraday rotation vs. applied z-direction magnetic field.

Download Full Size | PPT Slide | PDF

The MO sensor is commercially available in Matesy GmbH. According to the supplier, the sensor is made of a Bi-substituted yttrium iron garnet thin film crystal grown via liquid-phase epitaxy on a gadolinium gallium garnet substrate, and an additional process to optimize the MO sensor was applied after the epitaxy. The layer structure, whose thickness is 0.5 mm, including dielectric-distributed Bragg reflector layers with a stopband of 580–700 nm and a nonmetal protection layer, is shown on the right-hand side of Fig. 2(c). The thickness of the MO layer is speculated to be less than 10 µm, according to the supplied data sheet. Figure 2(d) shows the measured Faraday rotation characteristics at 25 °C for 590 nm light plotted against Hz, which was provided by the MO sensor supplier. The saturated rotation angle is approximately 7° around 1 kA/m or more and is expected to occur at a line electric current of 1.49 A for an assumed h value of 0.17 mm, which is given by the following geometrical consideration: Hz in plane Q is maximum at the points in the oblique directions of 45° from i in plane P (Fig. 1(b)).

The polarization stereoscopic microscope consists of an objective lens (Mitutoyo M Plan Apo 2×) with an NA of 0.055 and a focal length of 100 mm, a microscope unit (Mitutoyo VMU-V), a polarization unit (Mitutoyo 378–710) with a polarizer and an analyzer, an illuminating light source (Hayashi LA-100USW) with a halogen lamp, and a digital camera (HOZAN L-835) with 2592 × 1944 pixels. Optical images were taken under conditions of maximized contrast provided by the analyzer angle adjustment. The spectral peak wavelength of the halogen lamp was approximately 935 nm from its color temperature of 3100 K. Considering the wavelength-dependent MO sensor characteristics, an illumination light source with a shorter wavelength would lead to a higher sensitivity, and a single frequency illumination would give rise to contrast improvement.

Figure 3(a) shows the raw image, which is an MO image taken as a 24-bit RGB image with 1280 × 960 pixels under a condition of I = 0.5 A. The portion indicated by a yellow square is magnified and shown on the right. While the typical periodicity of the magnetic domains was measured at approximately 18 µm, their changes are apparent along the electric current path. In addition, saturated magnetization appears, i.e., disappearance of magnetic domains, inside the bend of the electric current path.

 

Fig. 3. (a) MO sensor image taken by a polarization stereoscopic microscope for an electric current of 0.5 A, whose yellow square is magnified in the right. (b) Fast Fourier transform power spectra of (a) with a magnified one around the origin in the right. The yellow circle indicates the cutoff frequency of the low pass filter. (c) Filtered image of (a) with a magnified one in the right. (d) Image profiles for the raw (a) and filtered (c) images along the horizontal yellow broken lines therein. The third line indicates a partial differential of the filtered profile.

Download Full Size | PPT Slide | PDF

A spatial low-pass filter was applied to the raw image to suppress the magnetic domain pattern effect. The left-hand side of Fig. 3(b) shows the power spectra of Fig. 3(a), which was calculated by the FFT of the image processing software ImageJ. The full wavenumber range K is 2872 mm−1. Major circular components appear at wavenumber radii of 454 and 965 mm−1, which correspond to wavelengths of 14 and 7 µm, respectively, in a fair agreement with the above-mentioned periodicity of magnetic domains. Its right-hand side is the 2D power spectrum in the 280 mm−1 wavenumber range K’ around the origin, where the electric current-oriented components are included. The yellow circle shows the high-frequency edge kr of the spatial low-pass filter, which is 33 mm−1 and indicates the elimination of image structures with characteristic sizes less than 0.19 mm. Figure 3(c) shows a filtered image and its magnification where the magnetic domain structures in Fig. 3(a) disappear. The effect of spatial frequency filtering is also indicated in the image profile in Fig. 3(d).

The bottom profile in Fig. 3(d) indicates the partial differential by −x for the filtered profile, which is given by the difference between profiles shifted by ± 2 pixels (± 4.4 µm) in the x direction. The resultant profile resembles the broken line in Fig. 1(c). The synthesized image differences are shown in Fig. 4(a).

