Abstract

A rigorous elliptical mirror focusing formula based on spherical wave transformation is derived as a kind of imaging technique with high NA for potential applications in molecule imaging, spectroscopy and industrial artifact microscopy. An apodization factor is given and used to compare the energy conversation rules in lens transmission and parabolic and elliptical mirror reflections. Simulation results indicate that the axial HFWHM of elliptical and parabolic mirrors is about 80% of the corresponding HFWHM of lens in case of NA = 1 and φs = 0, and the side lobe noise is also slightly lower than that of lens, but the transverse HFWHM of mirrors is comparatively wider despite the width of main lobe is still smaller. In comparison with parabolic mirror based system, an elliptical mirror based system is potentially promising in aberration control of incident beam when the aperture of mirror is enlarged to adapt a large stage or specimen container at a small beam shading ratio.

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  1. E. F. Borra, O. Seddiki, R. Angel, D. Eisenstein, P. Hickson, K. R. Seddon, and S. P. Worden, “Deposition of metal films on an ionic liquid as a basis for a lunar telescope,” Nature 447(7147), 979–981 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. C. Stanciu, M. Sackrow, and A. J. Meixner, “High NA particle- and tip-enhanced nanoscale Raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229(2), 247–253 (2008).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2009 (1)

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

2008 (3)

2007 (2)

E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “Aberration-free optical refocusing in high numerical aperture microscopy,” Opt. Lett. 32(14), 2007–2009 (2007).
[CrossRef] [PubMed]

E. F. Borra, O. Seddiki, R. Angel, D. Eisenstein, P. Hickson, K. R. Seddon, and S. P. Worden, “Deposition of metal films on an ionic liquid as a basis for a lunar telescope,” Nature 447(7147), 979–981 (2007).
[CrossRef] [PubMed]

2004 (1)

2003 (3)

2001 (2)

2000 (2)

1977 (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microsc. Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

1959 (2)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. A 253(1274), 358–379 (1959).
[CrossRef]

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

1920 (1)

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, 5 (1920).

Angel, R.

E. F. Borra, O. Seddiki, R. Angel, D. Eisenstein, P. Hickson, K. R. Seddon, and S. P. Worden, “Deposition of metal films on an ionic liquid as a basis for a lunar telescope,” Nature 447(7147), 979–981 (2007).
[CrossRef] [PubMed]

Booth, M.

Booth, M. J.

Borra, E. F.

E. F. Borra, O. Seddiki, R. Angel, D. Eisenstein, P. Hickson, K. R. Seddon, and S. P. Worden, “Deposition of metal films on an ionic liquid as a basis for a lunar telescope,” Nature 447(7147), 979–981 (2007).
[CrossRef] [PubMed]

Botcherby, E. J.

Brabec, C. J.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

Braun, K.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

Cagigal, M. P.

Canales, V. F.

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microsc. Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

Contag, C. H.

de Juana, D. M.

Debus, C.

Drechsler, A.

Egelhaaf, H.-J.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

Eisenstein, D.

E. F. Borra, O. Seddiki, R. Angel, D. Eisenstein, P. Hickson, K. R. Seddon, and S. P. Worden, “Deposition of metal films on an ionic liquid as a basis for a lunar telescope,” Nature 447(7147), 979–981 (2007).
[CrossRef] [PubMed]

Fleischer, M.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

Gannaway, J.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microsc. Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

Hennemann, L.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

Hickson, P.

E. F. Borra, O. Seddiki, R. Angel, D. Eisenstein, P. Hickson, K. R. Seddon, and S. P. Worden, “Deposition of metal films on an ionic liquid as a basis for a lunar telescope,” Nature 447(7147), 979–981 (2007).
[CrossRef] [PubMed]

Hintz, H.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

Ignatovsky, V. S.

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, 5 (1920).

Juskaitis, R.

Kern, D. P.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

Kino, G. S.

Lieb, M. A.

Liu, J.

Mandella, M. J.

