Abstract

Vector field electron tomography (VFET) reconstructs vector fields based on phase maps recorded from two or more orthogonal tilt series. The tomographic reconstruction of vector fields involves considerations beyond those involved in the reconstruction of scalar fields. Here we examine the effect of initial magnetization orientation on reconstruction errors. The orientation of a magnetic particle affects the contrast in the phase maps. This, in turn, affects the accuracy of the reconstructed vector fields. We derive expressions that model the dependence of reconstruction errors on initial specimen orientation when using a filtered backprojection algorithm to reconstruct a vector potential from two tilt series. We compare these analytical results with those from numerical simulations. Our results can inform experimental procedures, such as sample preparation techniques and the choice of tilt series orientations. Specimen orientation can be a significant source of error in VFET, and our results can provide the means to minimize these errors.

© 2016 Optical Society of America

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References

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2015 (1)

T. Tanigaki, Y. Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” Nano Lett. 15, 1309–1314 (2015).
[Crossref] [PubMed]

2014 (6)

B. Poornaprakash, S. Sambasivam, D. Amaranatha Reddy, G. Murali, R. P. Vijayalakshmi, and B. K. Reddy, “Dopant induced RTFM and enhancement of fluorescence efficiencies in spintronic ZnS: Ni nanoparticles,” Ceram. Int. 40, 2677–2684 (2014).
[Crossref]

L. Wu, J. Pierre-Olivier, D. Berman, W. Imaino, A. Nelson, H. Zhu, S. Zhang, and S. Sun, “Monolayer assembly of ferrimagnetic CoxFe3-xO4 nanocubes for magnetic recording,” Nano Lett. 14, 3395–3399 (2014).
[Crossref] [PubMed]

E. A. Lee, H. Yim, J. Heo, H. Kim, G. Jung, and N. S. Hwang, “Application of magnetic nanoparticle for controlled tissue assembly and tissue engineering,” Arch. Pharm. Res. 37, 120–128 (2014).
[Crossref]

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

A. G. Temiryazev, S. A. Saunin, V. E. Sizov, and M. P. Temiryazeva, “Magnetic force microscopy study of domain structures in magnetic films,” B. Russ. Acad. Sci. Phys. 78, 49–52 (2014).
[Crossref]

Z. D. C. Kemp, T. C. Petersen, D. M. Paganin, K. M. Spiers, M. Weyland, and M. J. Morgan, “Analysis of noise-induced errors in vector-field electron tomography,” Phys. Rev. A 90, 023859 (2014).
[Crossref]

2013 (1)

2011 (1)

R. P. Yu, M. J. Morgan, and D. M. Paganin, “Lorentz-electron vector tomography using two and three orthogonal tilt series,” Phys. Rev. A 83, 023813 (2011).
[Crossref]

2008 (3)

C. Phatak, M. Beleggia, and M. De Graef, “Vector field electron tomography of magnetic materials: Theoretical development,” Ultramicroscopy 108, 503–513 (2008).
[Crossref]

E. Snoeck, C. Gatel, L. M. Lacroix, T. Blon, S. Lachaize, J. Carrey, M. Respaud, and B. Chaudret, “Magnetic configurations of 30 nm iron nanocubes studied by electron holography,” Nano Lett. 8, 4293–4298 (2008).
[Crossref]

D. Shindo and Y. Murakami, “Electron holography of magnetic materials,” J. Phys. D Appl. Phys. 41, 183002 (2008).
[Crossref]

2005 (3)

A. Kohn, A. K. Petford-Long, and T. C. Anthony, “Magnetic potential in patterned materials determined using energy-dependent lorentz phase microscopy,” Phys. Rev. B 72, 014444 (2005).
[Crossref]

S. J. Lade, D. M. Paganin, and M. J. Morgan, “Electron tomography of electromagnetic fields, potentials and sources,” Opt. Commun. 253, 392–400 (2005).
[Crossref]

S. J. Lade, D. M. Paganin, and M. J. Morgan, “3-d vector tomography of doppler-transformed fields by filtered-backprojection,” Opt. Commun. 253, 382–391 (2005).
[Crossref]

2004 (2)

V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98, 271–281 (2004).
[Crossref] [PubMed]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc-Oxford 214, 51–61 (2004).
[Crossref]

2003 (1)

M. Beleggia and Y. Zhu, “Electron-optical phase shift of magnetic nanoparticles I. basic concepts,” Philos. Mag. 83, 1045–1057 (2003).
[Crossref]

2001 (1)

M. De Graef and Y. Zhu, “Quantitative noninterferometric Lorentz microscopy,” J. Appl. Phys. 89, 7177–7179 (2001).
[Crossref]

1998 (1)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586 (1998).
[Crossref]

1997 (1)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

1990 (1)

1987 (1)

1986 (1)

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B 34, 3397–3402 (1986).
[Crossref]

1983 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “Phase retrieval by iterated projections,” Optik 35, 237 (1972).

