Abstract

Nondestructive noncontact high-resolution optical technique for profiling soft or fluidic boundary of an opaque object is presented. Our technique utilizes the fact that the angle width, the angular separation between two adjacent intensity minima in the forward shadow diffraction, is inversely proportional to the projected width of the object in the same direction. An analytic formula for reconstructing the boundary shape is obtained for an object with two-fold symmetry in terms of the angle widths measured for various rotational angles of the object. The typical error in determining the object shape parameter is less than 0.2%, which corresponds to 20 nm of radial accuracy when applied to an object with a mean radius of 10 microns.

©2008 Optical Society of America

1. Introduction

In science and technology it is often necessary to know boundary shapes of two- and three-dimensional microscopic objects such as biological cells [1, 2, 3], micro-organisms [4, 5], scatterers in suspension [6], optical fibers [7], metallic wires, micropillars and microcavities [8, 9, 10]. There exist several imaging techniques for profiling shapes of two- or three-dimensional object. Nondestructive noncontact optical imaging techniques such as scanning microscopy in various forms can provide wavelength-order resolution in shape profiling. Although scanning electron microscopy, scanning tunneling microscopy, atomic force microscopy and their variations provide nanometer resolution on local surface structures, they are of little use if one needs to know the overall geometrical shape of an entire object unless the object is cut so that its cross sectional shape of boundary can be profiled with these techniques. This approach still may work if the object is a solid. If the object is soft matter such as biological or fluidic objects, these intrusive techniques simply cannot be used.

Recently, several groups have introduced imaging techniques based on phase measurement such as differential phase-contrast microscopy [1], optical coherent phase microscopy [2], Mach-Zender interferometer [3], scanning interferometric aperture-less microscopy [11], phase-shifting digital holography [12], fast Fourier phase microscopy [13], and so on. However, these techniques, based on interferometry, are most effective for probing one-dimensional objects such as layered or planar structures [1, 15, 16] or profiling mildly curved surfaces such as mirror or lens surfaces [17, 18, 19] while some providing the Angstrom-order resolution [14]. For two or three dimensional objects, most of these techniques provide geometric accuracy only in the order of wavelength although overall features of the object can be well reconstructed.

One of the research areas where profiling the geometrical shapes of two-dimensional microscopic objects accurately is critical is the microcavity science. Particularly, it is important to know the boundary shape of a deformed microcavity. Deformed microcavities [20, 21, 22] have drawn much attention due to their high output directionality and relatively high cavity quality factors compared to the conventional Fabry-Perot-type microcavities. Through extensive studies, it has been recognized that characteristics of the modes supported in these deformed microcavities depend sensitively on the exact shape of cavity boundary. For example, an elliptical cavity and a stadium-shape cavity exhibit totally different mode characteristics although their shapes are quite similar [20, 21].

A quadrupole-deformed microcavity (QDM), like a stadium-shape microcavity, is one of the nonintegrable systems where ray dynamics inside exhibit chaos. The chaos grows in general as the degree of deformation in a QDM is increased. However, a certain type of shape perturbation can delay the progress to chaos [23, 24]. A QDM that contains such perturbation can thus show a regular-system-like behavior, exhibiting quite different output directionality from that of a pure QDM even when the magnitude of perturbation is less than 1% of microcavity dimension. Therefore, it is critically important to know the actual shape of a deformed microcavity under investigation with an accuracy better than 1% of its typical dimension.

One important type of deformed microcavities is a two-dimensional liquid cavity made of a liquid jet column. This type of cavity has an unusual capability of changing its degree of deformation continuously [25], which is impossible for most other microcavities made of solid. In order to accurately profile the boundary shape of these liquid microcavities, a new nondestructive imaging technique applicable to soft matter is strongly demanded.

In this paper, we introduce a new surface profiling technique which is applicable to two-or three-dimensional soft-matter opaque object with a minimal two-fold symmetry. It is based on forward shadow diffraction of a plane wave by the opaque object, so it is based on phase. When applied to an object of tens of microns in size, its geometric accuracy is a few tens of nanometer, only about 0.2% of the object size.

2. Forward diffraction pattern by an opaque object

We begin by considering the diffraction pattern by a single slit of width d. For this textbook example, it is not well appreciated that the angular spacing between adjacent diffraction maxima/minima is inversely proportional to the projection width of the slit in the direction from the slit to the location of the maxima. This can be easily seen from the well known formula for the intensity distribution of the diffracted wave

I(θ)[sin(πdsinθ/λ)πdsinθ/λ]2,

where the angle θ is measured from the forward direction normal to the plane of the slit. The angular spacing between adjacent peaks is then

Δθ=λdcosθ,

where we recognize d cosθ to be the projection width of the slit in the direction of θ. The same result holds for the diffraction pattern by an opaque strip by Babinet’s complementary principle.

For another example, let us consider the diffraction pattern by an absorbing spherical cylinder. It is well known that the diffraction pattern by a transparent cylinder has somewhat irregular spacings between adjacent peaks, so as in Mie scattering. Obviously, the above result does not hold in this case. When the absorption is increased, however, so as to make the cylinder opaque, regular spacings emerge as shown in Fig. 1, where the incident plane wave is attenuated (intensity wise) across the cylinder by a transmittance factor T = exp(-2mr x/Qabs). Here Qabs is defined as

 figure: Fig. 1.

Fig. 1. Diffraction patterns of an absorbing cylindrical object with the size parameter x=180. The real part of its index of refraction is 1.361 while its imaginary part is 0.0022 (T=21%, Qabs=310), 0.0096 (T=0.1%, Qabs=71) or 0.019 (T=10-4%, Qabs=36). Also in animation. [Media 1]

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Qabc=mr2mi,

where mr and mi are the real and imaginary parts of index of refraction of the cylinder medium and the size parameter x is defined as

x=2πrλ

with r the radius of the cylinder.