Meixner, A. J.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

C. Stanciu, M. Sackrow, and A. J. Meixner, “High NA particle- and tip-enhanced nanoscale Raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229(2), 247–253 (2008).
[CrossRef] [PubMed]

A. Drechsler, M. A. Lieb, C. Debus, A. J. Meixner, and G. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9(12), 637–644 (2001).
[CrossRef] [PubMed]

M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001).
[CrossRef] [PubMed]

Oti, J. E.

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. A 253(1274), 358–379 (1959).
[CrossRef]

Ruckstuhl, T.

Sackrow, M.

C. Stanciu, M. Sackrow, and A. J. Meixner, “High NA particle- and tip-enhanced nanoscale Raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229(2), 247–253 (2008).
[CrossRef] [PubMed]

Schwertner, M.

Seddiki, O.

E. F. Borra, O. Seddiki, R. Angel, D. Eisenstein, P. Hickson, K. R. Seddon, and S. P. Worden, “Deposition of metal films on an ionic liquid as a basis for a lunar telescope,” Nature 447(7147), 979–981 (2007).
[CrossRef] [PubMed]

Seddon, K. R.

E. F. Borra, O. Seddiki, R. Angel, D. Eisenstein, P. Hickson, K. R. Seddon, and S. P. Worden, “Deposition of metal films on an ionic liquid as a basis for a lunar telescope,” Nature 447(7147), 979–981 (2007).
[CrossRef] [PubMed]

Seeger, S.

Sheppard, C. J. R.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microsc. Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

Stanciu, C.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

C. Stanciu, M. Sackrow, and A. J. Meixner, “High NA particle- and tip-enhanced nanoscale Raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229(2), 247–253 (2008).
[CrossRef] [PubMed]

Tan, J.

Tarrach, G.

Török, P.

Varga, P.

Wang, T. D.

Wang, X.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

Wang, Y.

Wilson, T.

Wolf, E.

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

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. A 253(1274), 358–379 (1959).
[CrossRef]

Worden, S. P.

E. F. Borra, O. Seddiki, R. Angel, D. Eisenstein, P. Hickson, K. R. Seddon, and S. P. Worden, “Deposition of metal films on an ionic liquid as a basis for a lunar telescope,” Nature 447(7147), 979–981 (2007).
[CrossRef] [PubMed]

Zhang, D.

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

Zhao, C.

Appl. Opt. (3)

J. Microsc. (1)

C. Stanciu, M. Sackrow, and A. J. Meixner, “High NA particle- and tip-enhanced nanoscale Raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229(2), 247–253 (2008).
[CrossRef] [PubMed]

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

J. Raman Spectrosc. (1)

D. Zhang, X. Wang, K. Braun, H.-J. Egelhaaf, M. Fleischer, L. Hennemann, H. Hintz, C. Stanciu, C. J. Brabec, D. P. Kern, and A. J. Meixner, “Parabolic mirror-assisted tip-enhanced spectroscopic imaging for non-transparent materials,” J. Raman Spectrosc. 40(10), 1371–1376 (2009).
[CrossRef]

Microsc. Opt. Acoust. (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microsc. Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

Nature (1)

E. F. Borra, O. Seddiki, R. Angel, D. Eisenstein, P. Hickson, K. R. Seddon, and S. P. Worden, “Deposition of metal films on an ionic liquid as a basis for a lunar telescope,” Nature 447(7147), 979–981 (2007).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (3)

Proc. R. Soc. A (2)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. A 253(1274), 358–379 (1959).
[CrossRef]

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

Trans. Opt. Inst. Petrograd (1)

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, 5 (1920).

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Figures (6)

Fig. 1
Fig. 1

Reflection on an elliptical mirror. Sm is the real surface of an elliptical mirror, Sw1 and Sw2 are two wavefront surfaces corresponding to inbound and outbound spherical waves with centers F1 and F2, respectively. Their focal lengths are a + c and a-c, respectively. α is the open aperture of a point-like source at focal point F1 . θ is the focusing angle at focal point F2 where there is a scanning stage or specimen container.

Fig. 2
Fig. 2

Comparison of apodization factors of lens, parabolic and elliptical mirrors. The illuminator at F1 is assumed to be the perfect point. Focusing angle θ is in radian, and a, b and c are 500mm, 400mm and 300mm, respectively.

