Optical trapping of a spherically symmetric rayleigh sphere:
a model for optical tweezers upon cells
Yi-Ren Chang
a,*, Long Hsu
b, Sien Chi
a,c aDepartment of Photonics and Institute of Electro-Optical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC
bDepartment of Electrophysics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC
cDepartment of Electrical Engineering, Yuan Ze University, 135, Far-East Road, Chung-Li, Taoyuan, Taiwan, ROC
Received 8 July 2004; received in revised form 4 October 2004; accepted 25 October 2004
Abstract
Optical tweezers have become a popular manipulation and force measurement tool in cellular and molecular biol-ogy. However, there is still a lack of a sophisticated model for optical tweezers on trapping cells. In this paper, we pre-sent a novel model for optical tweezers to calculate the stiffness of trapping force upon a spherically symmetric Rayleigh sphere, which stimulates a common biological cell. A numerical simulation of this model shows that the stiffness of an optical tweezers system in trapping a cell is significantly smaller than that in trapping a polystyrene bead of the same size. Furthermore, under a small variant condition of the refractive index, the proposed model provides an approximate method which requires only the radial distribution of the trapped cellÕs refractive index for calculating the stiffness. 2004 Elsevier B.V. All rights reserved.
PACS: 42.25.Fx; 42.50.Vk
Keywords: Optical tweezers; Rayleigh scattering; Biophotonics
1. Introduction
Since Ashkin et al. [1] first trapped a micron
size bead with a focused laser beam in 1986,
opti-cal tweezers have become a popular tool for
manipulation[2–5] and force measurement in
cel-lular and molecular biology [6–9]. Up to now,
there are two major models describing the trap-ping mechanism of optical tweezers; namely, a
ray-optics (RO) model by Ashkin [10] and an
electromagnetics (EM) model by Harada and
Asakura [11]. The RO model is valid as the
ra-dius of the trapped particle is larger than ten
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2004.10.066 *
Corresponding author. Tel. +8863571212156165; fax: +88635725230.
E-mail address: YRChang.eo91g@nctu.edu.tw (Y.-R.
Chang).
times the wavelength of the laser. However, the EM model is valid as the radius of the trapped particle is much smaller than the wavelength of the laser. Although there are some other models dealing with particle whose radius is near the
wavelength of the laser[12–14], all the mentioned
models assume that the trapped particle has a uniform refractive index. Unfortunately, this assumption is not true for biological cells. There-fore, it is desired to have a sophisticated model for optical tweezers on trapping a non-uniform particle like organelles and cells.
In this work, we assume the non-uniform parti-cle as a small Rayleigh sphere with spherically symmetric refractive index. To be more specific, we assume that this sphere is composed of multiple concentric layers with different refractive indices. In this regard, as the thickness of each layer ap-proaches infinitely small and the variation of the refractive indices of adjacent layers is small en-ough, the structure of this spherically symmetric Rayleigh sphere could be analogous to that of the common biological cells.
In Section 2, we first solve for the electric potential of the proposed cell-like sphere of mul-tiple concentric layers in a laser beam. Under the small variant condition of the refractive indices of adjacent layers as mentioned above, it can be shown that such a cell-like sphere functions as a single effective electric dipole induced by the sur-rounding electric field. Then, in Section 3, substi-tuting the induced electric dipole moment into Harada and AsakuraÕs EM model, we obtain a trapping force produced by an optical tweezers system upon the cell-like sphere. Lastly, in Sec-tion 4, we will compare the difference in magni-tude between the trapping force upon a uniform sphere and that upon a non-uniform cell of the same size. In this exercise, a Chinese hamster ovary (CHO) cell at an artificially reduced size is used as an example of the non-uniform cell. The same reduced-size CHO cell but of an artifi-cially single uniform layer, a silica bead and a polystyrene bead are used as the uniform sphere, separately. The selection of the four samples is simply due to the availability of the information on their structures and distributions of refractive indices.
2. Theoretical analysis
2.1. Electric dipole moment of a spherically symmetric sphere
To illustrate the induced electric dipole moment of a cell-like sphere by the electric field of a
fo-cused laser beam, we consider inFig. 1(a), N-layer
sphere of radius R in an electric field E*o. Suppose
that the radius rkand the index of refraction nkof
each concentric layer is known, where the sub-script k refers to the kth layer. For simplicity, we assume the cell-like sphere a Rayleigh particle, which means that it is such a very small particle that the electric field around it is nearly
un-changed. As shown inFig. 1, the electric field
out-side the sphere is assumed E*o¼ ^zEoþ D~Eo ^zEo,
where Eo is the average amplitude of the electric
field, and DE*o is a variation term.
