Light scattering on ultrathin capillaries

15 February 1999

Optics Communications 160 Ž1999. 201–206

Light scattering on ultrathin capillaries Jorg ¨ Enderlein

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Institute of Analytical Chemistry, Chemo- and Biosensors, UniÕersity of Regensburg, PF 10 10 42, D-93040 Regensburg, Germany Received 5 November 1998; revised 21 December 1998; accepted 22 December 1998

Abstract In recent experiments, very low amplitudes of light scattering were reported for a tightly focused laser beam on a glass microcapillary. This result may be of great importance for single molecule detection in liquids. In the present paper, a theoretical study of the problem of light scattering of a focused laser beam on a cylindrical glass capillary is given. The numerical calculations are based on an exact electrodynamic treatment of the whole problem, including vector effects of the electromagnetic field of the laser beam. It is shown that under favorite conditions Žinner and outer radius of the capillary, perfect adjustment of the laser beam. the intensity of the back-scattered light may be extremely low. q 1999 Elsevier Science B.V. All rights reserved. PACS: 87.64.N; 42.25.F Keywords: Single molecule detection; Laser light scattering

1. Introduction In recent years, the detection of single molecules in liquid solutions by laser induced fluorescence has seen a tremendous progress. For comprehensive reviews of this quickly expanding research field see Refs. w1–3x. For achieving a detection sensitivity sufficient to see individual fluorescing molecules in solution, one has to solve two problems. First, one has to obtain maximum light collection and detection using the experimental setup, and second, one has to minimize any kind of background source that could interfere with the molecule’s fluorescence signal. Improving the light collection efficiency is usually achieved by using microscope objectives with high numerical apertures, and single-photon sensitive opto-electric detectors with high quantum yields Že.g., single-photon avalanche diodes.. The background signal in single molecule detection experiments stems mostly from Rayleigh and Raman scattering of the fluorescence exciting laser beam. Minimization of this scatter background is usually done by using

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E-mail: [email protected]

highly efficient optical filters for the light detection and by minimizing the efficient detection volume via tight focusing of the laser beam and inserting apertures into the light detection channel. In addition to the small detection volume and optical filters, other methods of background rejection have been applied. One method is the application of pulsed laser excitation together with a time gate in the detection channel that is used to reject prompt scatter, see, e.g., Refs. w4,5x. In Ref. w6x, the use of a highly efficient narrow band metal vapor filter for blocking the laser light and its applicability in SMD were investigated. Another approach is the exploitation of two-photon excitation w7– 11x, which was found to be useful for the reduction of background. Soper et al. w12x are promoting the application of near-infrared dyes, since there is a strong decrease of light scattering intensity and fluorescence due to impurities at longer wavelengths. Besides the capability of being able to detect single molecules, another important aspect in many applications is the ability to direct all molecules of a sample solution through the detection volume. This is a necessary prerequisite for single molecule detection applications such as single molecule DNA sequencing, see, e.g., Ref. w13x. A straightforward way to achieve this would be the use of a

0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 6 8 5 - 3

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J. Enderleinr Optics Communications 160 (1999) 201–206

confining capillary directing the sample fluid through the optical detection region. However, it is well known that the presence of liquid–glass interfaces greatly enhances the strength of the scattered light signal, making it nearly impossible to extract any single molecule’s fluorescence from the measured data. The most promising approach, up to now, for directing single molecules through the optical detection region without confining the flow by glass walls, is the so-called hydrodynamic focusing, where the sample solution is injected through a capillary into a surrounding sheath flow which hydrodynamically focuses the sample solution around the center axis of the flow. Also, the use of highly viscous solutes for depressing the molecule’s diffusion has been discussed w14x. In all the approaches cited above, the detection volume was always situated inside the pure dye solution, avoiding the existence of any glass–liquid interface within the detection volume. In a recent paper, Zander et al. w15x reported single molecule detection within a conventional microcapillary ŽEppendorf. with extremely thin glass walls Ža few hundred nanometer.. Although the complete crosssection of the capillary was illuminated by the laser beam, and the detection region was larger than the capillary’s diameter, the measured signal strength of back-scattered light ŽRayleigh and Raman. was unusually low, which made the detection Žand identification. of single molecules readily possible. Besides other serious problems such as unspecific adsorption of the sample molecules at the glass walls, the prospect of using commercially available microcapillaries for confining the sample molecules to the detection region seems interesting and important enough to study in more detail the scattering behavior for such microcapillaries in a focused laser beam. In this paper, a theoretical study is presented of the scattering of a focused laser beam on a glass capillary. The calculations are done for the ideal case of a cylindrical capillary. The geometry of the studied setup is depicted in Fig. 1: A hollow glass capillary with its axis along the z-direction is illuminated by a laser beam propagating along the y-direction. In the experiments of Zander et al., the fluorescence was detected in an epifluorescence setup, along the negative y-direction. Section 2 is devoted to the theoretical approach used in calculating the scattering intensities. In Section 3, numerical results are presented and discussed for the back-scattering intensities of a tightly focused laser beam versus the inner and outer radii of the microcapillary.

