522 J. Opt. Soc. Am. A/Vol. 7, No. 3/March 1990

Three-dimensional transfer-function analysis of thetomographic capability of a confocal fluorescence

microscope

Osamu Nakamura

National Research Laboratory of Metrology, 1-1-4 Umezono, Tsukuba, Ibaraki 305, Japan

Satoshi Kawata

Department of Applied Physics, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565, Japan

Received July 26, 1989; accepted November 6, 1989

The three-dimensional optical transfer function of a confocal fluorescence microscope is derived. It has no missingcone and provides tomographic images of the sample. The derivation is based on the dispersion equation ofspherical waves and the diffraction limitation by an objective lens. Experimental results are shown to verify thederivation.

INTRODUCTION

Microscope tomography has been studied intensively duringthe past few years.'- 5 Agard and Sedat developed an opticalsectioning method that reconstructs the three-dimensional(3-D) structure of biological cells from focus series imagesobtained by a conventional fluorescence microscope.' Thebasic idea is a numerical inversion of the 3-D optical transferfunction (OTF) of conventional microscope imaging. How-ever, the 3-D OTF of a conventional microscope is angularlyband limited6-8 ; the 3-D spatial-frequency components out-side the angular band of the sample object cannot be recon-structed. In particular, the longitudinal resolution in a re-constructed 3-D structure is poor. This missing-cone prob-lem in a 3-D OTF is common to conventional fluorescence, 8

transmission, 3 phase-contrast, and confocal transmissionmicroscopes.9

For complete reconstruction of a 3-D structure, a digitalsuperresolution technique may be used. However, such atechnique is disadvantageous in terms of computation com-plexity and noise sensitivity. Therefore an optical tech-nique for recovering a missing cone is strongly recommend-ed.

It has been recognized that laser-scan confocal micro-scopes have the capability of depth discrimination.' 0 -'3 Thedepth-discrimination property is especially useful in surfaceprofile measurement by a confocal reflection microscope.The depth response of a confocal reflection microscope wasexamined both theoretically'4 and experimentally.' 5

Another application of the depth-discrimination propertyis the tomographic observation of thick fluorescent speci-mens. Wilson showed the sectioning property of the confo-cal fluorescence microscope by calculating the two-dimen-sional OTF's in several defocused planes.' 6 Since imagingin the observation of the internal structure of a thick sampleis 3-D1, the 3-D OTF should be used to appreciate the tomo-graphic capability of the confocal fluorescence system.

However, as far as the authors know, the 3-D OTF of aconfocal fluorescence microscope was not precisely derivedbefore now. In this paper we derive the 3-D OTF of aconfocal fluorescence microscope based on a weak-scatter-ing approximation in the specimen and on a diffraction limitdue to the objective-lens's numerical aperture. The derivedOTF of a confocal fluorescence microscope shows no missingcone and promises a tomographic-imaging capability thatdoes not require a complex digital method. Experimentalresults verify the derived theory.

THREE-DIMENSIONAL OPTICAL TRANSFERFUNCTION OF A CONFOCAL FLUORESCENCEMICROSCOPE

In this section we briefly describe the imaging principle ofthe confocal fluorescence microscope. Figure 1 is a sche-matic diagram of a confocal fluorescence microscope. Alight beam from a point source (usually a laser) is focused bythe objective lens onto a point of a fluorescent specimen andexcites the energy levels of a certain local position of thespecimen. Then the sample emits fluorescent light, gettingback to the ground energy state. This fluorescence is im-aged by the objective lens onto the detector plane, passingthrough a dichroic mirror. A pinhole is set in front of thedetector, and a single detector collects the fluorescent inten-sity. The specimen is mechanically scanned in three-di-mensions to form a 3-D image (or the excitation light beam isscanned in two dimensions by galvanomirrors, and the speci-men is mechanically scanned in one dimension along theoptical axis). The position of the pinhole corresponds to avirtual image point of the point source.