It can be shown that the electric potential Ukof
the kth layer of the multi-layer sphere for rk 1
< r < rkis usually given in the general form[15]
Ukðr; hÞ ¼ X1 l¼0 Al;krlþ Bl;krðlþ1Þ Plðcos hÞ; ð1Þ
where r and h are of spherical coordinates, Al,kand
Bl,kare two coefficients, and Pl(cos h) is the
Legen-dre polynomial of order l. Here, in our case, many
of the coefficients Al,k and Bl,k would vanish.
Firstly, as r approaches infinity, the boundary
condition UN + 1(r! 1) = Eoz =EorP1(cos h)
leads to the vanishment of all coefficients Al,N + 1
except for A1,N + 1=Eo. Secondly, as r
ap-Fig. 1. The structure of the proposed multi-layer spherically symmetric Rayleigh sphere.
proaches the origin, Bl,1= 0 because the potential
near the center of the sphere must be finite within the first layer where k = 1. Finally, the rest of the coefficients are left to be determined according to the following boundary conditions:
1 rk oUk oh r¼rk ¼ 1 rk oUkþ1 oh r¼rk ð2Þ and n2 k 1 rk oUk or r¼rk ¼ n2 kþ1 oUkþ1 or r¼rk : ð3Þ
Substituting Eq. (1) for UN + 1(r, h) into Eqs. (2)
and (3), we obtain Al;kþ Bl;k=r2lþ1k ¼ Al;kþ1þ Bl;kþ1=r2lþ1k ð4Þ and nk nkþ1 2 lAl;k nk nkþ1 2 ðl þ 1ÞBl;k=r2lþ1k ¼ lAl;kþ1 ðl þ 1ÞBl;kþ1=r2lþ1k : ð5Þ
From Eqs. (4) and (5), we can relate the
coeffi-cients Al,N + 1(=Eo) and Bl,N + 1of the space
out-side the sphere to the coefficients Al,1 and
Bl,1(= 0) of the first layer inside the sphere as
below.
For l = 1, we have
where mk” nk/nk + 1 and the matrix of M11, M12,
M21, and M22results from the N 1 inner
prod-ucts of N interface matrices of various functions of mk.
Similarly, for l6¼ 1, we have
Eq. (7) reveals that all Al,k and Bl,k vanish for
l6¼ 1. This leads to a simplified form for the
elec-tric potential outside the sphere, UN + 1(r, h) =
Eoz + B1,N + 1 z/r3, in which only the coefficient
B1,N + 1 is left to be solved. Eq. (6) indicates that
B1,N + 1 and, thus, the electric potential outside
the sphere, UN + 1(r, h), can be directly related to
A1,1 or U1(r, h) = A11z of the most inner layer via
M11, M12, M21 and M22 as given by
UNþ1ðr;hÞ ¼ Eorcos h Eo
M21cos h
M11r2
: ð8Þ
It can be seen that the overall electric potential outside the cell-like sphere is simply the sum of the
electric potential of the originally applied field Eo
and the electric potential of an electric dipole at the origin. Conventionally, the sphere is treated as a single effective electric dipole which is induced by the surrounding electric field. Classical electro-dynamics tells us that this effective electric dipole
will generate a dipole moment p = 4peoutEo(M21/
M11), where eout is the dielectric constant of the
medium outside the sphere. This results in an effec-tive polarizability a = p/(eoutEo) for the cell-like
sphere, as given by
a¼ 4pM21=M11: ð9Þ
It is worth to point out that the availability of this effective polarizability for a cell-like sphere is
due to the analogy between a cell and a spherically symmetric multi-layer sphere. As will be seen in the following section, this effective polarizability
a is a key factor in deriving the trapping force
produced by an optical tweezers system upon a E0 Bl;Nþ1 ¼ 2þm2 N 3 2ð1m2 NÞ 3r3 N ð1m2 NÞr 3 N 3 1þ2m2 N 3 2 4 3 5 2þm2 2 3 2ð1m2 2Þ 3r3 2 ð1m2 2Þr32 3 1þ2m2 2 3 2 4 3 5 2þm2 1 3 2ð1m2 1Þ 3r3 1 ð1m2 1Þr31 3 1þ2m2 1 3 2 4 3 5 A1;1 0 ¼ M11 M12 M21 M22 A1;1 0 ; ð6Þ 0 Bl;Nþ1 ¼ lþ1þlm2 N 2lþ1 ðlþ1Þð1m2 NÞ ð2lþ1Þr2lþ1 N ðlm2 NÞr 2lþ1 N 2lþ1 lþlðlþ1Þm2 N 2lþ1 2 4 3 5 lþ1þlm2 2 2lþ1 ðlþ1Þð1m2 2Þ ð2lþ1Þr2lþ1 2 ðlm2 2Þr 2lþ1 2 2lþ1 lþlðlþ1Þm2 2 2lþ1 2 4 3 5 lþ1þlm2 1 2lþ1 ðlþ1Þð1m2 1Þ ð2lþ1Þr2lþ1 1 ðlm2 1Þr 2lþ1 1 2lþ1 lþlðlþ1Þm2 1 2lþ1 2 4 3 5 A1;1 0 : ð7Þ