 

Fig. 4. (a) Electric current vector images: iy (left) and -ix (right). The yellow broken line corresponds to the profile shown in the lowest part of Fig. 3(d). (b) Superposition of images in (a) over the optical image of the PCB pattern in the right of Fig. 2(b). Red color was used to make the contrast higher. (c) Electric current diagonal vector images synthesized from Fig. 3(c): iv (left) and iu (right)

Download Full Size | PPT Slide | PDF

4. Results

Differential images in the x and y directions, i.e., electric current vector images, are shown in the left and right sides of Fig. 4(a), respectively. They are qualitatively in good agreement with predictions where the electric current is guided along the L-like electrode pattern. Contrasts in the electric current vector images were maximized by ImageJ, as shown hereafter. The spatial resolution is high enough to avoid possible interference even if nonzero current flows in the adjacent linear electrode pattern. However, some artifacts appear in the images. Periodic patterns on the background may originate from the initially non-uniform distribution of magnetic domains, whose causes are not yet clear. A few insignificant dotted patterns exist on the background, which are accidentally caused by dusts on the image sensor.

In Fig. 4(b), the electric current vector images of Fig. 4(a) are overlaid with a scale of red color on the optical image of the pattern in Fig. 2(b). Although the electric current path image matches the electrode pattern well, two portions with lower electric current densities appear in the outer parts of the two bends. This is possibly due to the electric current selection of the shortest paths, although a more detailed investigation with higher spatial resolutions is necessary, for which an S-like electrode pattern would be appropriate. Figure 4(c) indicates images diagonally differentiated in the u and v directions, which correspond well to the prediction. Particularly, in the right-hand side of the figure, a positive iu and negative iu simultaneously appear.

Figure 5 shows the electric current dependence in the present imaging method. A series of images, i.e., raw, filtered, and electric current vector images, shown in Figs. 3(a), 3(c), and 4(a) for I = 0.5 A, are tiled for I = 0.1 A in Fig. 5(a), and those for I = 2.0 A are shown in Fig. 5(b). In the electric current vector images in Fig. 5(a), the image signal is at the same level with the image noise. By contrast, the image SNR in Fig. 5(b) is considerably high. However, a region of saturated magnetization appears as a dark band inside the L-like electrode pattern in the raw and filtered images, which results in the black lines in the current images. This is probably due to the unknown anisotropy in the present method or to the magnetic field concentration effect for the inner areas of the electric current path or the degraded saturation magnetization by asymmetric heat generation by the electric current flowing closely to the MO sensor.

 

Fig. 5. Electric current dependence of the imaged results. (a) 0.1A and (b) 2.0A.

Download Full Size | PPT Slide | PDF

The SNR for the case of I = 0.5 A in Fig. 4(a) was evaluated as 16 dB via the analysis of the profile line at the bottom of Fig. 3(d). The estimated SNR value is 2 dB for a fifth (I = 0.1 A) of the electric current and 28 dB for a quadruple (I = 2.0 A). These SNR values are in good agreement with the images in Figs. 5(a) and 5(b). The range allowing effective imaging by the present setup is thus indicated in Fig. 5. The dynamic range was evaluated as 13 dB, which could be improved by improvements of the digital camera, MO sensor, and image processing.

The main factor restricting the spatial resolution is the size of the magnetic domains, which strongly depends on the MO material. In the present work, the cutoff of the spatial frequency filter was set to 0.19 mm, which is the spatial resolution in Figs. 4 and 5, where the effect of the magnetic domains disappears almost completely. The spatial resolution can be improved at the expense of noises caused by magnetic domain appearance.

Some hysteresis effects or residual magnetization effects were recognized during the raw image acquisitions, which originate from the Faraday rotation characteristics shown in Fig. 2(d). This feature is inherent in the present MO sensor. The dynamic imaging of electric current vectors suffers from the inversion velocity of magnetic domains [15] and speed of the image sensor. According to the MO sensor supplier, the response bandwidth of the MO sensor is higher than the MHz range. For the visualization of AC electric currents, rectifying or frequency-down conversion processing is necessary, as in the live electrooptic imaging technique [711].

