Rigorous theory on elliptical mirror focusing for point scanning microscopy

doi:10.1364/OE.20.006175

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Rigorous theory on elliptical mirror focusing for point scanning microscopy

Jian Liu, Jiubin Tan, Tony Wilson, and Cien Zhong »View Author Affiliations

Optics Express, Vol. 20, Issue 6, pp. 6175-6184 (2012)
http://dx.doi.org/10.1364/OE.20.006175

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▸ Article Outline
– Abstract
1 Introduction
2 Apodization factor
3 Vector theories
4 Discussions
5 Conclusions
– References and links

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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 φ_s = 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.

1 Introduction

Mirrors with low numerical aperture are fascinating for both wide spectrum imaging and design of astronomical telescopes [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]

–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]

]. But in the wide field of view devices, their usage is restricted by their poor off-axis properties and sensitive aberrations caused by assembly errors. This is one of the reasons why lenses are much more popular than mirrors in optical instruments in the past decades. However, this point of view is now changing gradually because of the fast development of precise engineering. The current optical process is so powerful to make a profile error under one tenth of a wavelength, or even better, and assembly error under several micrometers as well. Such an evolvement gradually reduces the disadvantage of a mirror in its conventional application and highlights the advantage of a mirror in wide spectrum imaging and short wave propagation, which is very useful for high resolution observation. Actually, mirror techniques have been used in microscopy as an approach to enlarge numerical aperture up to 1, the aperture limitation for lenses in case of no immersion [5

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]

]. This enlargement of aperture is potentially valuable for molecule imaging, spectroscopy and industrial artifact metrology with high resolution [6

M. Schwertner, M. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high NA adaptive optical microscopy,” Opt. Express 12(26), 6540–6552 (2004). [CrossRef] [PubMed]

–8

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]

The intensity distribution near the focal point of a high aperture focusing system is always very tight. Vectorial theory is therefore taken as a rigorous tool for the descriptions of intensity distribution and influence of polarization. An earlier independent discussion on focusing lens is from Richard and Wolf to represent how a linear polarized plane wave transformed being sphere wave, and the development of amplitude factor

\sqrt{\cos α}

is of significance for the calculation in high aperture focusing [9

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]

,10

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]

], where α is the focusing angle. But a mirror system differs from a lens because a different mirror formula corresponds to a different apodization factor. The earlier discussion on parabolic mirror with high aperture may date back to 1920 when V. S. Ignatovsky figured out

2 / (1 + \cos α)

, the apodization factor of a parabolic mirror, which is of significance for the calculation in high aperture focusing by assuming incident beam a plane wave [11

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

]. The time-averaged electric energy density near the focus of a parabolic mirror with wide open aperture is calculated and compared with that of an aplanatic lens system in ref [12

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]

], and it comes to the conclusion that the energy distribution in a mirror system is a greater departure from a circular symmetry. The solution of a parabolic mirror is derived from the Stratton–Chu integral by solving a boundary-value problem in ref [13

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000). [CrossRef] [PubMed]

], and the corresponding numerical results are published in ref [14

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. numerical results,” J. Opt. Soc. Am. A 17(11), 2090–2095 (2000). [CrossRef] [PubMed]

]. In addition, a parabolic mirror with high numerical aperture is explored for confocal imaging, spectroscopy of nano-particles and single molecule in ref [2

T. Ruckstuhl and S. Seeger, “Confocal total-internal-reflection fluorescence microscopy with a high-aperture parabolic mirror lens,” Appl. Opt. 42(16), 3277–3283 (2003). [CrossRef] [PubMed]

,15

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]

