October 1, 2005 / Vol. 30, No. 19 / OPTICS LETTERS
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Focusing astigmatic Gaussian beams through optical systems with a high numerical aperture Ingo Gregor and Jörg Enderlein Institutes for Biological Information Processing 1 & 2, Forschungszentrum Jülich, D-52425 Jülich, Germany Received May 3, 2005; revised manuscript received June 15, 2005; accepted June 15, 2005 We theoretically derive the electric field distribution of an astigmatic Gaussian laser beam after it is focused through a high-aperture objective. We show that astigmatism values that are hard to detect in the collimated laser beam can have a large effect after diffraction-limited focusing. Such astigmatic beams may be frequently encountered in fluorescence correlation measurements and in laser-scanning confocal microscopy. We present experimental measurements of the excitation intensity distribution measured by 3D scanning of single fluorescent molecules immobilized on a glass surface. © 2005 Optical Society of America OCIS codes: 050.1960, 180.1790.
Diffraction-limited focusing through objectives of high numerical aperture (NA) is an important element of many single-molecule fluorescence spectroscopy (SMS) techniques such as fluorescence correlation spectroscopy1–4 as well as in laser-scanning confocal microscopy (LSCM). To achieve minimum focus diameter one needs a perfectly Gaussian laser beam (TEM00 mode). However, such a laser beam can still be astigmatic, i.e., it is described by two principal planes perpendicular to each other, with the beam having different focus positions in these planes. Such astigmatism is easily introduced into a laser beam upon reflection by a mirror (e.g., the dichroic mirror of an epifluorescence microscope) with a slightly cylindrical surface, or due to an elliptical fiber profile. In this Letter, we derive a theoretical expression for the electric field distribution of a Gaussian laser beam focused though a high-NA objective. We compare the theoretical result with scanned fluorescence images of a single molecule in different planes along the optical axis. We start with the electric field amplitude of an astigmatic collimated laser beam: E0 ⬃
xˆ w1w2冑共1 + 12兲共1 + 22兲 −
y2共1 − i2兲 w22共1 + 22兲
册
,
冋
exp −
Eex共r兲 ⬀
冕
max
0
d sin 冑cos
冕
2
d E 0共 , 兲
0
⫻共eˆ pTp cos − eˆ sTs sin 兲exp共ikmsˆ · r兲.
共2兲
Unit vectors eˆ p, eˆ s, and sˆ , as well as angles and are shown in Fig. 1; r is the position where the electric field is calculated, and km is the wave vector amplitude in sample space (i.e., 2nm / , where nm is the sample solution’s refractive index). The Tp,s are the total transmission coefficients for plane p and s waves propagating at an angle with respect to the optical axis after exiting the objective’s front lens. The integration boundary max is defined by the NA of the focusing objective via the relation NA = n sin max, with n being the refractive index of the immersion medium. The relation between the angular variables and in Eq. (2) and the Cartesian coordinates x and y in Eq. (1) is given by tan = y / x and fn sin = 冑x2 + y2 (Abbe’s sine condition), where f is the focal length of the objective. In the following analysis, the Jacobi–Anger expansion [Eq. (3)] will be applied twice:
x2共1 − i1兲 w12共1 + 12兲 共1兲
where we have used Cartesian coordinates 兵x , y , z其 with z along the axis of propagation (optical axis), and x and y lying within the two principal planes of the beam. The electric field polarization is assumed to be along the x direction. The coordinates 1,2 are 2 , related to the z coordinate via 1,2 = 共z − z1,2兲 / w1,2 where the w1,2 and z1,2 are the beam waist diameters and positions in the two principal planes, respectively. In addition, z is the position of the objective’s back aperture, and is the laser wavelength in air. Following Refs. 5–9, the electric field amplitude in the sample after focusing through the objective is proportional to the plane wave representation: 0146-9592/05/192527-3/$15.00