1961 (1)

Y. Aharonov and D. Bohm, “Further considerations on electromagnetic potentials in the quantum theory,” Phys. Rev. 123, 1511 (1961).
[Crossref]

Aharonov, Y.

Y. Aharonov and D. Bohm, “Further considerations on electromagnetic potentials in the quantum theory,” Phys. Rev. 123, 1511 (1961).
[Crossref]

Akashi, T.

T. Tanigaki, Y. Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” Nano Lett. 15, 1309–1314 (2015).
[Crossref] [PubMed]

Amaranatha Reddy, D.

B. Poornaprakash, S. Sambasivam, D. Amaranatha Reddy, G. Murali, R. P. Vijayalakshmi, and B. K. Reddy, “Dopant induced RTFM and enhancement of fluorescence efficiencies in spintronic ZnS: Ni nanoparticles,” Ceram. Int. 40, 2677–2684 (2014).
[Crossref]

Anthony, T. C.

A. Kohn, A. K. Petford-Long, and T. C. Anthony, “Magnetic potential in patterned materials determined using energy-dependent lorentz phase microscopy,” Phys. Rev. B 72, 014444 (2005).
[Crossref]

Arii, T.

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B 34, 3397–3402 (1986).
[Crossref]

Asundi, A.

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc-Oxford 214, 51–61 (2004).
[Crossref]

Beleggia, M.

C. Phatak, M. Beleggia, and M. De Graef, “Vector field electron tomography of magnetic materials: Theoretical development,” Ultramicroscopy 108, 503–513 (2008).
[Crossref]

M. Beleggia and Y. Zhu, “Electron-optical phase shift of magnetic nanoparticles I. basic concepts,” Philos. Mag. 83, 1045–1057 (2003).
[Crossref]

Berman, D.

L. Wu, J. Pierre-Olivier, D. Berman, W. Imaino, A. Nelson, H. Zhu, S. Zhang, and S. Sun, “Monolayer assembly of ferrimagnetic CoxFe3-xO4 nanocubes for magnetic recording,” Nano Lett. 14, 3395–3399 (2014).
[Crossref] [PubMed]

Blon, T.

E. Snoeck, C. Gatel, L. M. Lacroix, T. Blon, S. Lachaize, J. Carrey, M. Respaud, and B. Chaudret, “Magnetic configurations of 30 nm iron nanocubes studied by electron holography,” Nano Lett. 8, 4293–4298 (2008).
[Crossref]

Bohm, D.

Y. Aharonov and D. Bohm, “Further considerations on electromagnetic potentials in the quantum theory,” Phys. Rev. 123, 1511 (1961).
[Crossref]

Carrey, J.

E. Snoeck, C. Gatel, L. M. Lacroix, T. Blon, S. Lachaize, J. Carrey, M. Respaud, and B. Chaudret, “Magnetic configurations of 30 nm iron nanocubes studied by electron holography,” Nano Lett. 8, 4293–4298 (2008).
[Crossref]

Chaudret, B.

E. Snoeck, C. Gatel, L. M. Lacroix, T. Blon, S. Lachaize, J. Carrey, M. Respaud, and B. Chaudret, “Magnetic configurations of 30 nm iron nanocubes studied by electron holography,” Nano Lett. 8, 4293–4298 (2008).
[Crossref]

Chen, Q.

De Graef, M.

C. Phatak, M. Beleggia, and M. De Graef, “Vector field electron tomography of magnetic materials: Theoretical development,” Ultramicroscopy 108, 503–513 (2008).
[Crossref]

M. De Graef and Y. Zhu, “Quantitative noninterferometric Lorentz microscopy,” J. Appl. Phys. 89, 7177–7179 (2001).
[Crossref]

Endo, J.

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B 34, 3397–3402 (1986).
[Crossref]

Gatel, C.

E. Snoeck, C. Gatel, L. M. Lacroix, T. Blon, S. Lachaize, J. Carrey, M. Respaud, and B. Chaudret, “Magnetic configurations of 30 nm iron nanocubes studied by electron holography,” Nano Lett. 8, 4293–4298 (2008).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “Phase retrieval by iterated projections,” Optik 35, 237 (1972).

Gureyev, T. E.

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

Heo, J.

E. A. Lee, H. Yim, J. Heo, H. Kim, G. Jung, and N. S. Hwang, “Application of magnetic nanoparticle for controlled tissue assembly and tissue engineering,” Arch. Pharm. Res. 37, 120–128 (2014).
[Crossref]

Hono, K.

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

Humphrey, E. M.