cell-like sphere. Yet the calculation for a is still dif-ficult, because it is complicated to calculate for the
matrix elements M11, M12, M21, and M22
Accord-ing to Eq. (6), calculating for the matrix elements
requires a series of inner production of the inter-face matrices of the N layers, involving various
functions of different mk, as shown above.
To overcome this problem, we further develop an approximate method for calculating a by apply-ing a small variant condition of the refractive indi-ces of adjacent layers. For this purpose, we assume
that the thickness of each layer, R/(N 1),
ap-proaches infinitely small, which implies N! 1.
Under this condition, the structure of such a Ray-leigh sphere of infinitely multiple layers could be analogous to that of a common cell. Consequently,
it can be shown that the four elements M11, M12,
M21, and M22 are simplified in the approximate
forms of M11 1 þ 2 3 limN!1 XN k¼1 dmk ¼ 1 þ2 3 limN!1 XN k¼1 1 nðk R=N Þ nðk R=N Þ nððk þ 1ÞR=N Þ R=N R N ¼ 1 þ2 3 ln½nð0Þ=nðRÞ; ð10Þ M12 4 3 limN!1 XN k¼1 dmk=r3k ¼4 3 Z R 0 n0ðr0Þ nðr0Þ r03dr 0; ð11Þ M21 2 3 limN!1 XN k¼1 dmkr3k ¼2 3 Z R 0 n0ðr0Þ:r03 nðr0Þ dr 0; ð12Þ and M22 1 þ 4 3 limN!1 XN k¼1 dmk ¼ 1 þ4 3 ln½nð0Þ=nðRÞ; ð13Þ
where n(0) and n(R) are the refractive indices of the layers at the center and the edge of the cell-like sphere, respectively.
Substituting Eqs.(10)–(13)into Eq.(6), we may
rewrite the coefficient A1,1and B1,N + 1 in terms of
the electric field outside the sphere, Eo, and the
radial distribution of the refractive indices of the sphere, n(r), as given by A1;1 Eo 1þ 2 3 ln nð0Þ=nðRÞ½ ð14Þ and B1;Nþ1 Eo 2 3 RR 0 n0ðr0Þr03 nðr0Þ dr0 1þ2 3 ln½nð0Þ=nðRÞ : ð15Þ
As a result, the electric potential outside the sphere is approximately in the form of
UðrÞ Eorcos h Eo 2 3 RR 0 n0ðr0Þr03 nðr0Þ dr0 1þ2 3 ln½nð0Þ=nðRÞ 1 r2cosh: ð16Þ
Consequently, the dipole moment p of the effective electric dipole is given by
p¼ 4peoutB1ðRÞ 4peoutEo 2 3 R0 R n0ðr0Þr03 nðr0Þ dr0 1þ2 3 ln½nð0Þ=nðRÞ : ð17Þ
And the effective polarizability a of the cell-like sphere is approximated as given by
a8p 3 R0 R n0ðr0Þr03 nðr0Þ dr0 1þ2 3 ln½nð0Þ=nðRÞ : ð18Þ
Eq.(18)signifies that only the radial distribution of
the refractive index n(r) of a cell is required for a. Obviously, the calculation for a becomes simple under the small variant condition of refractive in-dex. As will be shown in the following section, this approximate method for calculating a also simpli-fies the calculation for the trapping force produced by an optical tweezers system upon a spherical cell. 2.2. Light force
In the preceding section, we propose a Rayleigh sphere of multiple concentric layers to simulate a
common biological cell. In addition, we have proved that such a cell-like Rayleigh sphere is equivalent to a single effective electric dipole. This model enables us to further derive the trapping force induced by an optical tweezers system upon the cell-like sphere. This is done by substituting into Harada and AsakuraÕs model the dipole
mo-ment p* of this effective electric dipole, which is
induced in a focused laser beam of wavelength k,
as shown inFig. 2.