In the following derivation, we neglect multiscatteringinside the sample. In other words, the fluorescent materialin the specimen is sufficiently weak. The fluorescence in-tensity i(r,, z) of the object o(r, ), detected by the detectorthrough the pinhole, is given by

0740-3232/90/030522-05$02.00 © 1990 Optical Society of America

0. Nakamura and S. Kawata

Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. A 523

> Detector-~; - ' Till-'-k

Elh(:)icmirror

Ojlecuvclolls

3-Dobject

7,'-'U

SCeamring

Fig. 1. Schematic diagram of a confocal fluorescence microscope.

i(r8 , z8) = J o(r - r, z - z,)ex(r, z)em(r, z)drdz, (1)

where (r, z) represents the Cartesian coordinates at theobject space; z is the optical axis, and a vector r = (x, y) isperpendicular to z; (r8, z represents the current objectspace relative to the original object location shifted by eitheran object scan or a beam scan; ex(r, z) is a 3-D distribution ofthe laser-beam spot in the object space or a 3-D point-spreadfunction of the excitation system, and em(r, z) representsthe 3-D point-spread function of the emission system or thelight intensity at pinhole position when a point object at (r,z) of the object space emits fluorescence of unit intensity.In other words, em(r, z) equals the 3-D point-detectorspread function in object space.

From Eq. (1) the total 3-D point-spread function h(r, z) isfound as

h(r, z) = ex(r, z)em(r, z). (2)

The 3-D OTF H(p, ) of a confocal fluorescence system ishence given by

r

H(p, ) = EX(p, ) * EM(p, P)

= J J EX(p', ')EM(p - p', r - Odp'dR, (3)

where p and are the transaxial two-dimensional spatialfrequency and longitudinal frequency, respectively, andEX(p, 0) and EM(p, D) are 3-D OTF's of the excitation and'emission systems, respectively. * denotes 3-D convolution.Equation (3) indicates that the 3-D OTF of a confocal fluo-rescence microscope is given by a convolution of two OTF'sof the excitation and emission systems.

Let us discuss 3-D imaging and diffraction in a microscopeby using wave-vector analysis. The excitation light from apoint source can be regarded as a spherical wave. In thesource space, a spherical wave has wave vectors in all direc-tions. However, only a part of the wave vectors can passthrough the objective lens, owing to the finite size of thepupil, and form an image of the point source in the objectspace by the lens. Thus the distribution of wave vectors inobject space is given by

U(p, A) = P(P)6[ - (Xe§ 2 - P2)1/2], (4)

where P(p) is a pupil function of the objective lens and Xex isthe wavelength of the excitation light. The first term on theright-hand of Eq. (4) denotes the diffraction limit by thelens, and the second term denotes wave vectors of a sphericalwave. Figure 2 is an illustration of Eq. (4), where the boldarc shows the allowed range for u(p, I). (Actually it is a shellin 3-D space rather than an arc.) In the figure, a rotationallysymmetric pupil of radius pp is assumed.

Generally, the distribution of wave vectors of a light beamis a Fourier transform of a complex-amplitude point-spreadfunction of the beam. The 3-D OTF or the Fourier trans-form of intensity point-spread function is, hence, given byautocorrelation of the distribution of wave vectors in theFourier domain, because the transformation from amplitudeto intensity in object space corresponds to autocorrelation inFourier space. Thus the 3-D OTF of excitation system,EX(p, ), is given by

Fig. 2. Distribution of wave vectors of an excitation light in aconfocal fluorescence microscope.

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524 J. Opt. Soc. Am. A/Vol. 7, No. 3/March 1990

where * denotes 3-D autocorrelation. Figure 3(a) showsEX(p, A), and Fig. 3(b) shows the nonzero region of EX.EX(p, A) is rotationally symmetric around the taxis. In Fig.3(b) the unshaded region is an angularly band-limited miss-ing cone. This resultant missing cone is common to conven-tional incoherent microscopes. 3

The 3-D OTF of the emission system is derived similarlyto the derivation for the 3-D OTF of excitation system. Thesame objective lens as for the excitation is used for fluores-cence imaging. The only difference in derivation is thewavelength of the light; the wavelength of fluorescent light islonger than that of excitation light.

Substituting Eqs. (4) and (5) for excitation as well asemission into Eq. (3), we obtain the total 3-D OTF:

(a)

mrssing _co-e

p =(1/r)P

= (1/z)

(a)=4 P ,

B ((p)

(b)

Fig. 3. (a) 3-D OTF of the excitation system of a confocal fluores-cence microscope. (b) Nonzero region of (a). The nonzero regionof 3-D OTF of conventional incoherent microscopes has the sameshapes as in (b).