5. Conclusion

In this paper, a non-scanning method to image in-plane electric current vector distributions using an MO sensor in the form of a plate has been proposed theoretically and demonstrated experimentally for the first time. The present effectiveness and limitations of the method suggest a possible realization of the real-time electric current vector imaging by attaching an appropriate digital image signal processing unit to the system. In addition, the experimentally analyzed SNR is 2 to 28 dB depending on the electric current within a dynamic range of 13 dB. These values can be improved through the enhancement of the MO sensor characteristics such as higher degrees of sensitivity, proximity to current paths, saturation magnetization, etc. However, the author would like to emphasize that the system configuration presented in this paper is only an example, and the system can be better designed depending on its purposes and requirements. The present system configuration with the image signal processing unit might be applied to practical fault diagnoses of high electric current circuits. However, there exists a drawback that its proximity to electric current paths is inhibited by non-flat PCB surfaces and the spatial resolution is degraded. In addition, the capability of three-dimensional imaging has not been discussed. These subjects should be dealt with in future studies.

Acknowledgements

The author thanks Prof. J. Hamasaki, Prof. H. Kamimura, and Mrs. Kamimura for their heartfelt encouragement.

Disclosures

The author declares no conflicts of interest.

References

1. B. J. Roth, N. G. Sepulveda, and J. P. Wikswo, “Using a magnetometer to image a two-dimensional current distribution,” J. Appl. Phys. 65(1), 361–372 (1989). [CrossRef]  

2. Y. Tokura, T. Honda, K. Tsubaki, and S. Tarucha, “Noninvasive determination of the ballistic-electron current distribution,” Phys. Rev. 54(3), 1947–1952 (1996). [CrossRef]  

3. T. Sato, M. Iwanami, S. Hoshino, M. Kishi, and M. Tsuchiya, “Current profiling for pm-class planar circuit patterns by fiber-edge magneto-optic probe,” Proc. Intern. Topical Meet. on Microw. Photon. (MWP2003), pp. 341–344 (2003).

4. K. Kimura, Y. Mima, and N. Kimura, “Visualization of electric current density distribution—Application to nondestructive inspection for rechargeable batteries,” J. Inst. Elec. Eng. of Japan 135(7), 437–440 (2015).

5. J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, “Quantum imaging of current flow in graphene,” Sci. Adv. 3(4), e1602429 (2017). [CrossRef]  

6. L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019). [CrossRef]  

7. M. Tsuchiya and T. Shiozawa, “Photonics makes microwaves visible,” IEEE Photonics Society Newsletter 26(6), 9–17 (2012).

8. M. Tsuchiya, K. Sasagawa, A. Kanno, and T. Shiozawa, “Live electrooptic imaging of W-band waves,” IEEE Trans. Microwave Theory Tech. 58(11), 3011–3021 (2010). [CrossRef]  

9. M. Tsuchiya, S. Fukui, and M. Yorinaga, “Microscopic live electrooptic imaging,” Sci. Rep. 7(1), 7917 (2017). [CrossRef]  

10. M. Tsuchiya, S. Fukui, and M. Yorinaga, “Live electrooptic imaging with a low-coherence optical LO signal source for suppression of interference noises,” Opt. Lett. 44(5), 1178–1181 (2019). [CrossRef]  

11. M. Tsuchiya, “LEI camera,” http://lei-camera.nict.go.jp/ (2008) (Date of access: 12/09/2020).

12. . For example, T. Sueta and T. Okoshi, Ultrafast and Ultra-parallel Optoelectronics, (Ohmsha, Tokyo, 1995).

13. Q. Wu, T. Hewitt, and X. -C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69(8), 1026–1028 (1996). [CrossRef]  

14. Z. Jiang and X. -C. Zhang, “Terahertz imaging via electrooptic effect,” IEEE Trans. Microwave Theory Tech. 47(12), 2644–2650 (1999). [CrossRef]  

15. E. Yamazaki, S. Wakana, H. Park, M. Kishi, and M. Tsuchiya, “High-frequency magneto-optic probe based on BiRIG rotation magnetization,” IEICE Trans. Electron. E86(7), 1338–1344 (2003).