When the object is not opaque enough, not only diffraction but also wave refraction occur. In this case, the geometrical feature of the cylinder is encoded in a complicated way in a refracted wave, which is added to the diffracted wave, and results in irregularly spaced peaks. On the other hand, when the object is opaque, only the shadow diffraction occurs carrying the information on the projection width of the object. For the cylinder, the projection width is uniform, resulting in equally spaced peaks as shown in Fig. 1. It is instructive to note that irregularity still persist when T=0.1% (Qabs=71), whereas regularly spaced peaks are observed when T=10-4% (Qabs=36), approximately satisfying the condition of opaqueness.

A natural question is then whether the results in the above two special cases can be extended to other opaque objects of arbitrary shape. The answer is yes and its proof is given in Sec. 4. For now, we just take the answer without a proof and use it for devising a method to reconstruct the boundary profile of the object. We restrict ourselves with an opaque two-dimensional object of two- fold symmetry with its radial coordinate r(ψ) = r(ψ + π) satisfied by its boundary in polar coordinates with ψ the polar angle. We will briefly discuss in Sec. 4.4 what happens if the object does not have a two-fold symmetry.

 figure: Fig. 2.

Fig. 2. Measurement geometry for boundary profiling. Angle β is the rotation angle of the major axis of the object with respect to the forward direction and the polar angle α corresponds to the border between the shadow and illuminated regions on the object surface. The border angle ϕb measured from the forward direction is related to α by ϕb = αβ. Projected width w is also denoted.

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3. Surface profiling of a two-dimensional object

3.1. Description of measurement geometry

It is to be shown in Sec. 4.4 that the angular separation Δθ between two adjacent intensity minima in the forward diffraction by an opaque object is given by

Δθ=λ2w,

where w is the projected half width of the object in the forward direction. Suppose we repeatedly measure w for various rotational angle β of the object so that w = w(β ) is obtained. The measurement geometry is shown in Fig. 2, where the forward direction is rotated by β with respect to the major axis of the object. Polar angle α, also measured from the major axis of the object, corresponds to the border between the shadow and illuminated regions on the object surface. If we let r the radial coordinate of the object boundary, the projected width can then be written as

w(β)=r(α)sin(αβ).

3.2. Shape Reconstruction Algorithm

Our task now is to find the expression for α in terms of the quantities that we know, such as β, w(β) and its derivatives. Then, the radial coordinate r is given in terms of these quantities by Eq. (6) so that the geometry of the boundary is completely reconstructed.

We first note that the line of tangent to the boundary at polar angle α is in the direction of β, i.e., the direction of angle width (Δθ) observation. Therefore,

tanβ=d[r(α)sinα]d[r(α)cosα].

Differentiating Eq. (6) with respect to β yields

w(β)=wβ+wααβ=r(α)cos(αβ)+{d[r(α)sinα]cosβd[r(α)cosα]sinβ}.

Due to Eq. (7), the quantity in the curly bracket vanishes, and therefore

w(β)=r(α)cos(αβ).

From Eqs. (6) and (9), we obtain

α=βarctanw(β)w(β)

and therefore

r(βarctanw(β)w(β))=w(β)sin[arctanw(β)w(β)].

This formula gives the radius at polar angle α in terms of the projection width w(β ) and its derivative under the constraint that the tangent at the polar angle α is in the direction of β, the direction of angle width observation. This relation holds for arbitrary β covering all angles and thus the object boundary is completely specified.

3.3. Shape reconstruction examples: accuracy evaluation

In order to test both usefulness and accuracy of our shape reconstruction technique, we first applied the technique to a quadrupole-deformed opaque object whose boundary is given by the following expression.

r(ψ)=a(1+ηcos2ψ),

where η is a deformation parameter and the polar angle ψ is measured from the major axis of the object. We consider the case of η=16.00% with a size parameter 2πa/λ ka of 190. The reason for this example is that it approximates the general cross sectional shape of a liquid jet, which can serve as a shape-variable microcavity as in Ref. [25]. From the angle width function w(β) obtained from the calculated diffraction patterns for various β, we could reconstruct r(ϕ) well fit by Eq.(12) with η=15.95±0.05%.

Another test was done on a quadru-octapolar object whose shape is given by the expression:

r(ψ)=a(1+η1cos2ψ+η2cos4ψ).

For the ratio (quadrupole : octapole)=(94.00 : 6.00) with η 1=16.00% and η 2=1.02% and with a size parameter of 190.0, the result of reconstruction was η 1=15.90(±0.02)% and η 2=0.98(±0.05)% with a ratio of 94.2 : 5.8. The reconstructed boundary shape is shown in Fig. 3.We also applied the present technique to a quadru-octapole shape with the ratio (quadrupole : octapole)=(91.00 : 9.00) and η 1 +η 2=22.00%. From the calculated diffraction patterns for a size parameter of 190.0, we recovered η 1=20.06(±0.07)% and η 2=1.96(±0.08)% with a ratio of 91.1 : 8.9 and η 1 + η 2=(22.02±0.11)%.

We also applied the reconstruction technique to various pure quadrupoles with η varied from 2% to 19% and tried to fit the reconstructed shape with the quadru-octapole formula of Eq. (13) intentionally. The purpose of this test was to check the robustness and the sensitivity of the reconstruction algorithm. The results are summarized in Table 1. The erroneous octapole component was typically 0.2%, which can be considered as as a typical sensitivity in probing high order deformation components in our technique. For a mean radius 10 μm, it amounts to a shape error in the order of 20 nm (∼λ/30 for λ =600 nm) at most.

 figure: Fig. 3.