ing of a single plane wave on a hollow cylinder. The final result for the laser beam scattering is a superposition of these single plane wave scatterings. First, we assume for all electromagnetic field variables a time dependence of the kind expŽyi v t ., where v is the angular frequency of the electromagnetic field. For the Gaussian laser beam, we adopt the following vector plane wave representation of the electric field amplitude: EA

Hk qk Fk d k d k 2 x

2 z

qik z z y

x

2

w2 4

ˆ

z e 5 ,H

Ž k 2x qz2 .

(

exp ik x x q i k 2 y k 2x y k z2 y

,

Ž1.

(

where k x , k y s k 2 y k 2x y k z2 , k z are the components of the wave vector k, x, y, z are Cartesian coordinates Ž y being the main direction of the beam propagation., w is the beam waist parameter and eˆ 5,H is the unit polarization vector. This unit vector eˆ 5,H is itself a function of the wave vector k. With respect to the capillary geometry and the beam propagation, there are two physically distinct situations: in the first case Ždenoted with the symbol 5., one has eˆ 5 H eˆ x and eˆ 5 H k Žsituation shown in Fig. 1., and in the second case Ždenoted with the symbol H., one has eˆ HH eˆz and eˆ H H k. To study the scattering of a single plane wave on the capillary, one uses cylindrical vector functions, which can be conveniently defined in the following way. The starting

2. Theoretical background To study the electromagnetic scattering of a laser beam on a hollow cylinder, we will divide the problem into two parts. We describe the laser beam as a superposition of an infinite number of plane waves and study then the scatter-

Fig. 1. Geometry of the laser beam scattering on a capillary. The hyperboloid of revolution indicates a surface of constant intensity of an unperturbed incident laser beam with propagation direction k and main polarization E Ž5-polarization, see text.. Also shown are the geometrical meaning of the cylindrical coordinates Ž r , f , z . as used in the present paper.

J. Enderleinr Optics Communications 160 (1999) 201–206

point is the orthogonal solutions cq, n of the scalar Helmholtz equation: D c q k 2c s 0

Ž2.

203

Analogously, the plane waves in the plane wave decomposition of the H -polarized laser beam, eˆ HH eˆz and eˆ H H k, can be written as: `

in cylindrical coordinates Ž r , f , z .:

cq,Z n

E q, x A

Ž r , f , z . s Zn Ž q r . exp Ž in f . exp Ž iwz . ,

Ý

Ž3. B q, x A

Ý

(

Ž

Nq,Z n s rot

.

M qZ, n s

in

r

iwqZXn Zn

eˆr y

qZXn Zn

eˆf cqZ, n ,

Ž4.

eˆr y

r

2

e z cqZ, n .

eˆf q q ˆ

Ž5.

Let us now consider the scattering of a plane wave of type as in Eqs. Ž9a. and Ž9b. on the capillary. Within the capillary glass walls, the electric and magnetic field amplitudes can be assumed to have the following form: E qglX , x s

i ncq,J n Ž r , f y x , z . ,

Ž6.

ns y`

where the superscript J indicates that one has now to use Bessel functions of the first kind w17x. Applying the differential operator rot once and then twice to Eq. Ž6. yields: rot Ž Peˆz . s yiqP cos Ž f y x . eˆf q sin Ž f y x . eˆr , Ž 7 .

qwsin Ž f y x . eˆf q qeˆz .

ns y`

`

B qglX , x s

q´gl k 2 Cn4 M qYX , n ,

Ž 11b.

with qX s ´gl k 2r´out y q 2 , and ´gl Ž ´out . denoting the dielectric constant of the capillary’s glass Žcapillary’s surrounding.. The superscript Y refers to Bessel functions of the second kind, the Cn1 – 4 are unknown coefficients to be determined. Within the inner hollow cylinder of the capillary, the field amplitudes can be written as: `

E qinY , x s

Ý

Cn5 M qJY , n q Cn6 NqJY , n ,

Ž 12a.