E. M. Humphrey, “Three-dimensional magnetic field determination in magnetic nanoparticles using iterative reconstruction techniques,” Ph.D. thesis, Carnegie Mellon University (2013).

Hwang, N. S.

E. A. Lee, H. Yim, J. Heo, H. Kim, G. Jung, and N. S. Hwang, “Application of magnetic nanoparticle for controlled tissue assembly and tissue engineering,” Arch. Pharm. Res. 37, 120–128 (2014).
[Crossref]

Imaino, W.

L. Wu, J. Pierre-Olivier, D. Berman, W. Imaino, A. Nelson, H. Zhu, S. Zhang, and S. Sun, “Monolayer assembly of ferrimagnetic CoxFe3-xO4 nanocubes for magnetic recording,” Nano Lett. 14, 3395–3399 (2014).
[Crossref] [PubMed]

Jung, G.

E. A. Lee, H. Yim, J. Heo, H. Kim, G. Jung, and N. S. Hwang, “Application of magnetic nanoparticle for controlled tissue assembly and tissue engineering,” Arch. Pharm. Res. 37, 120–128 (2014).
[Crossref]

Kemp, Z. D. C.

Z. D. C. Kemp, T. C. Petersen, D. M. Paganin, K. M. Spiers, M. Weyland, and M. J. Morgan, “Analysis of noise-induced errors in vector-field electron tomography,” Phys. Rev. A 90, 023859 (2014).
[Crossref]

Kim, H.

E. A. Lee, H. Yim, J. Heo, H. Kim, G. Jung, and N. S. Hwang, “Application of magnetic nanoparticle for controlled tissue assembly and tissue engineering,” Arch. Pharm. Res. 37, 120–128 (2014).
[Crossref]

Kirkland, E. J.

E. J. Kirkland, Advanced Computing in Electron Microscopy (Springer, 2010), Chap. 3.
[Crossref]

Kohn, A.

A. Kohn, A. K. Petford-Long, and T. C. Anthony, “Magnetic potential in patterned materials determined using energy-dependent lorentz phase microscopy,” Phys. Rev. B 72, 014444 (2005).
[Crossref]

Lachaize, S.

E. Snoeck, C. Gatel, L. M. Lacroix, T. Blon, S. Lachaize, J. Carrey, M. Respaud, and B. Chaudret, “Magnetic configurations of 30 nm iron nanocubes studied by electron holography,” Nano Lett. 8, 4293–4298 (2008).
[Crossref]

Lacroix, L. M.

E. Snoeck, C. Gatel, L. M. Lacroix, T. Blon, S. Lachaize, J. Carrey, M. Respaud, and B. Chaudret, “Magnetic configurations of 30 nm iron nanocubes studied by electron holography,” Nano Lett. 8, 4293–4298 (2008).
[Crossref]

Lade, S. J.

S. J. Lade, D. M. Paganin, and M. J. Morgan, “Electron tomography of electromagnetic fields, potentials and sources,” Opt. Commun. 253, 392–400 (2005).
[Crossref]

S. J. Lade, D. M. Paganin, and M. J. Morgan, “3-d vector tomography of doppler-transformed fields by filtered-backprojection,” Opt. Commun. 253, 382–391 (2005).
[Crossref]

Lee, E. A.

E. A. Lee, H. Yim, J. Heo, H. Kim, G. Jung, and N. S. Hwang, “Application of magnetic nanoparticle for controlled tissue assembly and tissue engineering,” Arch. Pharm. Res. 37, 120–128 (2014).
[Crossref]

Matsuda, T.

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B 34, 3397–3402 (1986).
[Crossref]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc-Oxford 214, 51–61 (2004).
[Crossref]

Mihama, K.

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B 34, 3397–3402 (1986).
[Crossref]

Morgan, M. J.

Z. D. C. Kemp, T. C. Petersen, D. M. Paganin, K. M. Spiers, M. Weyland, and M. J. Morgan, “Analysis of noise-induced errors in vector-field electron tomography,” Phys. Rev. A 90, 023859 (2014).
[Crossref]

R. P. Yu, M. J. Morgan, and D. M. Paganin, “Lorentz-electron vector tomography using two and three orthogonal tilt series,” Phys. Rev. A 83, 023813 (2011).
[Crossref]

S. J. Lade, D. M. Paganin, and M. J. Morgan, “3-d vector tomography of doppler-transformed fields by filtered-backprojection,” Opt. Commun. 253, 382–391 (2005).
[Crossref]

S. J. Lade, D. M. Paganin, and M. J. Morgan, “Electron tomography of electromagnetic fields, potentials and sources,” Opt. Commun. 253, 392–400 (2005).
[Crossref]

Murakami, Y.