Fig. 3. (a) Reconstructed boundary profile (in black) obtained from calculated diffraction patterns for an opaque cylindrical object whose boundary (shown in red) is described by Eq. (13) with η 1=16.00% and η 2=1.02%. (b) Projected width w as a function of the object rotation angle β. The difference between the reconstructed width (in black) and the actual width (in red) is also shown (in blue). By fitting the reconstructed boundary of (a) with Eq. (13), we obtain η 1=15.90(±0.02)% and η 2=0.98(±0.05)%.

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Tables Icon

Table 1. Reconstructed shape parameters of various pure quadrupolar objects (2 ≤ η ≤ 19) with a common size parameter of 200. Reconstructed boundary profiles are intentionally fitted to the quadru-octapole shape of Eq. (13) in order to test the robustness and sensitivity of the present reconstruction technique. The resulting fit parameters are listed.

3.4. Experiment: application to a liquid-jet microcavity

We have applied the present technique to a two-dimensional liquid-jet microcavity and determined its boundary shape. The details of the liquid-jet microcavity setup is described elsewhere [25]. The liquid-jet assembly was mounted on a rotational stage so that the object rotation angle could be freely varied. An Ar-ion laser at λ=514 nm was collimated onto the liquid-jet column to induce a diffraction pattern. The liquid jet was doped with Rhodamine 590 dye at a concentration of 30 mM/L so as to make the transmission 5 × 10-4%, well satisfying the opaqueness condition. Angle widths were measured for various rotation angles β as shown in Fig. 4(a) and the results are shown in Fig. 4(b), from which the boundary shape of the microcavity shown in Fig. 4(c) could be reconstructed by using Eq. (11). By fitting the reconstructed shape to Eq. (13), we found that the boundary shape consists of a large quadrupole component and and a small octapole component with a ratio (quadrupole : octapole)=93.2 : 6.8 with a total deformation of 17.7(±0.3)% consisting of η 1=16.5(±0.3)% and η2=1.2(±0.1)%.

 figure: Fig. 4.

Fig. 4. (a) Scheme for measuring angle width Δθ for various rotational angle β of a liquid-jet microcavity (oval shape). A laser beam is indicated by a thick arrow. (b) Angle width versus object rotation angle measured for a liquid-jet microcavity of a mean radius of 14 μm with an Ar-ion laser (λ =514 nm). The angle widths were measured around at θ =5° in order to avoid the strong forward intensity peak. (c) Reconstructed boundary profile (black) and a fit (red). By fitting the reconstructed boundary with Eq. (13), we obtain (quadrupole : octapole)=93.2 : 6.8 with a total deformation of 17.7(±0.3)%. (a) and (b) also in animation. [Media 2]

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It is well known that the far-field emission pattern of a deformed microcavity is sensitively dependent on its boundary shape. We performed a separate experiment on the far-field emission distribution of this microcavity and have found that the observed far-field emission distributions are consistent with the obtained shape parameters, supporting the validity of our present technique. The details of this experimental study will be reported in a future publication.

4. Derivation of the forward shadow diffraction pattern by an opaque object

In Sec. 3, we used Eq. (5) without proof and proceeded to derive the surface profiling algorithm. Here, we provide a proof. We start by considering a plane electromagnetic wave of transverse-magnetic polarization incident on an opaque two-dimensional convex object which lies on the xy-plane as illustrated in Fig. 5, for which incident waves are written as E i = E 0 e ik0x and B i = (k 0 ×E 0/ω)e ik0x, where is a unit vector in z direction, k 0 is the wave vector in x direction with ∣k 0∣ =ω/c = 2π /λ and x is a position vector in x-y plane.

Starting from the Kirchhoff integral, we can approximately describe the diffracted amplitude with the following equations [26].

Ei4π[ωẑ(n̂×Bs)+ẑ(k×(n̂×Es)]eikr(ϕ)dl,

where is a unit vector outward normal to the surface, k is the diffracted wave vector, r(ϕ ) is the position vector at polar angle ϕ lying on the boundary surface of the object and E s and B s are the scattered electric and magnetic fields at the boundary surface, subject to the boundary conditions to be discussed below.

We are interested in the short wavelength limit specified as kr << 1. In this limit, the boundary surface is divided into two regions, a shadow region (k > 0) and an illuminated region (k < 0) with different boundary conditions to be satisfied [26, 27]. In the shadow region, the scattered fields E s and B s must be equal in magnitude and opposite in direction to the incident fields, respectively. However in the illuminated region, the following condition is satisfied: × E s = R × E i, × B s = −R × B i, where R is the Fresnel reflection coefficient for a TM wave.

We now divide the integral in Eq. (14) into two parts corresponding to the shadow (with subscript ‘sh’) and illuminated (with subscript ‘ill’) regions.

Esh=E04πiẑsh(n̂×(k0×ẑ)+k×(n̂×ẑ))ei(k0k)r(ϕ)dl
=E04πiẑsh(k0(n̂ẑ)ẑ(n̂k0)+n̂(k̂ẑ)ẑ(k̂n̂))ei(k0k)r(ϕ)dl
=E04πish(k+k0)n̂ei(k0k)r(ϕ)dl,
Eill=E04πiẑillR(n̂×(k0×ẑ)k×(n̂×ẑ))ei(k0k)r(ϕ)dl
=E04πiẑillR(k0(n̂ẑ)ẑ(n̂k0)n̂(kẑ)+ẑ(kn̂))ei(k0k)r(ϕ)dl
=E04πiillR(k0k)n̂ei(k0k)r(ϕ)dl

We introduce polar angles ϕ b1(> 0) and ϕ b2(< 0) corresponding to two borders between the shadow and illuminated regions. For k 0 aligned along the x-axis as shown in Fig. 5, we can represent the wave vector and the surface position vector as

k=x̂kcosθ+ŷksinθ,
r(ϕ)=x̂r(ϕ)cosϕ+ŷr(ϕ)sinϕ,

where θ is the angle between k and k 0. Accordingly, the differential length dl and the surface normal vector are written as

dl=[d(rcosϕ)]2+[d(rsinϕ)]2,
n̂=[x̂d(rsinϕ)ŷd(rcosϕ)]dl.