Undoubtedly, lens is the most popular and successfully developed imaging element for its insensitive off-axis property and lower fabrication cost, etc., but chromatic aberration is really a tough matter in wide spectrum imaging, and its numerical aperture is theoretically limited below 1 in case of no immersion. According to the above investigations, parabolic mirrors are mainly solved and widely discussed at present as a mirror technique for high aperture focusing. Certainly, it is understandable why much attention has been paid to parabolic mirrors because of their relatively easier fabrication and the focusing characteristics of parallel beams. However, as a stage scanning device with parallel beam illumination, a scanning stage or specimen container may result in serious beam shading which does reduce the contrast of image, and the scanning speed will also be slower. To reduce the beam shading ratio, the aperture of an incident beam need to be greatly enlarged relative to a given size of scanning stage or specimen container. But in this case it will result in a serious difficulty for the generation of a parallel beam by a large aperture collimator and to keep the wavefront error within a quarter of wavelength. Fortunately, the illumination aberration control in point scanning confocal microscopy will be relatively easy if a point-like source substitutes as the large aperture collimator. The feasibility of fine aspherical mirror fabrication has been proved in the applications of well-developed astronomical systems.

In comparison with a parabolic mirror based system, an elliptical mirror based system is potentially promising in the 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. As far as the difficulty of mirror fabrications, parabolic and elliptical mirrors are almost the same when their apertures are similar in sizes. But in the focusing of a parabolic mirror, a corresponding large aperture parallel beam is needed to fill in the incident pupil. Consequently, how to generate an aberration free incident beam becomes a new challenging task in a parabolic mirror based system since aberrations are always increased greatly when the aperture of collimator is enlarged as a necessary illumination. However, the size of a point-like illuminator in an elliptical mirror based system won’t be changed with the aperture of the focusing mirror, and this kind of point-like illuminator can be easily obtained by a general small aperture objective. Therefore, a focusing formula is established for an elliptical mirror with wide open aperture to highlight the potential application of an elliptical mirror.

2 Apodization factor

In vector theory, apodization factor is of great importance for the discovery of the transformation between inbound and outbound waves on a mirror or lens. For the focusing of a lens or a parabolic mirror, the incident wave is generally assumed to be a plane wave when apodization factor is discussed [10

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]

–14

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. numerical results,” J. Opt. Soc. Am. A 17(11), 2090–2095 (2000). [CrossRef] [PubMed]

]. But on the reflection of an elliptical mirror, it is theoretically different from that of a parabolic mirror because the incident wave is no longer a plane wave. As shown in Fig. 1 , point diffractive spherical wave S_w1 will be transformed into spherical wave S_w2 after the reflection of an elliptical mirror, and then S_w2 focuses at F₂. This propagation is passing by M, an incident point on real mirror surface S_m. |F₁M| + |MF₂| = 2a, the optical path from first focal point F₁ to second focal point F₂ is a well-known value when the mirror function is (x² + y²)/b² + (z-c)²/a² = 1. a = |OV| and b are the long and short axes of an elliptical mirror, respectively. c = |OF₁| = |OF₂| = (a²-b²)^1/2.

Fig. 1 Reflection on an elliptical mirror. S_m is the real surface of an elliptical mirror, S_w1 and S_w2 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.

Apodization factor l(θ) denotes the amplitude ratio of inbound and outbound beams on a mirror at focusing angle θ. For an ideal mirror, the energy loss caused by reflection and absorption is negligible, and so, l²(θ) = A₁²/A₂² = δS_w2/δS_w1 because the energy is kept constant in an infinitesimal area. A₁ and A₂ are the amplitudes of inbound and outbound beams. δS_w1 and δS_w2 are the infinitesimal surface elements on waves S_w1 and S_w2, respectively. From the definition, δS_w1 = 2πf₁²sinαdα and δS_w2 = 2πf₂²sinθdθ, f₁ = a + c and f₂ = a-c are the focal lengths of S_w1 and S_w2. By substituting δS_w1 and δS_w2,

l (θ) = \frac{a + c}{a - c} \sqrt{\frac{\sin α d α}{\sin θ d θ}},

(1)

where,

α = t g^{- 1} [(z - 2 c) t g θ / z]

z = (a^{2} c \cdot t g^{2} θ + a b^{2} \sqrt{1 + t g^{2} θ}) / (a^{2} t g^{2} θ + b^{2}) + c