EX(p, A) = U(p, A) * U(p, A)

JJ U(p-, r )U*(p + p", + r")dpdr"

= JP(P")bI.-"' - (ex2 -P2)1]

X P ) ( + ')bfr+ - -- (p + p") 2 J1/2j(b)

Fig. 4. (a) 3-D OTF of a confocal fluorescence microscope with488-nm excitation and 520-nm emission and (b) nonzero region of(a).

p

0. Nakamura and S. Kawata

(5)X d"dr",

Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. A 525

tP

(a) (b)

Fig. 5. 3-D spatial-frequency spectra of cultured cells of mousemyeloma observed (a) with a LSFM without a pinhole in front of thedetector plane and (b) with a LSFM with a pinhole.

(a)

5pm

Fig. 7. Focus series image of the same as in Fig. 6 observed by aLSFM with a pinhole. The distance between the two images is 1gum.

(b)

(c)

5pm

Fig. 6. Focus series image of Chironomus stained with propidiumiodide observed by a LSFM without a pinhole in front of the detec-tor. The distance between the two images is 1 gm.

H(p, A) = [Uex(p, ) * U e(P)

* [Uxem(P, A) * UXem(P, 1)]

J fJ f P(P")b[" - (Xe. 2_ p,/

2)1/

2]p*(pI + p)

X aW + " - Xej-2 - (p' + p") 2 ]1/2 Idp"dr"

X f J P(P')6W' - 1Xem 2 _ (P-)211/2}

X P*(P - + )W{-.r + t_ [hem -- (p - pi + p"') 2]l/2 jdp"'d~"'p"'dp'd.", (6)

where Xex and em are the wavelengths of excitation andfluorescence, respectively.

Figure 4(a) shows H(p, A) calculated by Eq. (6). Figure4(b) shows the nonzero region of Fig. 4(a), which is consis-tent with the derivation by Sheppard.' 7 We set Xem and Sexequal to 488 and 520 nm, respectively. In Figs. 4(a) and 4(b)there are no missing cones for the confocal fluorescencemicroscope, which implies a possibility of tomographic im-aging.

(a)

(b)

(c)

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526 J. Opt. Soc. Am. A/Vol. 7, No. 3/March 1990

The 3-D spatial-frequency spectrum observed by the con-focal fluorescence microscope is given by the true 3-D spec-trum of the specimen filtered by the 3-D OTF of the system.The 3-D structure of the specimen can be reconstructed bymultiplying the observation by the inverse of the 3-D OTF.

COMPARISON OF THE OPTICAL TRANSFERFUNCTIONS OF A CONVENTIONALFLUORESCENCE MICROSCOPE AND ACONFOCAL FLUORESCENCE MICROSCOPE

The nonzero region of the 3-D OTF of conventional incoher-ent (transmissions or fluorescence 8 ) microscopes has thesame shape as that of the excitation system (or emissionsystem) shown in Fig. 3(b). Compared with that in Fig. 3(b),the longitudinal bandwidth Br(p) of the confocal fluores-cence microscope in Fig. 4(b) is at least (1 + Xem/ex) timeswider than that of the conventional fluorescence microscopeat any lateral frequency p when Xem equals Nex. The differ-ence in the amount of Br(p) between the two microscopesbecomes evident as p approaches 0. At p = 0, Br(p) = 0 for aconventional fluorescence microscope, or it can pass only thedc component of longitudinal frequency, while the confocalfluorescence microscope passes longitudinal frequency com-ponents within a finite band extent for Br(O). This meansthat a conventional fluorescence microscope cannot imagean object with a p = 0 structure [uniformly distributed alongthe r axis (transaxial plane), such as a film, a board, or amultilayer], but a confocal microscope can.

The transaxial bandwidth BP of the confocal fluorescencemicroscope is twice as wide as that of a conventional micro-scope, which is consistent with the result of 2-D OTF the-ory.' 8

EXPERIMENTS

We carried out experiments to verify the developed theory.We observed 32 focus-series images of cultured cells ofmouse myeloma. Each image comprised 512 X 480 pixels,and the image-to-image distance was 1.0 Am. Figures 5(a)and 5(b) are the 3-D Fourier spectra of an image data setobserved by a laser-scan fluorescence microscope (LSFM)without a pinhole in front of the detector (type I system) andwith a pinhole (type II, confocal system), respectively. TheType I fluorescence microscope has the same shape in thenonzero region of the 3-D OTF as that of a conventionalnonscanning fluorescence microscope with uniform illumi-nation. In Fig. 5(a) the missing cone in the 3-D OTF is seennear the taxis, while there is no missing cone in Fig. 5(b); thetransaxial bandwidth BP of Fig. 5(b) along the p axis is widerthan that in Fig. 5(a).