It can be shown that an electric dipole moment induced in a non-uniform electric field will experi-ence a trapping force produced by the focused
laser beam due to a gradient dipole force[11]
F * gradð r * Þ ¼ ½p*ð r*Þ r* E*ð r*Þ ¼a 2 nout c r * Ið r*Þ; ð19Þ in which a is the effective polarizability of the
cell-like sphere as given in Eq.(18), noutis the
refrac-tive index of the medium surrounding the sphere,
c is the speed of light in vacuum, and Ið r*Þ is the
intensity distribution of the focused laser beam.
Substituting for a from Eqs.(18) and (19) for the
gradient dipole force F*gradð r * Þ can be rewritten in the form F * gradð r * Þ 4pnout 3c RR 0 n0ðr0Þr03 nðr0Þ dr0 1þ2 3 ln½nð0Þ=nðRÞ r*Ið r*Þ: ð20Þ Similar to Harada and AsakuraÕs EM model, we also consider a Gaussian laser beam of inten-sity as given by Ið r*Þ ¼ 2P pw0 1 1þ ð2~z=kwoÞ2 exp 2ð~x 2þ ~y2Þ 1þ ð2~z=kw0Þ2 " # ; ð21Þ
where P, w0, k = 2p/k are the power, the beam
waist, and the wave number of the laser beam, respectively. And ð~x; ~y; ~zÞ ¼ ðx=w0; y=w0; z=woÞ are
the normalized coordinates with respect to w0
Sub-stituting for Ið r*Þ from Eq.(21), the gradient dipole force F
* gradð r
*
Þ in Eq.(20)can be decomposed into
the following three components: F * grad;xð r * Þ ¼ ^xa nout c P pw2 0 4~x=w0 ½1 þ ð2~z=kw0Þ22 exp 2ð~x 2þ ~y2Þ 1þ ð2~z=kw0Þ2 " # ; ð22Þ F * grad;yð r * Þ ¼ ^ya nout c P pw2 0 4~y=w0 ½1 þ ð2~z=kw0Þ 2 2 exp 2ð~x 2þ ~y2Þ 1þ ð2~z=kw0Þ 2 " # ; ð23Þ and F * grad;zð r * Þ ¼ ^za nout c P pw2 0 16~z=k2w3 0 ½1 þ ð2~z=kw0Þ 2 2 1 2ð~x 2þ ~y2Þ 1þ ð2~z=kw0Þ2 " # exp 2ð~x 2þ ~y2Þ 1þ ð2~z=kw0Þ 2 " # : ð24Þ
Conventionally, optical tweezers are analogous to a three-dimensional optical spring. This is easily
seen from Eqs. (22)–(24) that, to the first order
approximation, each component of the trapping force is linearly proportional to its corresponding coordinate, which is indeed the displacement from the center of the trapped sphere to the center of the laser beam waist. Therefore, the three components of the stiffness of the 3-D optical spring are given by kx¼ ky¼ 4a nout c P pw4 0 ð25Þ
Fig. 2. The schematic of a Rayleigh sphere in a non-uniform electric field.
and kz¼ 16a nout c P pk2w6 0 ! : ð26Þ
Obviously, both the trapping force itself and the stiffness of the trapping force are linearly propor-tional to the induced palrizability a.
By the way, in addition to the gradient dipole force, there is an undesired scattering force exerted on the cell-like Rayleigh sphere. Essentially, the scattering force corresponds to a radiative-damping force. This results from the emission of electromag-netic radiation by the accelerating effective electric dipole, acting as an antenna, in the oscillating elec-tric field of the laser beam. According to the antenna theory, the scattering force is along the Poynting vector of the laser beam as given by
F * scatðR * Þ ¼ ^zð8p3=3ÞIa2 n out=c ^z512p 5n out 27c R0 R n0ðr0Þr03 nðr0Þ dr0 1þ2 3 ln½nð0Þ=nðRÞ 2 4 3 5 2 2P pw0 1 1þ ð2~z=kw0Þ2 exp 2ð~x þ ~y 2Þ 1þ ð2~z=kw0Þ 2 " # : ð27Þ
After all, the total light force F*tot exerted on the
cell-like Rayleigh sphere is the sum of the gradient
dipole force F*grad and the scattering force F
* scat.