With these expressions, and by using an identity

[(1cosθ)cosϕsinθsinϕ]=sinθ[sin(ϕθ)+sinϕ]
 figure: Fig. 5.

Fig. 5. A plane electromagnetic wave with a wave vector k 0 is incident on an opaque object on the xy-plane. The boundary of the object is described by r(ϕ ) with ϕ the polar angle and the unit vector normal to the boundary. The polar angle is measured with respect to the direction of k 0. The major axis of the object is rotated by β from the k 0 direction. The region of the object specified by the condition ϕ b2 < ϕ < ϕ b1 is shadowed. The diffracted wave with a wave vector k in the direction of θ is considered.

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and its counter part with θθ + π, the preceding integrals can be rewritten as

Esh=E0k4πiϕb2ϕb1eikr(ϕ)[(1cosθ)cosϕsinθsinϕ]d{r(ϕ)[sinϕ(1+cosθ)cosϕsinθ]}
=E0k4πiϕb2ϕb1e−ikr(ϕ)sinθ1+cosθ[sin(ϕθ)+sinϕ]d{r(ϕ)[sin(ϕθ)+sinϕ]},
Eill=E0k4πiϕb1ϕb2+2πR(ϕ)eikr(ϕ)((1cosθ)cosϕsinθsinϕ)d{r(ϕ)[sinϕ(1cosθ)+cosϕsinθ]}
=E0k4πiϕb1ϕb2+2πR(ϕ)eikr(ϕ)sinθ1+cosθ[sin(ϕθ)+sinϕ]d{r(ϕ)[sin(ϕθ)sinϕ]}.

For θ ≪ 1 (forward diffraction) we can approximate sinθ /(1 + cosθ θ/2, sin(ϕθ) + sinϕ ≃ 2sin(ϕθ/2) and sin(ϕθ )− sinϕθ cosϕ, and thus E ill, proportional to θ, gives only a small contribution to the total diffracted wave in general [26, 27]. Furthermore, a significant contribution comes from the edges, where R = −1 due to the grazing incidence of the incoming wave. Therefore, we can approximately set R = −1 in Eq. (19).

EshE0k2πiϕb2ϕb1eikθr(ϕ)sin(ϕθ2)d{r(ϕ)sin(ϕθ2)}E0k2πiX,
EillθE0k4πiϕb1ϕb2+2πeikθr(ϕ)sin(ϕθ2)d{r(ϕ)cosϕ}θE0k4πiY.

We can show that X and Y are of the same order but the former is real and the latter is purely imaginary, and thus the total intensity becomes

I=E2X2+θ2Y2X2

Therefore, we can safely neglect the contribution from E ill.

E(θ)E0k2πieikr(ϕb2)θsin(ϕb2θ2)eikr(ϕb1)θsin(ϕb1θ2)ikθ.

In addition, for two-fold symmetry with ϕ b2 =ϕ b1 − π and r(ϕ b2) = r(ϕ b1), it is further simplified to

E(θ)E0πisin[kr(ϕb)θsin(ϕbθ2)]θ.

and thus the intensity distribution becomes

I(θ)sin2[kr(ϕb)θsin(ϕbθ2)]θ2.

where we dropped the subscript ‘1’ in ϕ b1 for simplicity of notation.

We now show that Y in Eq. (20) is purely imaginary while it is in the same order as X. Since the major contribution to Y comes when ϕ′ ∼ ϕ b1,ϕ b2, Y is approximately equal to

eikθy2Δx2+eikθy1Δx1

where y i = rbi) sin(ϕbi − θ) and Δx i = Δ[rbi)cosϕbi] (i=1, 2). Assuming the two-fold symmetry with ϕb2b1 − π and rb2) = rb1), we have y 1 = − y 2 and δx 1 = −Δx 2 ∼ − ( )−1. Therefore, the second integral is proportional to

(eikθy1eikθy1)Δx1i2siny1

showing that it is purely imaginary and its magnitude is in the same order as X.

4.1. Relation between the angle width function and the projected width

Now let us examine what Eq. (23) means. First, when θ = 0, we identify the factor w 0r(ϕb) sinϕb is the projected width in the exact forward direction. In the forward shadow diffraction, however, the intensity is strongly peaked at θ = 0, masking the underlying diffraction pattern and thus it is necessary to measure the diffraction pattern at a slightly nonzero angle θ 0 (θ0 << 1). In this case, the quantity wr(ϕb) sin(ϕbθ 0/2) is equal to the projected width in the direction of θ 0/2. This is because in the limit of ∣θ 0∣ << 1 the following relation holds.

r(ϕb+θ02)sin(ϕb)r(ϕb)sin(ϕbθ02)θ022.

The proof of this relation is shown below. Therefore, wr(ϕb + θ 0/2) sin(ϕb), which is nothing but the projected width in the direction of θ 0/2.