,z is distance between M and F₂ along optical axis Z. In real mirror reflection, the optical path from F₁ to F₂ is |F₁M| + |MF₂|, which corresponds to a theoretical optical path

| F_{1} W_{1} | + | W_{2} F_{2} | = f_{1} + f_{2} =2a

Apodization factor is an indication of the change in the energy distribution on the exit pupil caused by reflection or transmission. For example, l(θ) will be close to 1 if θ is small enough to zero, which means the conversion of energy from wave S_w1 to S_w2 is approximately a direct projection, and the energy density remains constant before and after reflection. But if l(θ) is greater or less than 1, the incident wave within the infinitesimal area will be compressed or stretched after reflection or transmission, i.e. the energy density will increase or decrease as appropriate. For both parabolic and elliptical mirrors as shown in Fig. 2 , l(θ) increases from 1 toward 2 when θ increases up to 1.5 radian, about π/2. The increase in l(θ) means more energy distributes as high frequency in a mirror system rather than in a system of lenses. l(θ) of an elliptical mirror is slightly sharper than that of a parabolic mirror, and it helps the production of a compressed scanning spot. And even a more sharper curve of l(θ) is obtainable if c is close to a. But an additional fabrication cost cannot be avoided because an elliptical mirror will be much deeper in this case.

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.

In fact, the apodization factor of an elliptical mirror changes not only with θ, but also with a, b and c. This performance obviously differs from that of a parabolic mirror or a lens. A smaller c means focal points F₁ and F₂ are close to each other. The defined elliptical mirror will turn into a sphere-like mirror if c is small enough. As shown in Fig. 3 , for a different value of c, l(θ) shows the tendency of energy conversation caused by mirror reflection. This tendency can also be understood from Eq. (1) when l(θ) is constant 1 and α = θ. Either b or c can be used to describe the deepness of elliptical mirror, but only one of them is independent and necessary to be studied under a given a.

Fig. 3 l(θ) of elliptical mirror under different c (a = 500mm).

3 Vector theories

The two focal points of an elliptical mirror could be separately used for illumination and point scanning. The point-like source at F₁ is supposed to be a tight spot obtained by lens L with plane wave incidence. This consideration is more practical in a real optical system than a dipole emitter at F₁. A real illuminator is generally extended in size, but not a perfect point-like source. However, a dipole emitter is better assumed at F₂ while molecular imaging is concentrated in high resolution microscopy. That is an inverse process of focusing analysis. But this paper only aims at the focusing properties of an elliptical mirror.

To figure out the focusing formula of an elliptical mirror with high aperture focusing, a series of unit vectors are defined and shown in Fig. 4 to represent the electric field during reflection. An elliptical mirror is supposed large enough to match illumination angle α which corresponds to the numerical aperture of lens L. The largest θ is determined by parameters a, b and c, and numerical aperture of elliptical mirror NA, θ_m = arcsin(NA) in air propagation.

Fig. 4 A meridional plane of a ray and vectors definitions. g_L, g₀ and g₁ denote unit vectors perpendicular to the rays drawn in red lines and all in meridional plane F₁MF₂. s_L, s₀ and s₁ are unit vectors along the incident ray of Lens L, incident ray

\vec{F_{1} M}

of elliptical mirror and reflected ray

\vec{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.

\vec{m}

\vec{n}

and

\vec{k}

are unit vectors in the directions of Cartesian rectangular coordinate axis X, Y and Z. And then g₀ and g₁, s₀ and s₁ can be written as

g_{L} = \cos ϕ \vec{m} + \sin ϕ \vec{n},

(2)

\begin{matrix} g_{0} = \sin (α + π / 2) \cos ϕ \vec{m} + \sin (α + π / 2) \sin ϕ \vec{n} + \cos (α + π / 2) \vec{k} \\ = \cos α \cos ϕ \vec{m} + \cos α \sin ϕ \vec{n} - \sin α \vec{k}, \end{matrix}