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

D. Shindo and Y. Murakami, “Electron holography of magnetic materials,” J. Phys. D Appl. Phys. 41, 183002 (2008).
[Crossref]

Murali, G.

B. Poornaprakash, S. Sambasivam, D. Amaranatha Reddy, G. Murali, R. P. Vijayalakshmi, and B. K. Reddy, “Dopant induced RTFM and enhancement of fluorescence efficiencies in spintronic ZnS: Ni nanoparticles,” Ceram. Int. 40, 2677–2684 (2014).
[Crossref]

Nelson, A.

L. Wu, J. Pierre-Olivier, D. Berman, W. Imaino, A. Nelson, H. Zhu, S. Zhang, and S. Sun, “Monolayer assembly of ferrimagnetic CoxFe3-xO4 nanocubes for magnetic recording,” Nano Lett. 14, 3395–3399 (2014).
[Crossref] [PubMed]

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc-Oxford 214, 51–61 (2004).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586 (1998).
[Crossref]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

Ohkubo, T.

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc-Oxford 214, 51–61 (2004).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586 (1998).
[Crossref]

Paganin, D. M.

Z. D. C. Kemp, T. C. Petersen, D. M. Paganin, K. M. Spiers, M. Weyland, and M. J. Morgan, “Analysis of noise-induced errors in vector-field electron tomography,” Phys. Rev. A 90, 023859 (2014).
[Crossref]

R. P. Yu, M. J. Morgan, and D. M. Paganin, “Lorentz-electron vector tomography using two and three orthogonal tilt series,” Phys. Rev. A 83, 023813 (2011).
[Crossref]

S. J. Lade, D. M. Paganin, and M. J. Morgan, “3-d vector tomography of doppler-transformed fields by filtered-backprojection,” Opt. Commun. 253, 382–391 (2005).
[Crossref]

S. J. Lade, D. M. Paganin, and M. J. Morgan, “Electron tomography of electromagnetic fields, potentials and sources,” Opt. Commun. 253, 392–400 (2005).
[Crossref]

Park, H. S.

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

Pavel, M.

Peli, E.

Petersen, T. C.

Z. D. C. Kemp, T. C. Petersen, D. M. Paganin, K. M. Spiers, M. Weyland, and M. J. Morgan, “Analysis of noise-induced errors in vector-field electron tomography,” Phys. Rev. A 90, 023859 (2014).
[Crossref]

Petford-Long, A. K.

A. Kohn, A. K. Petford-Long, and T. C. Anthony, “Magnetic potential in patterned materials determined using energy-dependent lorentz phase microscopy,” Phys. Rev. B 72, 014444 (2005).
[Crossref]

Phatak, C.

C. Phatak, M. Beleggia, and M. De Graef, “Vector field electron tomography of magnetic materials: Theoretical development,” Ultramicroscopy 108, 503–513 (2008).
[Crossref]

C. Phatak, “On the use of phase reconstructed vector field electron tomography for the three-dimensional study of magnetic materials,” Ph.D. thesis, Carnegie Mellon University (2009).

Pierre-Olivier, J.

L. Wu, J. Pierre-Olivier, D. Berman, W. Imaino, A. Nelson, H. Zhu, S. Zhang, and S. Sun, “Monolayer assembly of ferrimagnetic CoxFe3-xO4 nanocubes for magnetic recording,” Nano Lett. 14, 3395–3399 (2014).
[Crossref] [PubMed]

Poornaprakash, B.

B. Poornaprakash, S. Sambasivam, D. Amaranatha Reddy, G. Murali, R. P. Vijayalakshmi, and B. K. Reddy, “Dopant induced RTFM and enhancement of fluorescence efficiencies in spintronic ZnS: Ni nanoparticles,” Ceram. Int. 40, 2677–2684 (2014).
[Crossref]

Reddy, B. K.

B. Poornaprakash, S. Sambasivam, D. Amaranatha Reddy, G. Murali, R. P. Vijayalakshmi, and B. K. Reddy, “Dopant induced RTFM and enhancement of fluorescence efficiencies in spintronic ZnS: Ni nanoparticles,” Ceram. Int. 40, 2677–2684 (2014).
[Crossref]

Reed Teague, M.

Respaud, M.

E. Snoeck, C. Gatel, L. M. Lacroix, T. Blon, S. Lachaize, J. Carrey, M. Respaud, and B. Chaudret, “Magnetic configurations of 30 nm iron nanocubes studied by electron holography,” Nano Lett. 8, 4293–4298 (2008).
[Crossref]

Riedl, T.

Sambasivam, S.

B. Poornaprakash, S. Sambasivam, D. Amaranatha Reddy, G. Murali, R. P. Vijayalakshmi, and B. K. Reddy, “Dopant induced RTFM and enhancement of fluorescence efficiencies in spintronic ZnS: Ni nanoparticles,” Ceram. Int. 40, 2677–2684 (2014).
[Crossref]

Sasaki, T. T.