References

  • View by:
  • |
  • |
  • |

  1. B. J. Roth, N. G. Sepulveda, and J. P. Wikswo, “Using a magnetometer to image a two-dimensional current distribution,” J. Appl. Phys. 65(1), 361–372 (1989).
    [Crossref]
  2. Y. Tokura, T. Honda, K. Tsubaki, and S. Tarucha, “Noninvasive determination of the ballistic-electron current distribution,” Phys. Rev. 54(3), 1947–1952 (1996).
    [Crossref]
  3. T. Sato, M. Iwanami, S. Hoshino, M. Kishi, and M. Tsuchiya, “Current profiling for pm-class planar circuit patterns by fiber-edge magneto-optic probe,” Proc. Intern. Topical Meet. on Microw. Photon. (MWP2003), pp. 341–344 (2003).
  4. K. Kimura, Y. Mima, and N. Kimura, “Visualization of electric current density distribution—Application to nondestructive inspection for rechargeable batteries,” J. Inst. Elec. Eng. of Japan 135(7), 437–440 (2015).
  5. J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, “Quantum imaging of current flow in graphene,” Sci. Adv. 3(4), e1602429 (2017).
    [Crossref]
  6. L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
    [Crossref]
  7. M. Tsuchiya and T. Shiozawa, “Photonics makes microwaves visible,” IEEE Photonics Society Newsletter 26(6), 9–17 (2012).
  8. M. Tsuchiya, K. Sasagawa, A. Kanno, and T. Shiozawa, “Live electrooptic imaging of W-band waves,” IEEE Trans. Microwave Theory Tech. 58(11), 3011–3021 (2010).
    [Crossref]
  9. M. Tsuchiya, S. Fukui, and M. Yorinaga, “Microscopic live electrooptic imaging,” Sci. Rep. 7(1), 7917 (2017).
    [Crossref]
  10. M. Tsuchiya, S. Fukui, and M. Yorinaga, “Live electrooptic imaging with a low-coherence optical LO signal source for suppression of interference noises,” Opt. Lett. 44(5), 1178–1181 (2019).
    [Crossref]
  11. M. Tsuchiya, “LEI camera,” http://lei-camera.nict.go.jp/ (2008) (Date of access: 12/09/2020).
  12. . For example, T. Sueta and T. Okoshi, Ultrafast and Ultra-parallel Optoelectronics, (Ohmsha, Tokyo, 1995).
  13. Q. Wu, T. Hewitt, and X. -C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69(8), 1026–1028 (1996).
    [Crossref]
  14. Z. Jiang and X. -C. Zhang, “Terahertz imaging via electrooptic effect,” IEEE Trans. Microwave Theory Tech. 47(12), 2644–2650 (1999).
    [Crossref]
  15. E. Yamazaki, S. Wakana, H. Park, M. Kishi, and M. Tsuchiya, “High-frequency magneto-optic probe based on BiRIG rotation magnetization,” IEICE Trans. Electron. E86(7), 1338–1344 (2003).

2019 (2)

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

M. Tsuchiya, S. Fukui, and M. Yorinaga, “Live electrooptic imaging with a low-coherence optical LO signal source for suppression of interference noises,” Opt. Lett. 44(5), 1178–1181 (2019).
[Crossref]

2017 (2)

J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, “Quantum imaging of current flow in graphene,” Sci. Adv. 3(4), e1602429 (2017).
[Crossref]

M. Tsuchiya, S. Fukui, and M. Yorinaga, “Microscopic live electrooptic imaging,” Sci. Rep. 7(1), 7917 (2017).
[Crossref]

2015 (1)

K. Kimura, Y. Mima, and N. Kimura, “Visualization of electric current density distribution—Application to nondestructive inspection for rechargeable batteries,” J. Inst. Elec. Eng. of Japan 135(7), 437–440 (2015).

2012 (1)

M. Tsuchiya and T. Shiozawa, “Photonics makes microwaves visible,” IEEE Photonics Society Newsletter 26(6), 9–17 (2012).

2010 (1)

M. Tsuchiya, K. Sasagawa, A. Kanno, and T. Shiozawa, “Live electrooptic imaging of W-band waves,” IEEE Trans. Microwave Theory Tech. 58(11), 3011–3021 (2010).
[Crossref]

2003 (1)

E. Yamazaki, S. Wakana, H. Park, M. Kishi, and M. Tsuchiya, “High-frequency magneto-optic probe based on BiRIG rotation magnetization,” IEICE Trans. Electron. E86(7), 1338–1344 (2003).