Forward diffracted intensity I ∝ ∣E2 then has local minima where the condition kwθ = (m is an integer) is satisfied. The angle width, the separation between two adjacent minima Δθ is then given by

Δθ=πkw

meaning that the angle width measured around a small forward diffraction angle θ 0 (<< 1) is inversely proportional to the projected width of the object in the direction of θ 0/2.

We now present the proof of Eq. (27). Consider an object whose boundary is described by r(ϕ) where ϕ is the polar angle measured from the direction of the incident wave as shown in Fig. 5. By the definition of the border angle ϕ b, we then have

ddϕr(ϕ)|ϕb=r´(ϕb)sinϕb+r(ϕb)cosϕb=0

By using this, the difference Dr(ϕb + θ) sinϕbr(ϕb) sin(ϕbθ) can be evaluated as

D[r(ϕb)+r´(ϕb)θ]sinϕbr(ϕb)[sin(ϕb)cosθcosϕbsinθ]
=r(ϕb)sinϕb(1cosθ)+r(ϕb)sinϕbθ+r(ϕb)cosϕbsinθ
r(ϕb)sinϕb(θ22)+[r(ϕb)sinϕb+r(ϕb)cosϕb]θr(ϕb)cosϕb(θ36)
=r(ϕb)sinϕb(θ22)r(ϕb)cosϕb(θ36)
2,

where a is the mean radius. Therefore, for small ∣θ∣ << 1, the difference D can be safely neglected. For example, in our experiment θ = θ 0/2 = 2.5° ≃ 0.044, and thus the error in this approximation is only 0.1% of the mean radius of the object. In fact, the size of this error is in the same order as the error observed in the shape reconstruction examples in Sec. 3.3.

4.2. Examples

As an example, we calculated the diffraction pattern of a quadrupole-deformed opaque object, whose boundary is given by Eq. (12). We chose the size parameter x=200 and calculated the diffraction patterns for various rotational angles (0 ≤ β ≤ 90°) of the object. Angle widths obtained from the diffraction patterns in the direction of θ 0 = 5° as a function of β are shown as dots in Fig. 6 for various values of the deformation parameter (4%≤ η ≤ 22%). For comparison, w −1(β), the inverse of the projection width measured in the direction of θ 0/2 as a function of β is also shown as a solid line for each deformation. We observe an excellent agreement, confirming the relation Eq. (5).

 figure: Fig. 6.

Fig. 6. Comparison of the simulated angle width Δθ (black dots) and the inverse w −1 of the projected width (red dots) as a function of object rotation angle β for a totally absorbing quadrupole-deformed object whose boundary is given by Eq. (12). The deformation parameter η is varied from 4% to 22% while the size parameter and the absorption Q are respectively fixed at 200 and 100. The angle widths are measured in the direction of θ 0=5° and the projected widths are measured in the direction of θ 0/2. We observe an excellent agreement between them.

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4.3. Projected width in the direction of angle width measurement

Comparing Figs. 2 and 5, we identify ϕb = α − β. For object rotation angle β, the projection width is now written as

w(β)=r(α)sin(αβθ02)r(α+θ02)sin(αβ),

where an implicit functional dependence α =α (β ) is assumed.

Note that we obtain the largest angle width and thus the smallest projection width when β = −θ 0/2 in the present convention of coordinates in Figs. 2 and 5. Explicitly, we obtain

w(θ02)=r(α+θ02)sin(α+θ02)=r(α)sinα,

where α′α + θ 0/2 is the polar angle at which the line of tangent at the boundary is in the direction of θ 0/2. Since the object is rotated by β = −θ 0/2, the angle α′ is the same as the polar angle ϕb in the exact forward direction when β = 0 and thus the projection width in this case is indeed the smallest.

Motivated by this observation we now make a formal substitution β′ = β + θ 0/2 and α′ = α + θ 0/2 so that

w(β)=r(α)sin(αβ).

Under this coordinate transformation, the resulting projected width is equivalent to the projected width in the exact forward direction in the shifted coordinate β′ = β + θ 0/2. Since only the function dependence of w(β′) matters for the surface profiling technique of Sec. 3, we can drop all primes in Eq. (32) and then have Eq. (6):

w(β)=r(α)sin(αβ).

4.4. What happens if the two-fold symmetry is not there

In order to answer this question, we go back to Eq. (22)

E(θ)E02πθ[eikr(ϕb2)θsin(ϕb2θ2)eikr(ϕb1)θsin(ϕb1θ2)],

which gives an intensity distribution of

I(θ)sin2[(w1+w2)2]θ2

where w 1 = r(ϕ b1) sin(ϕ b1θ/2) and w 2 = r(ϕ b2) sin(∣ϕ b2∣+ θ/2) (note ϕ b1 > 0,ϕ b2 < 0). By similar approximation to Eq. 27, we can show w 1r(ϕ b1 + θ/2) sin(ϕ b1) and w 2r(ϕ b2 + θ/2) sin(∣ϕ b2∣), which are nothing but the upper and lower projected widths or radii in the direction of θ/2, respectively. The angle width to be observed is then proportional to the inverse of the mean projected width w¯ ≡ (w 1 + w 2)/2.

If there is no two-fold symmetry, the projected widths w 1 and w 2 are not equal in general. Since only the mean projected width w¯ is measured as a function of the object rotational angle β in the present scheme, w 1 and w 2 are not separately known and thus there is no sufficient information to reconstruct the boundary shape. Of course, if w 1 and w 2 are separately known by some means, we can reconstruct the boundary shape by a similar algorithm to that in Sec. 3.2.