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

Saunin, S. A.

A. G. Temiryazev, S. A. Saunin, V. E. Sizov, and M. P. Temiryazeva, “Magnetic force microscopy study of domain structures in magnetic films,” B. Russ. Acad. Sci. Phys. 78, 49–52 (2014).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “Phase retrieval by iterated projections,” Optik 35, 237 (1972).

Shimakura, T.

T. Tanigaki, Y. Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” Nano Lett. 15, 1309–1314 (2015).
[Crossref] [PubMed]

Shindo, D.

T. Tanigaki, Y. Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” Nano Lett. 15, 1309–1314 (2015).
[Crossref] [PubMed]

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

D. Shindo and Y. Murakami, “Electron holography of magnetic materials,” J. Phys. D Appl. Phys. 41, 183002 (2008).
[Crossref]

Sizov, V. E.

A. G. Temiryazev, S. A. Saunin, V. E. Sizov, and M. P. Temiryazeva, “Magnetic force microscopy study of domain structures in magnetic films,” B. Russ. Acad. Sci. Phys. 78, 49–52 (2014).
[Crossref]

Snoeck, E.

E. Snoeck, C. Gatel, L. M. Lacroix, T. Blon, S. Lachaize, J. Carrey, M. Respaud, and B. Chaudret, “Magnetic configurations of 30 nm iron nanocubes studied by electron holography,” Nano Lett. 8, 4293–4298 (2008).
[Crossref]

Sperling, G.

Spiers, K. M.

Z. D. C. Kemp, T. C. Petersen, D. M. Paganin, K. M. Spiers, M. Weyland, and M. J. Morgan, “Analysis of noise-induced errors in vector-field electron tomography,” Phys. Rev. A 90, 023859 (2014).
[Crossref]

Sugawara, A.

T. Tanigaki, Y. Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” Nano Lett. 15, 1309–1314 (2015).
[Crossref] [PubMed]

Sun, S.

L. Wu, J. Pierre-Olivier, D. Berman, W. Imaino, A. Nelson, H. Zhu, S. Zhang, and S. Sun, “Monolayer assembly of ferrimagnetic CoxFe3-xO4 nanocubes for magnetic recording,” Nano Lett. 14, 3395–3399 (2014).
[Crossref] [PubMed]

Takahashi, Y.

T. Tanigaki, Y. Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” Nano Lett. 15, 1309–1314 (2015).
[Crossref] [PubMed]

Takeno, Y.

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

Tanigaki, T.

T. Tanigaki, Y. Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” Nano Lett. 15, 1309–1314 (2015).
[Crossref] [PubMed]

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

Temiryazev, A. G.

A. G. Temiryazev, S. A. Saunin, V. E. Sizov, and M. P. Temiryazeva, “Magnetic force microscopy study of domain structures in magnetic films,” B. Russ. Acad. Sci. Phys. 78, 49–52 (2014).
[Crossref]

Temiryazeva, M. P.

A. G. Temiryazev, S. A. Saunin, V. E. Sizov, and M. P. Temiryazeva, “Magnetic force microscopy study of domain structures in magnetic films,” B. Russ. Acad. Sci. Phys. 78, 49–52 (2014).
[Crossref]

Tonomura, A.

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B 34, 3397–3402 (1986).
[Crossref]

Tsuneta, R.

T. Tanigaki, Y. Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” Nano Lett. 15, 1309–1314 (2015).
[Crossref] [PubMed]

Vanderbeek, A.

Vijayalakshmi, R. P.

B. Poornaprakash, S. Sambasivam, D. Amaranatha Reddy, G. Murali, R. P. Vijayalakshmi, and B. K. Reddy, “Dopant induced RTFM and enhancement of fluorescence efficiencies in spintronic ZnS: Ni nanoparticles,” Ceram. Int. 40, 2677–2684 (2014).
[Crossref]

Volkov, V.

V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98, 271–281 (2004).
[Crossref] [PubMed]

Weyland, M.

Z. D. C. Kemp, T. C. Petersen, D. M. Paganin, K. M. Spiers, M. Weyland, and M. J. Morgan, “Analysis of noise-induced errors in vector-field electron tomography,” Phys. Rev. A 90, 023859 (2014).
[Crossref]

Wu, L.

L. Wu, J. Pierre-Olivier, D. Berman, W. Imaino, A. Nelson, H. Zhu, S. Zhang, and S. Sun, “Monolayer assembly of ferrimagnetic CoxFe3-xO4 nanocubes for magnetic recording,” Nano Lett. 14, 3395–3399 (2014).
[Crossref] [PubMed]

Yim, H.