1999 (1)

Z. Jiang and X. -C. Zhang, “Terahertz imaging via electrooptic effect,” IEEE Trans. Microwave Theory Tech. 47(12), 2644–2650 (1999).
[Crossref]

1996 (2)

Q. Wu, T. Hewitt, and X. -C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69(8), 1026–1028 (1996).
[Crossref]

Y. Tokura, T. Honda, K. Tsubaki, and S. Tarucha, “Noninvasive determination of the ballistic-electron current distribution,” Phys. Rev. 54(3), 1947–1952 (1996).
[Crossref]

1989 (1)

B. J. Roth, N. G. Sepulveda, and J. P. Wikswo, “Using a magnetometer to image a two-dimensional current distribution,” J. Appl. Phys. 65(1), 361–372 (1989).
[Crossref]

Ben-Shalom, M.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Birkbeck, J.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Broadway, D. A.

J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, “Quantum imaging of current flow in graphene,” Sci. Adv. 3(4), e1602429 (2017).
[Crossref]

Dontschuk, N.

J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, “Quantum imaging of current flow in graphene,” Sci. Adv. 3(4), e1602429 (2017).
[Crossref]

Ella, L.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Fukui, S.

Geim, A. K.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Hewitt, T.

Q. Wu, T. Hewitt, and X. -C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69(8), 1026–1028 (1996).
[Crossref]

Hollenberg, L. C. L.

J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, “Quantum imaging of current flow in graphene,” Sci. Adv. 3(4), e1602429 (2017).
[Crossref]

Honda, T.

Y. Tokura, T. Honda, K. Tsubaki, and S. Tarucha, “Noninvasive determination of the ballistic-electron current distribution,” Phys. Rev. 54(3), 1947–1952 (1996).
[Crossref]

Hoshino, S.

T. Sato, M. Iwanami, S. Hoshino, M. Kishi, and M. Tsuchiya, “Current profiling for pm-class planar circuit patterns by fiber-edge magneto-optic probe,” Proc. Intern. Topical Meet. on Microw. Photon. (MWP2003), pp. 341–344 (2003).

Ilani, S.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Iwanami, M.

T. Sato, M. Iwanami, S. Hoshino, M. Kishi, and M. Tsuchiya, “Current profiling for pm-class planar circuit patterns by fiber-edge magneto-optic probe,” Proc. Intern. Topical Meet. on Microw. Photon. (MWP2003), pp. 341–344 (2003).

Jiang, Z.

Z. Jiang and X. -C. Zhang, “Terahertz imaging via electrooptic effect,” IEEE Trans. Microwave Theory Tech. 47(12), 2644–2650 (1999).
[Crossref]

Kanno, A.

M. Tsuchiya, K. Sasagawa, A. Kanno, and T. Shiozawa, “Live electrooptic imaging of W-band waves,” IEEE Trans. Microwave Theory Tech. 58(11), 3011–3021 (2010).
[Crossref]

Kimura, K.

K. Kimura, Y. Mima, and N. Kimura, “Visualization of electric current density distribution—Application to nondestructive inspection for rechargeable batteries,” J. Inst. Elec. Eng. of Japan 135(7), 437–440 (2015).

Kimura, N.

K. Kimura, Y. Mima, and N. Kimura, “Visualization of electric current density distribution—Application to nondestructive inspection for rechargeable batteries,” J. Inst. Elec. Eng. of Japan 135(7), 437–440 (2015).

Kishi, M.

E. Yamazaki, S. Wakana, H. Park, M. Kishi, and M. Tsuchiya, “High-frequency magneto-optic probe based on BiRIG rotation magnetization,” IEICE Trans. Electron. E86(7), 1338–1344 (2003).

T. Sato, M. Iwanami, S. Hoshino, M. Kishi, and M. Tsuchiya, “Current profiling for pm-class planar circuit patterns by fiber-edge magneto-optic probe,” Proc. Intern. Topical Meet. on Microw. Photon. (MWP2003), pp. 341–344 (2003).

Mima, Y.

K. Kimura, Y. Mima, and N. Kimura, “Visualization of electric current density distribution—Application to nondestructive inspection for rechargeable batteries,” J. Inst. Elec. Eng. of Japan 135(7), 437–440 (2015).