(3)

\begin{matrix} g_{1} = \sin (π / 2 - θ) \cos (ϕ + π) \vec{m} + \sin (π / 2 - θ) \sin (ϕ + π) \vec{n} + \cos (π / 2 - θ) \vec{k} \\ = - \cos θ \cos ϕ \vec{m} - \cos θ \sin ϕ \vec{n} + \sin θ \vec{k}, \end{matrix}

(4)

s_{L} = \vec{k},

(5)

s_{0} = \sin α \cos ϕ \vec{m} + \sin α \sin ϕ \vec{n} + \cos α \vec{k},

(6)

\begin{matrix} s_{1} = \sin (π - θ) \cos (ϕ + π) \vec{m} + \sin (π - θ) \sin (ϕ + π) \vec{n} + \cos (π - θ) \vec{k} \\ = - \sin θ \cos ϕ \vec{m} - \sin θ \sin ϕ \vec{n} - \cos θ \vec{k} . \end{matrix}

(7)

In optical propagations, an electric field vector is always orthogonal to the ray. When electric field of the incident plane wave of lens L,

\hat{e_{L}^{0}}

is assumed a linear polarized beam, electric field of vector

\hat{e_{L}^{1}}

can be therefore expressed as shown below,

\hat{e_{L}^{1}} = \sqrt{\cos α} [\hat{e_{L - r}^{1}} g_{0} + \hat{e_{L - φ}^{1}} (g_{0} \times s_{0})],

(8)

where

\hat{e_{L - r}^{1}}

and

\hat{e_{L - φ}^{1}}

are radial and azimuthal components in the region of exit pupil of lens L. Their directions are along g₀ and g₀ × s₀.

\sqrt{\cos α}

is the apodization factor introduced by the transformation of a plane wave to a spherical wave through lens. Moreover,

\hat{e_{L - r}^{1}}

and

\hat{e_{L - φ}^{1}}

remains on the same side of the meridional plane [9

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]

,10

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]

]. This is the basic principle used to determine the radial and azimuthal components when a ray is focused by lens or reflected by mirror, and then

\begin{array}{l} \hat{e_{L - r}^{1}} = g_{0} \cdot \hat{e_{L}^{1}} = g_{L} \cdot \hat{e_{L}^{0}} = \cos ϕ, \\ \hat{e_{L - φ}^{1}} = (g_{0} \times s_{0}) \cdot \hat{e_{L}^{1}} = (g_{L} \times s_{L}) \cdot \hat{e_{L}^{0}} = \sin ϕ . \end{array}

(9)

As shown in Fig. 4, the spherical wave on the exit pupil of lens L will pass through focal point F₁, incident point M on an elliptical mirror and finally reach focal point F₂. This is a free space propagation to transform a focusing wave into a divergent spherical wave, and then turn into a focusing wave again. Amplitude factor f(a-c)/(a + c) must be introduced to describe the change of relative amplitude. f is the focal length of lens L, and a + c and a-c are the first and second focal lengths of an elliptical mirror, respectively. After being reflected by the mirror, apodization factor l(θ) is also introduced for the calculations of electric field. In addition, vector g₀ × s₀ will be inversed after the reflection [15

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]

]. It is of significance to show the phase jump in reflections. Through the same analysis on Eq. (9), electric field pointing to F₂,

\hat{e_{m}}

can be listed as shown below.

\hat{e_{m}} = f \sqrt{\frac{\sin 2 α d α}{2 \sin θ d θ}} [\hat{e_{m - r}} g_{1} + \hat{e_{m - φ}} (g_{1} \times s_{1})],

(10)

where,

\begin{array}{l} \hat{e_{m - r}} = g_{1} \cdot \hat{e_{m}} = \hat{e_{L - r}^{1}} = \cos ϕ, \\ \hat{e_{m - φ}^{1}} = (g_{1} \times s_{1}) \cdot \hat{e_{m}} = - \hat{e_{L - φ}^{1}} = - \sin ϕ . \end{array}

(11)