E. A. Lee, H. Yim, J. Heo, H. Kim, G. Jung, and N. S. Hwang, “Application of magnetic nanoparticle for controlled tissue assembly and tissue engineering,” Arch. Pharm. Res. 37, 120–128 (2014).
[Crossref]

Yu, R. P.

R. P. Yu, M. J. Morgan, and D. M. Paganin, “Lorentz-electron vector tomography using two and three orthogonal tilt series,” Phys. Rev. A 83, 023813 (2011).
[Crossref]

Yu, Y.

Zhang, S.

L. Wu, J. Pierre-Olivier, D. Berman, W. Imaino, A. Nelson, H. Zhu, S. Zhang, and S. Sun, “Monolayer assembly of ferrimagnetic CoxFe3-xO4 nanocubes for magnetic recording,” Nano Lett. 14, 3395–3399 (2014).
[Crossref] [PubMed]

Zhu, H.

L. Wu, J. Pierre-Olivier, D. Berman, W. Imaino, A. Nelson, H. Zhu, S. Zhang, and S. Sun, “Monolayer assembly of ferrimagnetic CoxFe3-xO4 nanocubes for magnetic recording,” Nano Lett. 14, 3395–3399 (2014).
[Crossref] [PubMed]

Zhu, Y.

V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98, 271–281 (2004).
[Crossref] [PubMed]

M. Beleggia and Y. Zhu, “Electron-optical phase shift of magnetic nanoparticles I. basic concepts,” Philos. Mag. 83, 1045–1057 (2003).
[Crossref]

M. De Graef and Y. Zhu, “Quantitative noninterferometric Lorentz microscopy,” J. Appl. Phys. 89, 7177–7179 (2001).
[Crossref]

Zuo, C.

Acta Mater. (1)

Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, and D. Shindo, “Magnetism of ultrathin intergranular boundary regions in Nd–Fe–B permanent magnets,” Acta Mater. 71, 370–379(2014).
[Crossref]

Arch. Pharm. Res. (1)

E. A. Lee, H. Yim, J. Heo, H. Kim, G. Jung, and N. S. Hwang, “Application of magnetic nanoparticle for controlled tissue assembly and tissue engineering,” Arch. Pharm. Res. 37, 120–128 (2014).
[Crossref]

B. Russ. Acad. Sci. Phys. (1)

A. G. Temiryazev, S. A. Saunin, V. E. Sizov, and M. P. Temiryazeva, “Magnetic force microscopy study of domain structures in magnetic films,” B. Russ. Acad. Sci. Phys. 78, 49–52 (2014).
[Crossref]

Ceram. Int. (1)

B. Poornaprakash, S. Sambasivam, D. Amaranatha Reddy, G. Murali, R. P. Vijayalakshmi, and B. K. Reddy, “Dopant induced RTFM and enhancement of fluorescence efficiencies in spintronic ZnS: Ni nanoparticles,” Ceram. Int. 40, 2677–2684 (2014).
[Crossref]

J. Appl. Phys. (1)

M. De Graef and Y. Zhu, “Quantitative noninterferometric Lorentz microscopy,” J. Appl. Phys. 89, 7177–7179 (2001).
[Crossref]

J. Microsc-Oxford (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc-Oxford 214, 51–61 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. D Appl. Phys. (1)

D. Shindo and Y. Murakami, “Electron holography of magnetic materials,” J. Phys. D Appl. Phys. 41, 183002 (2008).
[Crossref]

Nano Lett. (3)

L. Wu, J. Pierre-Olivier, D. Berman, W. Imaino, A. Nelson, H. Zhu, S. Zhang, and S. Sun, “Monolayer assembly of ferrimagnetic CoxFe3-xO4 nanocubes for magnetic recording,” Nano Lett. 14, 3395–3399 (2014).
[Crossref] [PubMed]

E. Snoeck, C. Gatel, L. M. Lacroix, T. Blon, S. Lachaize, J. Carrey, M. Respaud, and B. Chaudret, “Magnetic configurations of 30 nm iron nanocubes studied by electron holography,” Nano Lett. 8, 4293–4298 (2008).
[Crossref]

T. Tanigaki, Y. Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” Nano Lett. 15, 1309–1314 (2015).
[Crossref] [PubMed]

Opt. Commun. (3)

S. J. Lade, D. M. Paganin, and M. J. Morgan, “Electron tomography of electromagnetic fields, potentials and sources,” Opt. Commun. 253, 392–400 (2005).
[Crossref]

S. J. Lade, D. M. Paganin, and M. J. Morgan, “3-d vector tomography of doppler-transformed fields by filtered-backprojection,” Opt. Commun. 253, 382–391 (2005).
[Crossref]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

Opt. Express (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “Phase retrieval by iterated projections,” Optik 35, 237 (1972).