3. Numerical results
3.1. Data of refractive indices and sizes of Chinese hamster ovary cell
Practically, detailed refractive indices of organ-elles and bacterial cells are rarely characterized. Here, we only find a rough estimate on the refrac-tive index of mammalian cells in Brunsting and
MullaneyÕs work[16]. They modeled the CHO cell
as a coated sphere in an optical and morphological fashion. In their model, the nucleus of the mam-malian suspension cell is surrounded by its cytoplasm.
On one hand, the magnitudes of the refractive
index of the cytoplasm, ncytand the refractive
in-dex of the nucleus, nnuc were measured to be
1.3703 and 1.392 ± 0.005, respectively. On the other hand, they found a linear relationship
be-tween the radius of the nucleus, rnucand the radius
of the cell, R, for many CHO cells of various sizes as given by
R¼ ð1:38 0:02Þrnucþ ð0:03 0:05Þ: ð28Þ
In average, the magnitudes ofÆrnucæ and ÆRæ were
measured to be 8.1 ± 0.9 and 11.2 ± 0.5 lm, respectively.
Unfortunately, the sizes of CHO cell and even mammalian cells are too large to fit in Rayleigh re-gime. However, these data are still meaningful in the following simulation in Section 3.2 for two artificially reduced-size CHO cells for comparison. Here, we take an artificial CHO cell at a reduced size with a radius of R = 50 nm as an example to our cell-like model. A similar assumption was once
adopted in [17]. According to Eq. (28), this
sug-gests that the radius of its nuclear is approximately
rnuc= 36.2 ± 0.5 nm. Presumably, we further
as-sume the radial distribution of refractive index of this artificial CHO cell as given by
nðrÞ ¼ 1:392 0:005 for 0 6 r < rnuc ðcytoplasmÞ
1:3703 for rnuc6r 6 R ðnuclearÞ:
ð29Þ In this regard, n(0) = 1.392 ± 0.005 and n(R) = 1.3703.
Then, we take another CHO cell at the same re-duced size but of an artificially uniform refractive index of 1.379 as an example of uniform sphere to our cell-like model. Note that the value of 1.379 is the averaged refractive index of the CHO cell.
3.2. Numerical analysis
The validity of our model is numerically exam-ined by comparing the difference in magnitude be-tween the effective polarizabilities a of the trapped samples as well as the resulting stiffnesses k of opti-cal tweezers for a uniform sphere and those for a non-uniform cell of the same size. In this exercise, other than the uniform artificial CHO cell, a silica
bead and a polystyrene bead are also ideal to be used as the uniform sphere, because they are often used in cellular and molecular biology experiments with optical tweezers. And the non-uniform artifi-cial CHO cell seems an adequate example of the non-uniform cell. Surely, the radii of the two beads are taken to be the same as that of the two artificial CHO cells, which is 50 nm. Note that the refractive indices of uniform silica bead and polystyrene bead are 1.45 and 1.56, respectively. The samples are trapped in water, whose refractive index is 1.33. Additionally, the optical tweezers system is assumed consisting of a 10-mW Nd:YAG laser at a wavelength of k = 1064 nm and a microscope objective of N.A. = 1.25.
On the one hand, by substituting the appropriate
values into Eqs.(9) and (18)for the effective
polar-izabilities of the non-uniform artificial CHO cell of 50 nm in radius, we obtain an exact solution, anon-uni-CHO,exact= (3.798 ± 0.263)· 1023m3, and
an approximate solution, anon-uni-CHO,appr=
(3.691 ± 0.221)· 1023 m3, separately. Similarly,
we obtain, auni-CHO= 3.833· 1023 m3, asilica=
9.291· 1023 m3 and apoly= 17.48· 1023 m
3
for the uniform artificial CHO cell, the silica bead and the polystyrene bead, respectively.
On the other hand, by substituting the
appro-priate values into Eqs. (25) and (26), we obtain
the values of the transverse and the longitudinal components of the optical tweezersÕ stiffness result-ing from the effective polarizabilities, as given above. All data for the two kinds of artificial CHO cells, the silica bead, and the polystyrene
bead are listed inTable 1.