There are several ideas to separate w 1 and w 2. One is to employ a reference plane wave with a known tilt angle with respect to the incident plane wave in order to encode a phase shift information in the diffraction pattern in the case of displaced object center. Another idea, applicable to a transparent object, is to utilize refraction for decoding the information on the object center. This idea is rooted in the fact that the angle width in a near forward direction (θ << 1) is still inversely proportional to the projected width even when the object is semitransparent. This fact can be observed in Fig. 1, where the distance between the first and third off-axis intensity minima is almost independent of the absorption coefficient of the object. Since refraction depends on the object shape, one might be able to decode the information on the object center from the total diffraction/refraction pattern. In addition, the above observation also suggests that the requirement on opaqueness is not very stringent for the present technique. Quantitative investigation on these subjects is left for a future study.

5. Conclusion

We have developed a high-resolution boundary profiling technique for an opaque two- or three-dimensional convex object with a two-fold symmetry. Our technique is based on the fact that the angle width or the angular separation between two intensity minima in the forward direction is inversely proportional to the projected width of the object in the same direction. We derived an analytic formula with which the object boundary can be reconstructed from the angle widths observed for various object rotation angles. For an object with a mean radius of 10 μm, the typical accuracy of our technique is less than 0.2% of the object size, amounting to 20 nm only. Since our technique is nondestructive and noncontact, it can be applied to determine the boundary shapes of soft matter or fluidic objects such as biological samples and liquid droplets and columns, for which the conventional techniques such as confocal scanning microscopy or other interferometry techniques are not effective.

Acknowledgments

This work was supported by National Research Laboratory Grant and by KRF Grant (2005-070-C00058). SWK was supported by KRF Grant (2006-005-J02804) and by KOSEF Grant (R01-2005-000-10678-0).

References and links

1. C. G. Rylander, D. P. Davé, T. Akkin, E. Milner, K. R. Diller, and A. J. Welch, “Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy,” Opt. Lett. 29, 1509–1511 (2004). [CrossRef]   [PubMed]  

2. C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. 30, 2131–2133 (2005). [CrossRef]   [PubMed]  

3. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature Methods 4, 717–719 (2007). [CrossRef]   [PubMed]  

4. T. Noda and S. Kawata, “Separation of phase and absorption images in phase-contrast microscopy,” J. Opt. Soc. Am. A 9, 924–931 (1992). [CrossRef]  

5. H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, “Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory,” Phys. Rev. E 55, 4458–4474 (1997). [CrossRef]  

6. G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, “Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results,” Appl. Opt. 37, 7310–7319 (1998). [CrossRef]  

7. Z. Chen, “Study of a dynamic-shape-curve function for a fused tapering optical fiber,” Appl. Opt. 45, 6914–6918 (2006). [CrossRef]   [PubMed]  

8. S. V. Boriskina, P. Sewell, and T. M. Benson, “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A 21, 393–402 (2004). [CrossRef]  

9. S. G. L. Harald, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramaticape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21, 923–934 (2004). [CrossRef]  

10. T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, “Classical Phase Space Revealed by Coherent Light,” Phys. Rev. Lett. 98, 033902 (2007). [CrossRef]   [PubMed]  

11. F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, “Scanning Interferometric Apertureless Microscopy: Optical Imaging at 10 Angstrom Resolution,” Science 269, 1083–1085 (1995). [CrossRef]   [PubMed]  

12. I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image Formation in Phase-Shifting Digital Holography and Applications to Microscopy,” Appl. Opt. 40, 6177–6186 (2001). [CrossRef]  

13. N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. 46, 1836–1842 (2007). [CrossRef]   [PubMed]  

14. B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985). [CrossRef]   [PubMed]  

15. M. R. Atkinson, A. E. Dixon, and S. Damaskinos, “Surface-profile reconstruction using reflection differential phase-contrast microscopy,” Appl. Opt. 31, 6765–6771 (1992). [CrossRef]   [PubMed]  

16. H. Liang, M. G. Cid, R. G. Cucu, G. M. Dobre, A. G. Podoleanu, J. Pedro, and D. Saunders, “En-face optical coherence tomography ? a novel application of non-invasive imaging to art conservation,” Opt. Express 13, 6133–6144 (2005). [CrossRef]   [PubMed]  

17. M. Yokota, A. Asaka, and T. Yosino, “Stabilization Improvements of Laser-Diode Closed- oop Heterodyne Phase-Shifting Interferometer for Surface Profile Measurement,” Appl. Opt. 42, 1805–1808 (2003). [CrossRef]   [PubMed]  

18. S. Sonozaki, K. Iwata, and Y. Iwahashi, “Measurement of Profiles along a Circle on Two Flat Surfaces by Use of a Fizeau Interferometer with No Standard,” Appl. Opt. 42, 6853–6858 (2003). [CrossRef]   [PubMed]  

19. P. Z. Takacs, E. L. Church, C. J. Bresloff, and L. Assoufid, “Improvements in the Accuracy and the Repeatability of Long Trace Profiler Measurements”, Appl. Opt. 38, 5468–5479 (1999). [CrossRef]  

20. J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities, ” Nature 385, 45–47 (1997). [CrossRef]  

21. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, “High-Power Directional Emission from Microlasers with Chaotic Resonators,” Science 280, 1556–1564 (1998). [CrossRef]   [PubMed]  

22. S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers”, Phys. Rev. Lett. 88, 033903 (2002). [CrossRef]   [PubMed]  

23. S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007). [CrossRef]  

24. J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, “Uncertainty-limited turnstile transport in deformed microcavities,” Phys. Rev. Lett. 100, 174102 (2008). [CrossRef]   [PubMed]  

25. J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77, 083103 (2006). [CrossRef]  

26. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York,1998).

27. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

References

  • View by:

  1. C. G. Rylander, D. P. Davé, T. Akkin, E. Milner, K. R. Diller, and A. J. Welch, “Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy,” Opt. Lett. 29, 1509–1511 (2004).
    [Crossref] [PubMed]
  2. C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. 30, 2131–2133 (2005).
    [Crossref] [PubMed]
  3. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature Methods 4, 717–719 (2007).
    [Crossref] [PubMed]
  4. T. Noda and S. Kawata, “Separation of phase and absorption images in phase-contrast microscopy,” J. Opt. Soc. Am. A 9, 924–931 (1992).
    [Crossref]
  5. H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, “Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory,” Phys. Rev. E 55, 4458–4474 (1997).
    [Crossref]
  6. G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, “Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results,” Appl. Opt. 37, 7310–7319 (1998).
    [Crossref]
  7. Z. Chen, “Study of a dynamic-shape-curve function for a fused tapering optical fiber,” Appl. Opt. 45, 6914–6918 (2006).
    [Crossref] [PubMed]
  8. S. V. Boriskina, P. Sewell, and T. M. Benson, “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A 21, 393–402 (2004).
    [Crossref]
  9. S. G. L. Harald, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramaticape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21, 923–934 (2004).
    [Crossref]
  10. T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, “Classical Phase Space Revealed by Coherent Light,” Phys. Rev. Lett. 98, 033902 (2007).
    [Crossref] [PubMed]
  11. F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, “Scanning Interferometric Apertureless Microscopy: Optical Imaging at 10 Angstrom Resolution,” Science 269, 1083–1085 (1995).
    [Crossref] [PubMed]
  12. I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image Formation in Phase-Shifting Digital Holography and Applications to Microscopy,” Appl. Opt. 40, 6177–6186 (2001).
    [Crossref]
  13. N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. 46, 1836–1842 (2007).
    [Crossref] [PubMed]
  14. B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985).
    [Crossref] [PubMed]
  15. M. R. Atkinson, A. E. Dixon, and S. Damaskinos, “Surface-profile reconstruction using reflection differential phase-contrast microscopy,” Appl. Opt. 31, 6765–6771 (1992).
    [Crossref] [PubMed]
  16. H. Liang, M. G. Cid, R. G. Cucu, G. M. Dobre, A. G. Podoleanu, J. Pedro, and D. Saunders, “En-face optical coherence tomography ? a novel application of non-invasive imaging to art conservation,” Opt. Express 13, 6133–6144 (2005).
    [Crossref] [PubMed]
  17. M. Yokota, A. Asaka, and T. Yosino, “Stabilization Improvements of Laser-Diode Closed- oop Heterodyne Phase-Shifting Interferometer for Surface Profile Measurement,” Appl. Opt. 42, 1805–1808 (2003).
    [Crossref] [PubMed]
  18. S. Sonozaki, K. Iwata, and Y. Iwahashi, “Measurement of Profiles along a Circle on Two Flat Surfaces by Use of a Fizeau Interferometer with No Standard,” Appl. Opt. 42, 6853–6858 (2003).
    [Crossref] [PubMed]
  19. P. Z. Takacs, E. L. Church, C. J. Bresloff, and L. Assoufid, “Improvements in the Accuracy and the Repeatability of Long Trace Profiler Measurements”, Appl. Opt. 38, 5468–5479 (1999).
    [Crossref]
  20. J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities, ” Nature 385, 45–47 (1997).
    [Crossref]
  21. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, “High-Power Directional Emission from Microlasers with Chaotic Resonators,” Science 280, 1556–1564 (1998).
    [Crossref] [PubMed]
  22. S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers”, Phys. Rev. Lett. 88, 033903 (2002).
    [Crossref] [PubMed]
  23. S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
    [Crossref]
  24. J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, “Uncertainty-limited turnstile transport in deformed microcavities,” Phys. Rev. Lett. 100, 174102 (2008).
    [Crossref] [PubMed]
  25. J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77, 083103 (2006).
    [Crossref]
  26. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York,1998).
  27. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

2008 (1)

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, “Uncertainty-limited turnstile transport in deformed microcavities,” Phys. Rev. Lett. 100, 174102 (2008).
[Crossref] [PubMed]

2007 (4)

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature Methods 4, 717–719 (2007).
[Crossref] [PubMed]

T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, “Classical Phase Space Revealed by Coherent Light,” Phys. Rev. Lett. 98, 033902 (2007).
[Crossref] [PubMed]

N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. 46, 1836–1842 (2007).
[Crossref] [PubMed]

2006 (2)

Z. Chen, “Study of a dynamic-shape-curve function for a fused tapering optical fiber,” Appl. Opt. 45, 6914–6918 (2006).
[Crossref] [PubMed]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77, 083103 (2006).
[Crossref]

2005 (2)

2004 (3)

2003 (2)

2002 (1)

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers”, Phys. Rev. Lett. 88, 033903 (2002).
[Crossref] [PubMed]

2001 (1)

1999 (1)

1998 (2)

G. N. Constantinides, D. Gintides, S. E. Kattis, K. Kiriaki, C. A. Paraskeva, A. C. Payatakes, D. Polyzos, S. V. Tsinopoulos, and S. N. Yannopoulos, “Computation of Light Scattering by Axisymmetric Nonspherical Particles and Comparison with Experimental Results,” Appl. Opt. 37, 7310–7319 (1998).
[Crossref]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, “High-Power Directional Emission from Microlasers with Chaotic Resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

1997 (2)

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, “Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory,” Phys. Rev. E 55, 4458–4474 (1997).
[Crossref]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities, ” Nature 385, 45–47 (1997).
[Crossref]

1995 (1)

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, “Scanning Interferometric Apertureless Microscopy: Optical Imaging at 10 Angstrom Resolution,” Science 269, 1083–1085 (1995).
[Crossref] [PubMed]

1992 (2)

1985 (1)

Akkin, T.