Philos. Mag. (1)

M. Beleggia and Y. Zhu, “Electron-optical phase shift of magnetic nanoparticles I. basic concepts,” Philos. Mag. 83, 1045–1057 (2003).
[Crossref]

Phys. Rev. (1)

Y. Aharonov and D. Bohm, “Further considerations on electromagnetic potentials in the quantum theory,” Phys. Rev. 123, 1511 (1961).
[Crossref]

Phys. Rev. A (2)

R. P. Yu, M. J. Morgan, and D. M. Paganin, “Lorentz-electron vector tomography using two and three orthogonal tilt series,” Phys. Rev. A 83, 023813 (2011).
[Crossref]

Z. D. C. Kemp, T. C. Petersen, D. M. Paganin, K. M. Spiers, M. Weyland, and M. J. Morgan, “Analysis of noise-induced errors in vector-field electron tomography,” Phys. Rev. A 90, 023859 (2014).
[Crossref]

Phys. Rev. B (2)

A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mihama, “Holographic interference electron microscopy for determining specimen magnetic structure and thickness distribution,” Phys. Rev. B 34, 3397–3402 (1986).
[Crossref]

A. Kohn, A. K. Petford-Long, and T. C. Anthony, “Magnetic potential in patterned materials determined using energy-dependent lorentz phase microscopy,” Phys. Rev. B 72, 014444 (2005).
[Crossref]

Phys. Rev. Lett. (1)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586 (1998).
[Crossref]

Ultramicroscopy (2)

C. Phatak, M. Beleggia, and M. De Graef, “Vector field electron tomography of magnetic materials: Theoretical development,” Ultramicroscopy 108, 503–513 (2008).
[Crossref]

V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98, 271–281 (2004).
[Crossref] [PubMed]

Other (3)

E. M. Humphrey, “Three-dimensional magnetic field determination in magnetic nanoparticles using iterative reconstruction techniques,” Ph.D. thesis, Carnegie Mellon University (2013).

C. Phatak, “On the use of phase reconstructed vector field electron tomography for the three-dimensional study of magnetic materials,” Ph.D. thesis, Carnegie Mellon University (2009).

E. J. Kirkland, Advanced Computing in Electron Microscopy (Springer, 2010), Chap. 3.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Diagram showing the phase shift (represented by the length of the arrows) in the electron beam after passing through a magnetic vector potential.
Fig. 2
Fig. 2 (a) Geometry used for the acquisition of tilt series. In the α series, the object �� is rotated about the x-axis and, in the θ series, it is rotated about the z-axis. (b) Spherical polar coordinate system used to describe magnetization directions. Here m(γ, γ′) is the magnetization vector of the particle, which is assumed to be uniformly magnetized, γ is the polar angle measured from the negative y-axis, and γ′ is the azimuthal angle, measured from the x-axis to the projection of m onto the xz plane.
Fig. 3
Fig. 3 A 1 voxel (3.125 nm) thick slice through the origin of each component of A for a 100 nm diameter magnetite sphere reconstructed from simulated micrographs with a defocus of 100 μm. Top row: The exact simulated vector potential. Middle row: Reconstruction with m(γ, γ′) chosen to maximize rmsE. Bottom row: Reconstruction with m(γ, γ′) chosen to minimize rmsE.
Fig. 4
Fig. 4 (a) Grayscale plots of error as a function of specimen orientation for the Δf = 25 μm case, using simulations (left) and analytical results (right). The errors here are shown as a function of γ and γ′, and the table shows the values chosen for each constant in the analytical model. (b) The spheres show the results mapped back onto spherical coordinates, with the axes underneath indicating the orientation of these spheres in Cartesian coordinates.
Fig. 5
Fig. 5 Scatter plots showing the rms errors in the reconstructed vector potential as a function of initial specimen orientation and for a range of defoci, using simulated micrographs (top) and the analytical estimate (bottom). Plots on the left and right show slices at γ′ = 0 and γ = π/2, respectively. For each defocus, in these results, the rms error varies by an approximate factor of two as a function of γ.