According to Eq. (10) and (11),

\hat{e_{m}}

is independent from parameters a, b and c because of the assumption that the aperture of an elliptical mirror is large enough to accept all the incident rays within a given α. In this case, the electric field of focusing spot only depends on working apertureθ _m. But the relationship between θ and α is indirectly influenced by a, b and c. The formula of vector field amplitude

\hat{e_{F}}

on F₂ could be given by referencing Eq. (2).2) in ref [9

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]

]. And for sake of brevity, aberration is neglected,

\hat{e_{F}} = - \frac{i k}{2 π} \int_{0}^{2 π} \int_{0}^{θ_{m}} \hat{e_{m}} (θ, ϕ) e^{i k [s_{1} \cdot r]} \sin θ d θ d ϕ,

(12)

where,

r = (r_{s x}, r_{s y}, r_{s z}) = (ρ_{s} \cos φ_{s}, ρ_{s} \sin φ_{s}, z_{s})

is a cylindrical coordinate vector in vicinity of F₂, ρ_s and φ_s is the polar radius in the focal plane, z_s is the defocus distance near F₂.

i = \sqrt{- 1}

, k = 2π/λ, λ is the wavelength.

By substituting r and s₁,

s_{1} \cdot r = - [ρ_{s} \sin θ \cos (ϕ - φ_{s}) + z_{s} \cos θ]

. And

\hat{e_{F}}

can also be simplified by using

\begin{array}{l} \int_{0}^{2 π} \sin (n ϕ) e^{i ρ \cos (ϕ - φ_{s})} d ϕ = 2 π i^{n} J_{n} (ρ) \sin (n φ_{s}), \\ \int_{0}^{2 π} \cos (n ϕ) e^{i ρ \cos (ϕ - φ_{s})} d ϕ = 2 π i^{n} J_{n} (ρ) \cos (n φ_{s}), \end{array}

(13)

to yield

\hat{e_{F}}

= [

\hat{e_{F}^{x}}

\hat{e_{F}^{y}}

\hat{e_{F}^{z}}

]^T, where

\begin{array}{l} \hat{e_{F}^{x}} = i A (I_{0} + I_{2} \cos 2 φ_{s}), \\ \hat{e_{F}^{y}} = i A I_{2} \sin 2 φ_{s}, \\ \hat{e_{F}^{z}} = - 2 A I_{1} \cos φ_{s}, \end{array}

(14)

and,

\begin{array}{l} I_{0} = \int_{0}^{θ_{m}} E (θ) \sin θ (1 + \cos θ) J_{0} (\frac{v \sin θ}{N A}) e^{\frac{- i u \cos θ}{N A^{2}}} d θ, \\ I_{1} = \int_{0}^{θ_{m}} E (θ) \sin^{2} θ J_{1} (\frac{v \sin θ}{N A}) e^{\frac{- i u \cos θ}{N A^{2}}} d θ, \\ I_{2} = \int_{0}^{θ_{m}} E (θ) \sin θ (1 - \cos θ) J_{2} (\frac{v \sin θ}{N A}) e^{\frac{- i u \cos θ}{N A^{2}}} d θ, \end{array}

(15)

A = k f / 2 \sqrt{2}

E (θ) = \sqrt{\sin 2 α d α / \sin θ d θ}

v = k N A ρ_{s}

and

u = k N A^{2} z_{s}

. v and u are lateral and longitudinal dimensionless optical coordinates.

Equation (14) is a vectorial expression of amplitude for the focusing spot at focal point F₂ with a point-like source at F₁. Compared with the formulas for parabolic mirror and lens, E(θ) is informative to show the particular characteristics of an elliptical mirror in high aperture focusing since apodization factor l(θ) is involved. Theoretically, Eq. (14) is available for the open aperture near π/2, or even over π/2. But to consider the beam shading of stage in a real application, the open aperture has to be kept under π/2, so that NA will not be over 1 in numerical calculations.