Fig. 3
Fig. 3

l(θ) of elliptical mirror under different c (a = 500mm).

Fig. 4
Fig. 4

A meridional plane of a ray and vectors definitions. gL, g0 and g1 denote unit vectors perpendicular to the rays drawn in red lines and all in meridional plane F1MF2. sL, s0 and s1 are unit vectors along the incident ray of Lens L, incident ray F 1 M of elliptical mirror and reflected ray M F 2 , respectively. α, θ, β and γ are the corresponding polar angles of s0, s1, g0 and g1 with respect to axis Z. β = α + π/2 and γ = π/2-θ. The largest angles of α and θ are restricted by the numerical aperture of lens L and the elliptical mirror. M is an incident point on the elliptical mirror, ϕ denotes the angle of plane F1MF2 with respect to axis X.

Fig. 5
Fig. 5

Electric field intensity e F ^ e F ^ * of elliptical mirror compared with that of lens and parabolic mirror (φs = 0, a = 500mm, b = 400mm and c = 300mm. In sub-Figs. 5(a), 5(c) and 5(b), 5(d), θm is π/3 and π/2, respectively.)

Fig. 6
Fig. 6

Contour map of electric field intensity in the focal plane on F2 in polar coordinates (φs and v are drawn as polar angle and polar radius. u = 0, θm = π/2).

Tables (1)

Tables Icon

Table 1 Comparison of Critical Sizes of Focusing Spots

Equations (15)

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l( θ )= a+c ac sinαdα sinθdθ ,
g L =cosϕ m +sinϕ n ,
g 0 =sin( α+π/2 )cosϕ m +sin( α+π/2 )sinϕ n +cos( α+π/2 ) k =cosαcosϕ m +cosαsinϕ n sinα k ,
g 1 =sin( π/2 θ )cos( ϕ+π ) m +sin( π/2 θ )sin( ϕ+π ) n +cos( π/2 θ ) k =cosθcosϕ m cosθsinϕ n +sinθ k ,
s L = k ,
s 0 =sinαcosϕ m +sinαsinϕ n +cosα k ,
s 1 =sin( πθ )cos( ϕ+π ) m +sin( πθ )sin( ϕ+π ) n +cos( πθ ) k =sinθcosϕ m sinθsinϕ n cosθ k .
e L 1 ^ = cosα [ e Lr 1 ^ g 0 + e Lφ 1 ^ ( g 0 × s 0 ) ],
e Lr 1 ^ = g 0 · e L 1 ^ = g L · e L 0 ^ =cosϕ, e Lφ 1 ^ =( g 0 × s 0 )· e L 1 ^ =( g L × s L )· e L 0 ^ =sinϕ.
e m ^ =f sin2αdα 2sinθdθ [ e mr ^ g 1 + e mφ ^ ( g 1 × s 1 ) ],
e mr ^ = g 1 · e m ^ = e Lr 1 ^ =cosϕ, e mφ 1 ^ =( g 1 × s 1 )· e m ^ = e Lφ 1 ^ =sinϕ.
e F ^ = ik 2π 0 2π 0 θ m e m ^ ( θ,ϕ ) e ik[ s 1 ·r ] sinθdθdϕ ,
0 2π sin(nϕ) e iρcos(ϕ φ s ) dϕ=2π i n J n (ρ)sin(n φ s ) , 0 2π cos(nϕ) e iρcos(ϕ φ s ) dϕ=2π i n J n (ρ)cos(n φ s ) ,
e F x ^ =iA( I 0 + I 2 cos2 φ s ), e F y ^ =iA I 2 sin2 φ s , e F z ^ =2A I 1 cos φ s ,
I 0 = 0 θ m E( θ )sinθ(1+cosθ) J 0 ( vsinθ NA ) e iucosθ N A 2 dθ , I 1 = 0 θ m E( θ ) sin 2 θ J 1 ( vsinθ NA ) e iucosθ N A 2 dθ , I 2 = 0 θ m E( θ )sinθ(1cosθ) J 2 ( vsinθ NA ) e iucosθ N A 2 dθ ,

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