Equations (38)

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

φ m ( r ) = e A ( r , z ) d z ,
φ e ( r ) = π E λ V ( r , z ) d z ,
φ = φ e + φ m .
k I 0 z = ( I 0 φ 0 ) .
I 0 z I ( r , z ) z | z = 0 .
I 0 z I + I 2 Δ f ,
I 0 I + + I 2 ,
φ 0 = k 4 π 2 1 { k | k | 2 { 1 I 0 1 { { I 0 z } k | k | 2 } } } ,
φ m = φ φ 2 ,
T θ ( x , z ) = e φ m ( x , z , θ ) ,
T α ( y , x ) = e φ m ( y , x , α ) ,
A ( x , y , z ) = 0 π T ˜ θ | k r θ | k r θ k x 2 + k y 2 + k z 2 [ k y 2 + k z 2 k y k x k x k z k y ] e 2 π i [ k r θ ( x cos θ + y sin θ ) + k z z ] d k r θ d k z d θ + 0 π T ˜ α | k r α | k r α k x 2 + k y 2 + k z 2 [ k x k z k y k z k x 2 + k y 2 k y ] e 2 π i [ k r α ( y cos α + z sin α ) + k x x ] d k r α d k x d α .
γ = arccos ( γ x 2 + y 2 + z 2 ) ,
γ = arctan ( z x ) .
k I 0 z = I 0 2 φ 0 .
I + I 0 Δ f = I 0 k 2 φ 0 .
I + = I 0 Δ f k 2 φ 0 + I 0 .
K = 1 I 0 2 ( M 2 1 ) i , j ( I i , j + I 0 ) 2
= 1 ( M 2 1 ) i , j [ Δ f k 2 ( φ m ( r ) ) ] 2 ,
φ m = e A d z = μ 0 e 2 π ( m × r ) z ^ x 2 + y 2 ,
2 ( φ m ) = μ 0 e 2 π 2 ( ( z ^ × m ) r x 2 + y 2 + τ 2 )
= 4 μ 0 e π ( τ 2 ( z ^ × m ) r ( x 2 + y 2 + τ 2 ) 3 ) ,
K = 4 μ 0 e Δ f | m | π k M 2 1 i , j ( τ 2 ( z ^ × m ^ ) r ( x 2 + y 2 + τ 2 ) 3 ) 2
= 4 μ 0 e τ 2 Δ f | m | | z ^ × m ^ | π k M 2 1 i , j ( cos ζ ( x 2 + y 2 + τ 2 ) 3 ) 2 ,
K | z ^ × m ^ | .
K 2 K max 2 = | z ^ × m ^ | 2
= 1 ( m ^ z ^ ) 2 .
E trunc ( Δ f ) 2 6 3 I 0 z 3 .
E rms = i , j , k | A i , j , k rec A i , j , k | 2 i , j , k | A i , j , k | 2 ,
E ¯ φ 2 rms ( ( m ^ z ^ ) 2 , ( Δ f ) 4 ) = E 2 ( 1 , 0 ) + 1 2 [ 2 ( E ¯ φ 2 rms ) p 2 ( ( m ^ z ^ ) 2 1 ) 2 + 2 2 ( E ¯ φ 2 rms ) p q ( ( m ^ z ^ ) 2 1 ) ( Δ f ) 4 + 2 ( E ¯ φ 2 rms ) q 2 ( Δ f ) 8 ] ,
A = 2 p 2 ( E ¯ φ 2 rms ) , B = 2 p q ( E ¯ φ 2 rms ) , and C = 1 2 2 q 2 ( E ¯ φ 2 rms ) .
E ¯ φ 2 rms ( ( m ^ z ^ ) 2 , ( Δ f ) 4 ) = E 2 min + ( B ( Δ f ) 4 A ) ( m ^ z ^ ) 2 B ( Δ f ) 4 + 1 2 A ( m ^ z ^ ) 4 + C ( Δ f ) 8 ,
E ¯ φ 2 rms ( α , γ , γ ) = ( m y sin ( α ) + m z cos ( α ) ) 2 ( B α ( Δ f ) 4 A α ) B α ( Δ f ) 4 + A α 2 ( m y sin ( α ) + m z cos ( α ) ) 4 + C α ( Δ f ) 8 + E α 2 min
E ¯ φ 2 rms ( θ , γ , γ ) = ( m x sin ( θ ) + m y cos ( θ ) ) 2 ( B θ ( Δ f ) 4 A θ ) B θ ( Δ f ) 4 + A θ 2 ( m x sin ( θ ) + m y cos ( θ ) ) 4 + C θ ( Δ f ) 8 + E θ 2 min
m ^ = sin γ cos γ x ^ + cos γ y ^ + sin γ sin γ z ^ .
E ¯ α 2 rms = π 2 ( m y 2 + m z 2 ) ( B α ( Δ f ) 4 A α ) π B α ( Δ f ) 4 + 3 π A α 16 ( m y 2 + m z 2 ) 2 + π C α ( Δ f ) 8 + π E α 2 min
E ¯ θ 2 rms = π 2 ( m y 2 + m x 2 ) ( B θ ( Δ f ) 4 A θ ) π B θ ( Δ f ) 4 + 3 π A θ 16 ( m y 2 + m x 2 ) 2 + π C θ ( Δ f ) 8 + π E θ 2 min .
E ¯ rms = E ¯ α rms + E ¯ θ rms

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