4 Discussions

Focusing characteristics are usually evaluated by the spot size and the amplitude of side lobe after the intensity of electric field is normalized to unit 1, and the spot size could also be detailed by the width of main lobe or full width at half maximum (FWHM). Point spread function based evaluations are well developed in super-resolution engineering as a powerful theory for comparisons on focusing tight spot [16

J. Liu, J. Tan, and C. Zhao, “Convex objective function-based design method developed for minimizing side lobe,” Appl. Opt. 47(22), 4061–4067 (2008). [CrossRef] [PubMed]

–18

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28(8), 607–609 (2003). [CrossRef] [PubMed]

]. It is also adapted to show the performance of an elliptical mirror in high aperture focusing.

As shown in Fig. 5 , the electric field intensities generated by conventional lens, and parabolic and elliptical mirrors are compared both in transverse and longitudinal directions. The formulas for calculations of conventional lens and parabolic mirror are defined in ref [12

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]

–14

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. numerical results,” J. Opt. Soc. Am. A 17(11), 2090–2095 (2000). [CrossRef] [PubMed]

]. To make main and side lobes informative, parts of the curves are sub-plotted in Fig. 5(a), 5(c) and 5(d). As shown in Fig. 5(a) and 5(c), the three curves are almost overlapped each other if θ_m = π/3. But in the sub-figures, the differences can be identified so that the size of main lobe generated by mirrors is smaller in both v and u directions. But the amplitudes of lateral side lobe of mirror focal spots are stronger than that of the lens as shown in the sub-figure of Fig. 5(a). More critical sizes of the focusing spot are listed in Table 1 .

Fig. 5 Electric field intensity

\hat{e_{F}} \cdot {\hat{e_{F}}}^{*}

of elliptical mirror compared with that of lens and parabolic mirror (φ_s = 0, a = 500mm, b = 400mm and c = 300mm. In sub-Figs. 5(a), 5(c) and 5(b), 5(d), θ_m is π/3 and π/2, respectively.)

Table 1 Comparison of Critical Sizes of Focusing Spots

θ_m = π/3	Transverse direction (v)			Longitudinal direction (u)
θ_m = π/3	M_s	G_s	HFWHM	M_s	G_s	HFWHM
Lens	0.01673	4.73	2.00	0.04781	9.59	4.25
Parabolic mirror	0.02598	4.53	1.97	0.04719	9.44	4.17
Elliptical mirror	0.02529	4.54	1.97	0.04697	9.43	4.18
θ_m = π/2
θ_m = π/2	M_s	G_s	HFWHM	M_s	G_s	HFWHM
Lens	0.03098	4.70	2.37	0.04869	7.75	3.25
Parabolic mirror	0.08688	4.27	2.54	0.04719	6.26	2.78
Elliptical mirror	0.10200	4.18	2.55	0.04613	6.31	2.79

In addition, it is interesting to state the difference of focusing performance between parabolic and elliptical mirrors. Since the difference of electric field intensity as shown in Fig. 5 is so slight that it is very difficult to say which mirror is better in focusing. But as indicated in Fig. 3, a sharper l(θ) of elliptical mirror is obtainable by increasing the value of c to distribute more energy as high frequency, which helps the narrowing of a diffractive spot. However, a greatly increased c will result in a serious fabrication cost. Hence compared with a parabolic mirror, the advantage of an elliptical mirror is not a highlighted focusing performance, but an easy aberration control in point-like illuminator.

As shown in Fig. 5, open aperture θ_m increases up to π/2. As for the transverse electric field intensity of elliptical and parabolic mirror focusing spot, the width of main lobes is always smaller than that of lens as shown in Fig. 5(a) and 5(b). But the half of FWHM (HFWHM) of mirrors is only slightly smaller than that of lens in Fig. 5(a). The relationship between HFWHM of mirrors and lens is inversed in Fig. 5(b) when θ_m = π/2, and the mirror focused spots are comparatively wider in the close areas of central peak points, and the side lobes grow up obviously at the same time. The reason why the simulated focusing spots of parabolic and elliptical mirrors are almost overlapped to each other, but obviously differed with that of lens, is due to the similar l(θ) as shown in Fig. 2. This is a typical example of the actual effect of apodization factor. The largest apodization factor of parabolic mirror of 2 can be very easily determined from 2/(1 + cosα). But for elliptical mirror the acceptable limit of apodization factor is restricted by both c and θ, it cannot be directly given using Eq. (1), which merits attention in the future.