Fig. 1. Geometry of light focusing: One cone of light is shown exiting the objective’s front lens with propagation angle with respect to the optical axis. Each interface refracts the light and the propagation angles , g, and m are related by Snell’s law. The unit vectors of polarization are also shown for one plane p and s wave as well as the related unit vector of propagation sˆ , as used for the calculation of the excitation intensity distribution in the sample solution. © 2005 Optical Society of America
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OPTICS LETTERS / Vol. 30, No. 19 / October 1, 2005 ⬁
e
ix cos
=
兺 iqJq共x兲eiq , q=−⬁
共3兲
The coefficients in Eq. (10) are given by 1 H0 =
where the Jq represent Bessel functions of the first kind of order q.10 This allows us to expand the exponent in Eq. (1) into the series
冋
x2共1 − i1兲
exp −
⫻
w12共1 + 12兲
冋
y2共1 − i2兲
−
w22共1 + 22兲
共1 − i1兲cos2 w12共1 + 12兲
冉
= exp −
f2n2 sin2 22
+
册 再 册冎
冊兺
2
=
⍀=
w12共1
+
=
w22共1 + 22兲
12兲
f2n2 sin2 2
w12共1 + 12兲
冋
2 + i
共4兲
w22共1 + 22兲
max
d sin 冑cos J2q共km sin m兲
0
冕
max
共11b兲
d sin 冑cos J2q−1共km sin m兲
⫻h1关iJq共⍀兲 − Jq−1共⍀兲兴, 共s兲 H2q
共5兲
,
−
冕
共11a兲
0
i−qJq共⍀兲ei2q ,
1 − i2
+
2
共c兲 = iq H2q−1
q=−⬁
1 − i1
d sin 冑cos J0共km sin m兲
0
⫻兵ih2关Jq−1共⍀兲 − Jq+1共⍀兲兴 + 2h0Jq共⍀兲其,
where the abbreviations 1
iq
共1 − i2兲sin2 ⬁
max
⫻关h0J0共⍀兲 − ih2J1共⍀兲兴, 共c兲 H2q
= exp − f2n2 sin2
2
冕
1 + i w12共1 + 12兲
册
iq+1 =
2
冕
max
共11c兲
d sin 冑cos J2q共km sin m兲
0
⫻h2关Jq−1共⍀兲 + Jq+1共⍀兲兴,
共11d兲
using the abbreviations 共6兲
were introduced. Furthermore, the product sˆ · r in the exponent of Eq. (2) is approximated by sˆ · r = − sin m cos共 − 兲 + z cos m ,
共7兲
冉
h0 = 共Tp cos m + Ts兲exp −
冉
h1 = Ts sin m exp −
f2n2 sin2 22
f2n2 sin2 22
冉
冊
+ ikmz cos m ,
冊
+ ikmz cos m ,
f2n2 sin2
冊
where 共 , , z兲 are the cylindrical coordinates of vector r. Furthermore, taking into account the explicit expression
h2 = 共Tp cos m − Ts兲exp −
epTp cos − esTs sin
The derived representation of the electric field in the focus is versatile and takes care of several optical effects. The effect of cover slide thickness variation is accounted for by the transmission coefficients Tp,s, containing the phase factor exp共i␦kg cos g兲, where ␦ is the glass thickness variation and kg and g are the wave vector and the propagation angle of the plane wave components in glass, respectively. It should be noted that all propagation angles , g, and m are related by Snell’s law of refraction. Possible mismatch of the sample solution’s refractive index is taken into account by using the sample solution’s wave vector km in the exponent in Eq. (2). The resulting formula is easily generalized for alternative polarizations. A y-polarized beam will generate a field distribution that can be obtained from Eq. (10) by replacing Ex with −Ey, Ey with Ex, and with − / 2. More complex excitation polarizations can be represented as a linear superposition of x- and y-polarized beams. The experimental setup was described in detail in Ref. 11. The linearly polarized laser light of a diode laser with 635 nm wavelength (PDL 800, PicoQuant) is sent through a polarization-maintaining singlemode glass fiber and subsequently collimated to form a beam with a Gaussian beam profile of 3.5 mm beam waist radius (1 / e2 maximum intensity). Vertical mo-
1 =
2
冦
Tp cos m + Ts + 共Tp cos m − Ts兲cos 2 共Tp cos m − Ts兲sin 2 2Tp sin m cos
冧
共8兲
and again using the Jacobi–Anger expansion, exp关− ikm sin m cos共 − 兲兴 ⬁
=
i−qJq共km sin m兲exp关iq共 − 兲兴, 兺 q=−⬁
共9兲
the integration over can be carried out explicitly, yielding the final result:
Eex共, ,z兲 =
冦
⬁
H0 +
兺 H2q共c兲 cos 2q
q=1 ⬁
兺 H2q共s兲 sin 2q q=1 ⬁
共c兲 H2q−1 cos共2q − 1兲 兺 q=1
冧
.