4. Discussion
From Table 1, it can be seen that the effective polarizabilities a and the three components of the optical tweezersÕ stiffness k of a silica bead and of a polystyrene bead are about 2.5–4.5 times those of either artificial CHO cell. The difference in a and k between the beads and the artificial CHO cells mainly results from the difference in the refractive indices between them. According to
Eq.(20), this relationship should be applicable to
the trapping forces, the gradient dipole forces pro- Ta
ble 1 Nu merical result of eff ective polariz abilities and stiffness Non-u niform artifi cial CHO cell Uniform artifi cial CHO cell Silica be ad Polysty rene bead Accur ate result (matrix calculation ) Appro xim ate result (s mall varia nt cond ition) Refra ctive index nnuc = 1.392 ± 0.005 (0 6 r <36.2 nm ) 1.379 1.45 1.56 ncyt = 1.370 3 (36.2 nm 6 r 6 50 nm) Polar izabilit y (10 23 m 3) 3.798 ± 0.263 3.691 ± 0.221 3.833 9.291 17.48 Tra nsverse stiffness, kx and ky (10 9 N/ m) 9.426 ± 0.653 9.359 ± 0.548 9.514 23.06 43.39 Long itud inal st iffness, kz (10 9 N/m) 0.566 8 ± 0.0393 0.562 7 ± 0.033 0 0.572 1 1.387 2.609 The eff ective polariz abilities of fou r 50-n m-in-radius sample s: includin g a non-u niform artifi cial CHO cell, a unif orm artifi cial CHO cell, a silica b ead , and a pol ystyrene be ad, whic h are follo wed by their corres pondin g trans verse and longitu dinal compo nents of the resulting stiffness o f a n optical tweezers syste m usin g a 10-m W Nd:YAG lase r a t a w avelengt h o f k = 1064 nm .
duced by the optical tweezers upon them. This predication somehow agrees with the experimental
result in Liang et al.Õs work [18]. They found that
the trapping force upon a polystyrene bead is three to four times that upon an Escherichia coli cell. Similarly, we also found in our measurement that the trapping force upon a 1-lm-in-diameter poly-styrene bead is approximately 3.8 times that upon a Klebsiella pneumoniane cell whose long axis is 1.2 lm and short axis is 0.98 lm. It is worth to note that even the sizes and shapes of the uniform bead and the K. pneumoniane cell are not really in the Rayleigh regime, similar relationship of the trap-ping forces between them remains the same.
In general, it seems that the larger the refractive index is, the larger the corresponding a, k, and trapping force are. However, on the contrary, the uniform artificial CHO cell gives rise to a slightly larger a and k than the non-uniform artificial
CHO cell of the same size, as shown in Table 1.
Note that the refractive index of the former is only the average of that of the latter. Most importantly, this particular case emphasizes the dependence of trapping force upon the distribution of refractive index of the trapped cell.
Moreover, we see that the exact solution
3.798 ± 0.263 and the approximate solution
3.691 ± 0.221 to the effective polarizability of the non-uniform artificial CHO cell are nearly the same, only differing by 2.5%. This proves the valid-ity of the approximate method in our model for the effective polarizability under the small variant condition of the refractive index.
5. Conclusion
As a summary, we have successfully developed a model to simulate a common biological cell as a spherically symmetrical Rayleigh sphere. This model is an extension of Harada and AsakuraÕs EM model from a uniform sphere to a non-uniform cell. In this model, only the radial distribution of the refractive index of a cell is required for the fol-lowing calculations: the effective polarizability of the cell, the resulting trapping force produced by an optical tweezers system upon the trapped cell, and the stiffness of the optical tweezers.
The numerical simulation of this model analyt-ically predicts a three- to fivefold difference in trapping forces between an artificial CHO cell at a reduced size and a commonly used bead of the same size, which is mostly due to the differ-ence in refractive index. On the one hand, the trapping force is dependent upon the distribution of refractive index of the trapped cell. The exact value of the trapping force can be derived by applying the accurate method of this model, which requires a series of matrix calculation. On the other hand, it is simple and convenient to estimate the trapping force by applying the approximate method of this model under the small variant condition of refractive index. Addi-tionally, it is appropriate as well to estimate the trapping force by approximating the trapped cell as a uniform cell with an effectively averaged refractive index of the cell.
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