Cn5 NqJY , n q ´ in k 2 Cn6 M qJY , n ,

Ž 12b.

ns y`

Ý ns y`

Ž8.

where qY s ´ in k 2r´out y q 2 , and ´ in denotes the dielectric constant of the inner capillary space. Again, Cn5,6 are yet unknown coefficients. Finally, for the scattered field amplitudes outside the capillary, one has:

(

` sc E q, xs

1

Ý

1

Cn7 M qH, n q Cn8 NqH, n ,

Ž 13a.

ns y`

i n w cos x M qJ, n q i sin x NqJ, n ,

Ž 9a .

` sc Bq, xs

ns y`

where the relation between the integration variables k x , k y in Eq. Ž1. and the subscripts q, x in the last equation is given by tan x s k xrk y and q s k 2x q k 2y . By using the additional relation rot Nq,Z n s k 2 M q,Z n , the corresponding magnetic field amplitude has the form:

(

` ns y`

Cn1 NqJX , n q Cn2 NqYX , n q ´gl k 2 Cn3 M qJX , n

ns y`

`

Ý

Ý

B qinY , x s

Let us first consider a laser beam with 5-polarization, eˆ 5 H eˆ x and eˆ 5 H k. Taking into account that e x s cosŽ f . er y sinŽ f . ef , eˆ x P rotŽ Peˆz . s iPq sin x and eˆ x P rot rotŽ Peˆz . s yPwq cos x , one obtains for the plane waves in the decomposition of Eq. Ž1. the following series expansion Žup to a normalizing constant.:

B q, x A

Cn1 M qJX , n q Cn2 M qYX , n q Cn3 NqJX , n q Cn4 NqYX , n ,

`

rot rot Ž Peˆz . s qP yw cos Ž f y x . eˆr

Ý

Ý

(

`

E q, x A

Ž 10b.

Ž 11a.

P ' exp w iwz q iq r cos Ž f y x . x

Ý

i n NqJ, n .

ns y`

`

nw

Here, eˆr , f , z are unit vectors along the corresponding coordinate lines. Next, one has to find a representation of a plane electromagnetic wave as a series of these cylindrical vector functions. For a scalar plane wave P, one has the following decomposition into cylindrical functions cq, n w16x:

s

Ž 10a.

`

where n g Z and q labels the orthogonal solutions, w is determined by w s k 2 y q 2 with Im w G 0 and Zn are integer Bessel functions. The cylindrical vector functions M q,Z n and Nq,Z n are then defined the following way: M q,Z n s rot cqZ, n eˆz s

i n M qJ, n ,

ns y`

i n ik 2 sin x M qJ, n q w cos x NqJ, n .

Ž 9b.

1

Ý

1

Cn7 NqH, n q k 2 Cn8 M qH, n ,

Ž 13b.

ns y`

with the additional unknown coefficients Cn7,8, and H 1 being Bessel functions of the third kind ŽHankel functions., Hn1 s Jn q iYn. Thus, for every integer value of n, we have eight unknown constants Cnj which can be determined by imposing as boundary conditions smoothness of the tangential components of the electric and magnetic field amplitudes

204

J. Enderleinr Optics Communications 160 (1999) 201–206

at the outer and inner walls of the capillary. Knowledge of the values of Cnj determines completely the electromagnetic fields for the plane wave scattering, and performing a corresponding calculation for every plane wave in the representation Ž1. and integrating the results yields the final result for the whole laser beam scattering. It remains to determine the scattering amplitude in a specific angular direction. In doing so, one has to consider two field components: scattered fields with electric field polarization within the plane formed by the scattering direction and the capillary axis Žscattering amplitude S 5 .,

and scattered fields with polarization perpendicular to the capillary axis Žscattering amplitude S H.. A simple way to derive the angular distribution of the scattered field is to use the asymptotic plane wave representation of the Hankel function: 1 pr2 p Hn1 Ž q r . ™ dh exp iq r cos h y in h q . p yp r2 2

H

ž

/

Ž 14. For obtaining the scattering intensity into direction uˆ s Žsin u cos f , sin u sin f , cos u ., one replaces all Hankel

Fig. 2. Ža. Contour plot of the back-scattering intensity, S 5 Ž0, ypr2., for a 5-polarized laser beam versus the values of the capillary radii, r1 and r 2 . The radii are given in units of the vacuum wavelength l. The scattering intensities are normalized by the laser beam intensity along direction u s 0, f s pr2. Attention should be paid to the logarithmic scale of the plot. Žb. Same as in Fig. 2a, but S H Ž0, ypr2. for a H -polarized laser beam.