Okoshi, T.

. For example, T. Sueta and T. Okoshi, Ultrafast and Ultra-parallel Optoelectronics, (Ohmsha, Tokyo, 1995).

Park, H.

E. Yamazaki, S. Wakana, H. Park, M. Kishi, and M. Tsuchiya, “High-frequency magneto-optic probe based on BiRIG rotation magnetization,” IEICE Trans. Electron. E86(7), 1338–1344 (2003).

Perello, D.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Roth, B. J.

B. J. Roth, N. G. Sepulveda, and J. P. Wikswo, “Using a magnetometer to image a two-dimensional current distribution,” J. Appl. Phys. 65(1), 361–372 (1989).
[Crossref]

Rozen, A.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Sasagawa, K.

M. Tsuchiya, K. Sasagawa, A. Kanno, and T. Shiozawa, “Live electrooptic imaging of W-band waves,” IEEE Trans. Microwave Theory Tech. 58(11), 3011–3021 (2010).
[Crossref]

Sato, T.

T. Sato, M. Iwanami, S. Hoshino, M. Kishi, and M. Tsuchiya, “Current profiling for pm-class planar circuit patterns by fiber-edge magneto-optic probe,” Proc. Intern. Topical Meet. on Microw. Photon. (MWP2003), pp. 341–344 (2003).

Sepulveda, N. G.

B. J. Roth, N. G. Sepulveda, and J. P. Wikswo, “Using a magnetometer to image a two-dimensional current distribution,” J. Appl. Phys. 65(1), 361–372 (1989).
[Crossref]

Shiozawa, T.

M. Tsuchiya and T. Shiozawa, “Photonics makes microwaves visible,” IEEE Photonics Society Newsletter 26(6), 9–17 (2012).

M. Tsuchiya, K. Sasagawa, A. Kanno, and T. Shiozawa, “Live electrooptic imaging of W-band waves,” IEEE Trans. Microwave Theory Tech. 58(11), 3011–3021 (2010).
[Crossref]

Simpson, D. A.

J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, “Quantum imaging of current flow in graphene,” Sci. Adv. 3(4), e1602429 (2017).
[Crossref]

Stacey, A.

J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, “Quantum imaging of current flow in graphene,” Sci. Adv. 3(4), e1602429 (2017).
[Crossref]

Sueta, T.

. For example, T. Sueta and T. Okoshi, Ultrafast and Ultra-parallel Optoelectronics, (Ohmsha, Tokyo, 1995).

Sulpizio, J. A.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Taniguchi, T.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Tarucha, S.

Y. Tokura, T. Honda, K. Tsubaki, and S. Tarucha, “Noninvasive determination of the ballistic-electron current distribution,” Phys. Rev. 54(3), 1947–1952 (1996).
[Crossref]

Tetienne, J.-P.

J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, “Quantum imaging of current flow in graphene,” Sci. Adv. 3(4), e1602429 (2017).
[Crossref]

Tokura, Y.

Y. Tokura, T. Honda, K. Tsubaki, and S. Tarucha, “Noninvasive determination of the ballistic-electron current distribution,” Phys. Rev. 54(3), 1947–1952 (1996).
[Crossref]

Tsubaki, K.

Y. Tokura, T. Honda, K. Tsubaki, and S. Tarucha, “Noninvasive determination of the ballistic-electron current distribution,” Phys. Rev. 54(3), 1947–1952 (1996).
[Crossref]

Tsuchiya, M.

M. Tsuchiya, S. Fukui, and M. Yorinaga, “Live electrooptic imaging with a low-coherence optical LO signal source for suppression of interference noises,” Opt. Lett. 44(5), 1178–1181 (2019).
[Crossref]

M. Tsuchiya, S. Fukui, and M. Yorinaga, “Microscopic live electrooptic imaging,” Sci. Rep. 7(1), 7917 (2017).
[Crossref]

M. Tsuchiya and T. Shiozawa, “Photonics makes microwaves visible,” IEEE Photonics Society Newsletter 26(6), 9–17 (2012).

M. Tsuchiya, K. Sasagawa, A. Kanno, and T. Shiozawa, “Live electrooptic imaging of W-band waves,” IEEE Trans. Microwave Theory Tech. 58(11), 3011–3021 (2010).
[Crossref]

E. Yamazaki, S. Wakana, H. Park, M. Kishi, and M. Tsuchiya, “High-frequency magneto-optic probe based on BiRIG rotation magnetization,” IEICE Trans. Electron. E86(7), 1338–1344 (2003).