In the axial direction, the HFWHM of elliptical and parabolic mirror is about 80% of the spot size of an objective lens in dimensionless optical coordinate. The amplitude of side lobe M_s and the first zero point coordinate of diffractive spot G_s of the mirrors are both kept lower than those of lens as shown in Fig. 5(d) and Table 1. This is valuable for high axial resolution application in scanning microscopy.Additionally, the focusing performance of an elliptical mirror with very low aperture can also be verified through simulations and comparison with scalar result. For example, to give θ_m = π/6, the first zero point coordinate of a diffractive spot given using Eq. (14) is also 3.83 in a dimensionless optical coordinates well-defined as v which is in agreement with the well-known conclusion in scalar theory.

It may be interesting to mention parameter c. Since 2c is the distance between F₁ and F₂, the defined elliptical mirror will change into a sphere-like mirror if c is supposed to be 0. And its influence on energy exchange has also been illustrated in Fig. 3 so that there is no doubt to conclude the intensity of electric field will also be influenced by a given c. Fortunately, it can be analytically answered by Eq. (1) and (14). When c is assumed ‘0’, and then α = θ,

E (θ) = \sqrt{\cos θ}

, which means Eq. (14) exactly equals the conventional lens formula. The focusing character of an elliptical mirror will be intervenient when the value of c increases from 0 to 300mm, or more.

At last, in the high aperture focusing of an elliptical mirror, it is of significance to state the asymmetry of electric field intensity. As shown in Fig. 6 , the electric field intensity in the focal plane on F₂ is not homogeneous when φ_s is altered from 0 to π/2, which corresponds to a different meridional plane. The orthogonal meridional plane at φ_s = 0 and φ_s = π/2 corresponds to maximum or minimum size of main lobe. But this asymmetry is not the peculiar character of an elliptical mirror, but a general character which belongs to lenses and parabolic mirrors when their open aperture is close to π/2.

Fig. 6 Contour map of electric field intensity in the focal plane on F₂ in polar coordinates (φ_s and v are drawn as polar angle and polar radius. u = 0, θ_m = π/2).

5 Conclusions

A rigorous elliptical mirror focusing formula based on spherical wave transformation is derived as a kind of high aperture imaging technique for potential applications in molecule imaging, spectroscopy and industrial artifact microscopy etc.. Simulation results indicate that in mirror reflections more energy distributes as high frequency and the HFWHM of elliptical and parabolic mirror is about 80% of the size of longitudinal spot of objective lens with NA = 1 and φ_s = 0 whilst the noise of longitudinal side lobe is slightly lower than that of lens. This is valuable to obtain a high axial resolution in scanning microscopy. But the transverse HFWHM of mirrors is comparatively wider despite the width of main lobe is still smaller.

The elliptical mirror based focusing technique is highlighted because the absence of chromatic aberration is really promising for wide spectrum imaging and the two aberration free focal points inside of elliptical mirror might be helpful to construct a fast point scanning microscopy by using well developed scanning techniques at F₁. In addition, compared 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. Finally, off-axis performance and homogeneous or inhomogeneous polarized illumination are worthy of analyzing for practical mirror system design and molecules imaging in future.

Acknowledgments

We thank Prof. Konyakhin Igor from National Research University of Information Technologies, Mechanics and Optics (Saint-Petersburg, Russia) for providing the copies of papers by V. S. Ignatovsky, and National Nature Science Foundation of China (Grant No. 50905048) for its fund.