J. Enderleinr Optics Communications 160 (1999) 201–206

functions in Eqs. Ž13a. and Ž13b. by Eq. Ž14., and sums all factors multiplying the corresponding plane wave factor expŽ ikuˆ P r .. Taking into account that for a plane wave with amplitude A, the Poynting vector amplitude is proportional to < A < 2 , the results for S 5 and S H then read Žup to a constant factor.: S 5 Ž u , f . A k 4 sin2u

Hy''k kykyk 2

=

2

2

`

2 z

2 z

dkx

Cn8 e i n Žf y xy p r2 . ,

Ý ns y`

Ž 15a. 2

2

S H Ž u , f . A k sin u

'k =H y 'k yk 2

2

2 z

dkx

Ý

Cn7e i n Žf y xy p r2 .

cal applications, it is important to notice the existence of bands of nearly zero scattering intensity dividing islands of scattering maxima. The commercially available microcapillaries are not ideal cylinders but rather conically shaped. Thus, the present calculations are only an approximation of the experimental situation. But the numerical results suggest that, by scanning the laser beam along the conically shaped capillary, one has a good chance to find a position with values of inner and outer capillary radii that will result in very low back-scattering. In a real experiment, the observing microscope objective collects not only light scattered back at exactly u s 0, f s ypr2, but light within a cone defined by:
2

`

yk 2z

,

ns y`

205

Ž 16.

where umax is defined by the objective numerical aperture, NA, and its refractive index, n obj , via sin umax s NA. To

Ž 15b.

(

u , tan x s k xr k y k 2x y k 2z , that the coefficients Cn7,8 are

where k z s k tan has to remember functions of k x , k z .

2

and one implicit

3. Numerical results and discussion In single molecule spectroscopic experiments, a glass capillary Žrefraction index of 1.5. is filled with a watery dye solution and suspended into water Žrefraction index of 1.33.. Thus, numerical calculations were performed for the scattering of a focused laser beam on a glass capillary Ž ´gl s 1.5 2 . in water filled with water Ž ´ in,out s 1.33 2 .. The capillary axis is positioned at x, y s 0. As indicated in Eq. Ž1., the laser beam is propagating along the y-direction and its beam waist is centered at y s 0. The laser beam waist w was set equal to the light wavelength in air, comparable with typical values in usual experimental setups. The dependence of the back-scattering intensities, u s 0 and f s ypr2 in Eqs. Ž15a. and Ž15b., upon the values of the inner and outer capillary radii are shown in Fig. 2. For every pair of radii values, the intensity of the back-scattered light is given. Fig. 2a shows the values of S 5 Ž0, ypr2. for a 5-polarized laser beam Ž S H Ž0, f . identically zero in this case. and Fig. 2b shows the values of S H Ž0, ypr2. for a H -polarized laser beam Ž S 5 Ž0, f . identically zero in this case.. First, one can see that the back-scattering intensities for the 5-polarization are greater than for the H -polarization, thus, favoring an excitation by a H -polarized laser beam. Second, the scattering intensities, on average, increase with increasing dimensions of the capillary, even for equal wall thickness, explaining partially the low observed scattering signal for the small dimensions of capillaries as used in Ref. w15x. The most interesting result is the island-like structure of the scattering intensities’ dependence upon the capillary’s radii, reflecting destructive and constructive interference effects of the scattered laser light. For practi-

Fig. 3. Ža. Detected scattering intensities for a H-polarized beam and for different numerical apertures of the objective and different values of outer capillary radius, r 2 . From the lowest curve through the highest, the numerical aperture changes from 0.6 to 1.2 in steps of 0.2. All curves are normalized by the total laser beam intensity. Žb. Same as in Fig. 3a, but for 5-polarized beam.

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J. Enderleinr Optics Communications 160 (1999) 201–206

check the influence of different numerical apertures on the detected amount of scattered light, numerical calculations were done for the case of fixed inner radius, r 1 s lr2, where l denotes the vacuum wavelength of the laser light. The result for different numerical apertures Ž n obj s n gl s 1.5. is shown in Fig. 3. As can be seen, the position of the maxima and minima is not changed, and the difference between maximum and minimum intensities of detected scattered light remains significant. This indicates that the back-scattering amplitudes S 5,H are indeed a good indicator for finding optimal capillary radii with a minimum of light scattering into the observing objective. Summarizing, it can be said that an optimal positioning of the microcapillary with micropositioners may significantly reduce the measured scattering signal and thus optimize observing conditions for single molecule detection.

Acknowledgements The author thanks Christian Zander ŽSiegen University. for bringing to his attention the experimental results of ŽRegensburg University. for Ref. w15x and Martin Bohmer ¨ many helpful hints and enlightening discussions. Richard Ansell ŽRegensburg University. is also acknowledged for his linguistic support.

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