M. Tsuchiya, “LEI camera,” http://lei-camera.nict.go.jp/ (2008) (Date of access: 12/09/2020).

T. Sato, M. Iwanami, S. Hoshino, M. Kishi, and M. Tsuchiya, “Current profiling for pm-class planar circuit patterns by fiber-edge magneto-optic probe,” Proc. Intern. Topical Meet. on Microw. Photon. (MWP2003), pp. 341–344 (2003).

Wakana, S.

E. Yamazaki, S. Wakana, H. Park, M. Kishi, and M. Tsuchiya, “High-frequency magneto-optic probe based on BiRIG rotation magnetization,” IEICE Trans. Electron. E86(7), 1338–1344 (2003).

Watanabe, K.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Wikswo, J. P.

B. J. Roth, N. G. Sepulveda, and J. P. Wikswo, “Using a magnetometer to image a two-dimensional current distribution,” J. Appl. Phys. 65(1), 361–372 (1989).
[Crossref]

Wu, Q.

Q. Wu, T. Hewitt, and X. -C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69(8), 1026–1028 (1996).
[Crossref]

Yamazaki, E.

E. Yamazaki, S. Wakana, H. Park, M. Kishi, and M. Tsuchiya, “High-frequency magneto-optic probe based on BiRIG rotation magnetization,” IEICE Trans. Electron. E86(7), 1338–1344 (2003).

Yorinaga, M.

Zhang, X. -C.

Z. Jiang and X. -C. Zhang, “Terahertz imaging via electrooptic effect,” IEEE Trans. Microwave Theory Tech. 47(12), 2644–2650 (1999).
[Crossref]

Q. Wu, T. Hewitt, and X. -C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69(8), 1026–1028 (1996).
[Crossref]

Zultak, J.

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Appl. Phys. Lett. (1)

Q. Wu, T. Hewitt, and X. -C. Zhang, “Two-dimensional electro-optic imaging of THz beams,” Appl. Phys. Lett. 69(8), 1026–1028 (1996).
[Crossref]

IEEE Photonics Society Newsletter (1)

M. Tsuchiya and T. Shiozawa, “Photonics makes microwaves visible,” IEEE Photonics Society Newsletter 26(6), 9–17 (2012).

IEEE Trans. Microwave Theory Tech. (2)

M. Tsuchiya, K. Sasagawa, A. Kanno, and T. Shiozawa, “Live electrooptic imaging of W-band waves,” IEEE Trans. Microwave Theory Tech. 58(11), 3011–3021 (2010).
[Crossref]

Z. Jiang and X. -C. Zhang, “Terahertz imaging via electrooptic effect,” IEEE Trans. Microwave Theory Tech. 47(12), 2644–2650 (1999).
[Crossref]

IEICE Trans. Electron. (1)

E. Yamazaki, S. Wakana, H. Park, M. Kishi, and M. Tsuchiya, “High-frequency magneto-optic probe based on BiRIG rotation magnetization,” IEICE Trans. Electron. E86(7), 1338–1344 (2003).

J. Appl. Phys. (1)

B. J. Roth, N. G. Sepulveda, and J. P. Wikswo, “Using a magnetometer to image a two-dimensional current distribution,” J. Appl. Phys. 65(1), 361–372 (1989).
[Crossref]

J. Inst. Elec. Eng. of Japan (1)

K. Kimura, Y. Mima, and N. Kimura, “Visualization of electric current density distribution—Application to nondestructive inspection for rechargeable batteries,” J. Inst. Elec. Eng. of Japan 135(7), 437–440 (2015).