共10兲
22
+ ikmz cos m . 共12兲
October 1, 2005 / Vol. 30, No. 19 / OPTICS LETTERS
Fig. 2. (Online color) Calculated and measured intensity distributions 共⬃E2x 兲 in three different planes along the optical axis (position values shown to the left of the panels). The experimentally observed distribution (rightmost panels) is best reproduced by distributions calculated for an astigmatism value of ⍀max = 0.3⫻ 2.
tion of the focusing objective (water immersion, 1.2 NA, 60⫻, Olympus) is achieved with submicrometer precision using a piezoactuator (PIFOC P-721-20, Physik Instrumente). Lateral scanning of the sample is accomplished using a piezo table (PI P-527.2CL, Physik Instrumente). The scanning step size was 40 nm, defining the edge size of one pixel in the final intensity distribution images. Excitation power was adjusted to 1 W with a gray filter (EAM, Laser2000). Low density distributions of immobilized fluorescing molecules were prepared by spin coating a diluted solution of the dye Atto655 onto a glass cover slide and drying the sample. The thickness of the cover slide was measured by a Dial Thickness Gauge (FZ1101/30, Käfer), and the objective was adjusted to that thickness. We scanned single-molecule fluorescence in three planes along the optical axis: at −250, 0, and +250 nm with respect to the plane showing circular focus. We calculated intensity distributions for the same optical parameters as those used in the experiment for varying values of beam astigmatism. Due to the limited signal-to-noise ratio of our single-molecule measurement, only molecules with their excitation dipole axis nearly aligned with the polarization of the excitation light could be observed easily. Consequently, the measured intensity
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distribution can be adequately described by the calculated distribution of the x component of the electric field. As a measure of the astigmatism we employed the maximum value of ⍀ in Eq. (6). We assume that the laser beam incident onto the objective is well collimated and has a circular beam profile at the objective’s back entry. This is modeled by setting w1 = w2 ⬅ w0 and 2 = −1 ⬅ in Eq. (6), leading to a circular beam profile at the position of the objective’s back aperture 共z = 0兲 with beam diameter w = w0冑1 + 2. Thus, we find the relation ⍀max = f2NA2 / w2 between ⍀max and the unknown . A comparison between calculated and measured intensity distributions is shown in Fig. 2. In the calculations, we used the expansions of Eq. (10) up to the order q = 3. Adding terms with higher values of q did not lead to any visible changes in the calculated patterns. It should be emphasized that, with the exception of the astigmatism parameter ⍀max, there was no other adjustable parameter in the calculations. There is good agreement between measured and calculated patterns for ⍀max (values near 0.3⫻ 2). It should be noted that we used a standard epifluorescence setup and care was taken to ensure that the dichroic mirror was not bent when mounting it in its holder. We observed similar but different astigmatism values when using different monomode fibers, both polarization conserving and nonpolarization conserving. Due to the fact that optical fibers are commonly used for mode cleaning and for conducting excitation light in confocal microscopes, laser beam astigmatism may be of wider concern than previously realized. We are much obliged to Benjamin Kaupp for his generous support of our work. We thank Thomas Ruckstuhl for many helpful discussions and Conor Burke for his linguistic help. Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. J. Enderlein’s e-mail address is
[email protected] (www.joerg-enderlein.de). References 1. N. L. Thompson, in Topics in Fluorescence Spectroscopy 1, J. R. Lakowicz, ed. (Plenum, 1991), pp. 337–378. 2. J. Widengren and Ü. Mets, in Single-Molecule Detection in Solution—Methods and Applications, C. Zander, J. Enderlein, and R. A. Keller, eds. (WileyVCH, 2002), pp. 69–95. 3. R. Rigler and E. Elson, eds., Fluorescence Correlation Spectroscopy (Springer, 2001). 4. J. Enderlein, I. Gregor, D. Patra, and J. Fitter, Curr. Pharm. Biotechnol. 5, 155 (2004). 5. P. Török, Z. Varga, G. R. Laczik, and J. Booker, J. Opt. Soc. Am. A 12, 325 (1995). 6. P. Török and P. Varga, Appl. Opt. 36, 2305 (1997). 7. A. Egner, M. Schrader, and S. W. Hell, Opt. Commun. 153, 211 (1998). 8. O. Haeberlé, Opt. Commun. 235, 1 (2004). 9. O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, Opt. Express 11, 2964 (2003). 10. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Harry Deutsch, 1984). 11. M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, Rev. Sci. Instrum. 72, 4145 (2001).