Figure 6 shows a focus series image of giant chromosomesof Chironomus stained by propidium iodide observed by theLSFM without the pinhole; Fig. 7 shows the same samples asFig. 6 imaged by the same LSFM but with the pinhole.Figures 6(a), 6(b), and 6(c) correspond to the same focusposition images as Figs. 7(a), 7(b), and 7(c), respectively. InFig. 7, only sections of this thick sample are imaged, and out-of-focus images disappear. The stripe structure inside thechromosomes can be seen clearly in Fig. 7(a) but not in Fig.6(a).

SUMMARY AND DISCUSSION

We have derived a 3-D OTF of the confocal fluorescencemicroscope, using wave-vector analysis of the excitation andfluorescent lights and a diffraction-limit theory. The theo-retical result was verified by the practical experiments, andthe tomographic capability of the confocal fluorescence mi-croscope was proved.

The theoretical resolution of a confocal fluorescence mi-croscope according to our theory is 0 .3 1 5 Am in the longitudi-nal axis and 0.158 m in the transaxis for a N.A. = 0.8objective lens, 488-nm excitation, and 520-nm fluorescencedetection. In practical cases, the finite sizes of the pinholeand the source may degrade resolution.

ACKNOWLEDGMENTS

This research was performed when both authors were withthe Department of Applied Physics, Osaka University, Osa-ka, Japan. The authors thank S. Minami of Osaka Universi-ty for his valuable advice and guidance. This also acknowl-edge the cooperation of S. Fujita and T. Takamatsu of KyotoPrefectural University of Medicine in the experiments. Theexperiments were performed at the Kyoto Prefectural Uni-versity of Medicine.

REFERENCES1. D. A. Agard and J. W. Sedat, "Three-dimensional architecture

of a polytene nucleus," Nature (London) 302, 676-681 (1983).2. A. Erhardt, G. Zinser, D. Komitowski, and J. Bille, "Recon-

structing 3-D light-microscopic images by digital image process-ing," Appl. Opt. 24, 194-200 (1985).

3. N. Streibl, "Three-dimensional imaging by a microscope," J.Opt. Soc. Am. A 2, 121-127 (1985).

4. S. Kawata, 0. Nakamura, and S. Minami, "Optical microscopetomography. I. Support constraint," J. Opt. Soc. Am. A 4,292-297 (1987).

5. 0. Nakamura, S. Kawata, and S. Minami, "Optical microscopetomography. II. Nonnegative constraint by a gradient-projec-tion method," J. Opt. Soc. Am. A 5, 554-561 (1988).

6. L. Mertz, Transformations in Optics (Wiley, New York, 1965),p. 101.

7. B. R. Frieden, "Optical transfer of the three-dimensional ob-ject," J. Opt. Soc. Am. 57, 56-66 (1967).

8. N. Streibl, "Fundamental restrictions for 3-D light distribu-tions," Optik 66, 341-354 (1984).

9. 0. Nakamura, S. Kawata, and S. Minami, "Three-dimensionalimaging characteristics of confocal laser-scanning microscope,"Oyo Buturi 57, 784-791 (1988).

10. T. Wilson and C. Sheppard, Theory and Practice of ScanningOptical Microscopy (Academic, London, 1984), Chap. 5.

11. T. Wilson and L. Balk, eds, Scanning Imaging Technology,Proc. Soc. Photo-Opt. Instrum. Eng. 809 (1987).

12. H. T. M. van der Voort, G. J. Brakenhoff, and G. C. A. M.Jansson, "Determination of the 3-dimensional optical proper-ties of a confocal scanning laser microscope," Optik 78, 48-53(1988).

13. T. Takamatsu and S. Fujita, "Microscopic tomography by laserscanning microscopy and its three-dimensional reconstruc-tion," J. Microsc. 149, 167-174 (1988).

14. T. Wilson and A. R. Carlini, "Depth discrimination criteria inconfocal optical system," Optik 76, 164-166 (1987).

15. T. R. Corle, C. H.Chou, and G. S. Kino, "Depth response ofconfocal optical microscopes," Opt. Lett. 11, 770-772 (1986).

16. T. Wilson, "Optical sectioning in confocal fluorescent micro-scopes," J. Microsc. 154, 143-156 (1989).

17. C. J. R. Sheppard, "The spatial frequency cut-off in three-dimensional imaging II," Optik 74, 128-129 (1986).

18. I. J. Cox, C. J. R. Sheppard, and T. Wilson, "Super-resolution byconfocal fluorescent microscopy," Optik 60, 391-396 (1982).

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