Nat. Nanotechnol. (1)

L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, and J. A. Sulpizio, “Simultaneous voltage and current density imaging of flowing electrons in two dimensions,” Nat. Nanotechnol. 14(5), 480–487 (2019).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (1)

Y. Tokura, T. Honda, K. Tsubaki, and S. Tarucha, “Noninvasive determination of the ballistic-electron current distribution,” Phys. Rev. 54(3), 1947–1952 (1996).
[Crossref]

Sci. Adv. (1)

J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, “Quantum imaging of current flow in graphene,” Sci. Adv. 3(4), e1602429 (2017).
[Crossref]

Sci. Rep. (1)

M. Tsuchiya, S. Fukui, and M. Yorinaga, “Microscopic live electrooptic imaging,” Sci. Rep. 7(1), 7917 (2017).
[Crossref]

Other (3)

M. Tsuchiya, “LEI camera,” http://lei-camera.nict.go.jp/ (2008) (Date of access: 12/09/2020).

. For example, T. Sueta and T. Okoshi, Ultrafast and Ultra-parallel Optoelectronics, (Ohmsha, Tokyo, 1995).

T. Sato, M. Iwanami, S. Hoshino, M. Kishi, and M. Tsuchiya, “Current profiling for pm-class planar circuit patterns by fiber-edge magneto-optic probe,” Proc. Intern. Topical Meet. on Microw. Photon. (MWP2003), pp. 341–344 (2003).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1. (a) In-plane electric current element i = (ix, iy, 0) at a point A in the z = 0 plane P is indicated by an arrow. Here, a case of ix = 0 is depicted as an example. (b) Side view of i with the magnetic field H generated by i, a planar magneto-optical (MO) sensor placed in the z = h plane Q, and polarization optics. The broken lines indicate the oblique directions of 45° from i. The polarization of the illuminating light is rotated by the MO sensor in accordance with the z component of the magnetic field Hz. DBR: distributed Bragg reflector. (c) Normalized integrant in the right-hand side of Eq. (2) plotted in the x or y directions. The solid line shows the first term, and the broken and dotted-dashed lines are the summation of all the terms in the x and y directions, respectively. (d) The same normalized integrant in Eq. (2) is plotted in the u or v directions. The solid line is the same as in (c), and the broken and dotted-dashed lines shows the third term.
Fig. 2.
Fig. 2. (a) Commercially available patterned printed circuit board connected to an external circuit supplying DC electric current I. (b) Left: a magnified picture of the yellow square in (a). Right: a magnified area taken by a stereoscopic microscope corresponding to the yellow rectangle in the left. (c) Commercially available planar magneto-optical (MO) sensor and a schematic of its layer structure. DBR: distributed Bragg reflector. (d) Faraday rotation vs. applied z-direction magnetic field.
Fig. 3.
Fig. 3. (a) MO sensor image taken by a polarization stereoscopic microscope for an electric current of 0.5 A, whose yellow square is magnified in the right. (b) Fast Fourier transform power spectra of (a) with a magnified one around the origin in the right. The yellow circle indicates the cutoff frequency of the low pass filter. (c) Filtered image of (a) with a magnified one in the right. (d) Image profiles for the raw (a) and filtered (c) images along the horizontal yellow broken lines therein. The third line indicates a partial differential of the filtered profile.
Fig. 4.
Fig. 4. (a) Electric current vector images: iy (left) and -ix (right). The yellow broken line corresponds to the profile shown in the lowest part of Fig. 3(d). (b) Superposition of images in (a) over the optical image of the PCB pattern in the right of Fig. 2(b). Red color was used to make the contrast higher. (c) Electric current diagonal vector images synthesized from Fig. 3(c): iv (left) and iu (right)
Fig. 5.
Fig. 5. Electric current dependence of the imaged results. (a) 0.1A and (b) 2.0A.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

H ( x , y , h ) = d x P d y P i × R 4 π R 3 ,
x H z ( x , y , h ) = 1 4 π h 3 d x P d y P i y + i y [ 2 ( x x P h ) 2 ( y y P h ) 2 ] 3 i x ( x x P h ) ( y y P h ) ( R / h ) 5 .
y H z ( x , y , h ) = 1 4 π h 3 d x P d y P i x + i x [ ( x x P h ) 2 2 ( y y P h ) 2 ] + 3 i y ( x x P h ) ( y y P h ) ( R / h ) 5 .
x H z ( x , y , h ) = i y ( x , y , 0 ) C x / 4 π h 3 .
y H z ( x , y , h ) = i x ( x , y , 0 ) C y / 4 π h 3

Metrics