References and links

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]
2.	T. Ruckstuhl and S. Seeger, “Confocal total-internal-reflection fluorescence microscopy with a high-aperture parabolic mirror lens,” Appl. Opt. 42(16), 3277–3283 (2003). [CrossRef] [PubMed]
3.	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]
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]
5.	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]
6.	M. Schwertner, M. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high NA adaptive optical microscopy,” Opt. Express 12(26), 6540–6552 (2004). [CrossRef] [PubMed]
7.	T. D. Wang, M. J. Mandella, C. H. Contag, and G. S. Kino, “Dual-axis confocal microscope for high-resolution in vivo imaging,” Opt. Lett. 28(6), 414–416 (2003). [CrossRef] [PubMed]
8.	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]
9.	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]
10.	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]
11.	V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, 5 (1920).
12.	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]
13.	P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000). [CrossRef] [PubMed]
14.	P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. numerical results,” J. Opt. Soc. Am. A 17(11), 2090–2095 (2000). [CrossRef] [PubMed]
15.	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]
16.	J. Liu, J. Tan, and C. Zhao, “Convex objective function-based design method developed for minimizing side lobe,” Appl. Opt. 47(22), 4061–4067 (2008). [CrossRef] [PubMed]
17.	J. Liu, J. Tan, and Y. Wang, “Synthetic complex superresolving pupil filter based on double-beam phase modulation,” Appl. Opt. 47(21), 3803–3807 (2008). [CrossRef] [PubMed]
18.	D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28(8), 607–609 (2003). [CrossRef] [PubMed]

OCIS Codes
(180.1790) Microscopy : Confocal microscopy
(180.5810) Microscopy : Scanning microscopy
(230.4040) Optical devices : Mirrors

ToC Category:
Microscopy

History
Original Manuscript: January 19, 2012
Revised Manuscript: February 18, 2012
Manuscript Accepted: February 26, 2012
Published: March 1, 2012

Virtual Issues
Vol. 7, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Jian Liu, Jiubin Tan, Tony Wilson, and Cien Zhong, "Rigorous theory on elliptical mirror focusing for point scanning microscopy," Opt. Express 20, 6175-6184 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6175

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References

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,” Nature447(7147), 979–981 (2007). [CrossRef] [PubMed]
T. Ruckstuhl and S. Seeger, “Confocal total-internal-reflection fluorescence microscopy with a high-aperture parabolic mirror lens,” Appl. Opt.42(16), 3277–3283 (2003). [CrossRef] [PubMed]
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. Express9(12), 637–644 (2001). [CrossRef] [PubMed]
M. Schwertner, M. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high NA adaptive optical microscopy,” Opt. Express12(26), 6540–6552 (2004). [CrossRef] [PubMed]
T. D. Wang, M. J. Mandella, C. H. Contag, and G. S. Kino, “Dual-axis confocal microscope for high-resolution in vivo imaging,” Opt. Lett.28(6), 414–416 (2003). [CrossRef] [PubMed]
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]
B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. A253(1274), 358–379 (1959). [CrossRef]
E. Wolf, “Electromagnetic diffraction in optical systems. I. an integral representation of the image field,” Proc. R. Soc. A253(1274), 349–357 (1959). [CrossRef]
V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd1, 5 (1920).
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]
P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. theory,” J. Opt. Soc. Am. A17(11), 2081–2089 (2000). [CrossRef] [PubMed]
P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. numerical results,” J. Opt. Soc. Am. A17(11), 2090–2095 (2000). [CrossRef] [PubMed]
M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express8(7), 458–474 (2001). [CrossRef] [PubMed]
J. Liu, J. Tan, and C. Zhao, “Convex objective function-based design method developed for minimizing side lobe,” Appl. Opt.47(22), 4061–4067 (2008). [CrossRef] [PubMed]
J. Liu, J. Tan, and Y. Wang, “Synthetic complex superresolving pupil filter based on double-beam phase modulation,” Appl. Opt.47(21), 3803–3807 (2008). [CrossRef] [PubMed]
D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett.28(8), 607–609 (2003). [CrossRef] [PubMed]

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