An, K.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, “Uncertainty-limited turnstile transport in deformed microcavities,” Phys. Rev. Lett. 100, 174102 (2008).
[Crossref] [PubMed]

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
[Crossref]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77, 083103 (2006).
[Crossref]

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers”, Phys. Rev. Lett. 88, 033903 (2002).
[Crossref] [PubMed]

Asaka, A.

Assoufid, L.

Atkinson, M. R.

Badizadegan, K.

Ben-Messaoud, T.

Benson, T. M.

Bhushan, B.

Boriskina, S. V.

Bresloff, C. J.

Capasso, F.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, “High-Power Directional Emission from Microlasers with Chaotic Resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

Cense, B.

Chang, J.-S.

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers”, Phys. Rev. Lett. 88, 033903 (2002).
[Crossref] [PubMed]

Chang, R. K.

Chen, Z.

Cho, A. Y.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, “High-Power Directional Emission from Microlasers with Chaotic Resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

Choi, W.

Church, E. L.

Cid, M. G.

Constantinides, G. N.

Cucu, R. G.

Damaskinos, S.

Dasari, R. R.

Davé, D. P.

de Boer, J. F.

Diller, K. R.

Dixon, A. E.

Döbereiner, H.-G.

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, “Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory,” Phys. Rev. E 55, 4458–4474 (1997).
[Crossref]

Dobre, G. M.

Evans, E.

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, “Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory,” Phys. Rev. E 55, 4458–4474 (1997).
[Crossref]

Faist, G. J.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, “High-Power Directional Emission from Microlasers with Chaotic Resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

Fang-Yen, C.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Feld, M. S.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Fukushima, T.

T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, “Classical Phase Space Revealed by Coherent Light,” Phys. Rev. Lett. 98, 033902 (2007).
[Crossref] [PubMed]

Gintides, D.

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, “High-Power Directional Emission from Microlasers with Chaotic Resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

Harald, S. G. L.

Harayama, T.

T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, “Classical Phase Space Revealed by Coherent Light,” Phys. Rev. Lett. 98, 033902 (2007).
[Crossref] [PubMed]

Hentschel, M.

T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, “Classical Phase Space Revealed by Coherent Light,” Phys. Rev. Lett. 98, 033902 (2007).
[Crossref] [PubMed]

Ikeda, T.

Iwahashi, Y.

Iwata, K.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York,1998).

Joo, C.

Kato, J.

Kattis, S. E.

Kawata, S.

Kim, S. W.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, “Uncertainty-limited turnstile transport in deformed microcavities,” Phys. Rev. Lett. 100, 174102 (2008).
[Crossref] [PubMed]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77, 083103 (2006).
[Crossref]

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers”, Phys. Rev. Lett. 88, 033903 (2002).
[Crossref] [PubMed]

Kim, S.-W.

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
[Crossref]

Kiriaki, K.

Koliopoulos, C. L.

Kraus, M.

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, “Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory,” Phys. Rev. E 55, 4458–4474 (1997).
[Crossref]

Lee, H.-W.

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
[Crossref]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77, 083103 (2006).
[Crossref]

Lee, J.-H.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, “Uncertainty-limited turnstile transport in deformed microcavities,” Phys. Rev. Lett. 100, 174102 (2008).
[Crossref] [PubMed]

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
[Crossref]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77, 083103 (2006).
[Crossref]

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers”, Phys. Rev. Lett. 88, 033903 (2002).
[Crossref] [PubMed]

Lee, S. B.

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers”, Phys. Rev. Lett. 88, 033903 (2002).
[Crossref] [PubMed]

Lee, S.-B.

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, “Uncertainty-limited turnstile transport in deformed microcavities,” Phys. Rev. Lett. 100, 174102 (2008).
[Crossref] [PubMed]

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
[Crossref]

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77, 083103 (2006).
[Crossref]

Lee, S.-Y.

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J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, “Uncertainty-limited turnstile transport in deformed microcavities,” Phys. Rev. Lett. 100, 174102 (2008).
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J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77, 083103 (2006).
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J. Opt. Soc. Am. A (2)

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Nature (1)

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities, ” Nature 385, 45–47 (1997).
[Crossref]

Nature Methods (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

S.-B. Lee, J.-B. Shim, J. Yang, S. Moon, S.-W. Kim, H.-W. Lee, J.-H. Lee, and K. An, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
[Crossref]

Phys. Rev. E (1)

H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, “Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory,” Phys. Rev. E 55, 4458–4474 (1997).
[Crossref]

Phys. Rev. Lett. (3)

T. Tanaka, M. Hentschel, T. Fukushima, and T. Harayama, “Classical Phase Space Revealed by Coherent Light,” Phys. Rev. Lett. 98, 033902 (2007).
[Crossref] [PubMed]

J.-B. Shim, S.-B. Lee, S. W. Kim, S.-Y. Lee, H. Yang, S. Moon, J.-H. Lee, and K. An, “Uncertainty-limited turnstile transport in deformed microcavities,” Phys. Rev. Lett. 100, 174102 (2008).
[Crossref] [PubMed]

S. B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers”, Phys. Rev. Lett. 88, 033903 (2002).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77, 083103 (2006).
[Crossref]

Science (2)

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, “Scanning Interferometric Apertureless Microscopy: Optical Imaging at 10 Angstrom Resolution,” Science 269, 1083–1085 (1995).
[Crossref] [PubMed]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, G. J. Faist, D. L. Sivco, and A. Y. Cho, “High-Power Directional Emission from Microlasers with Chaotic Resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

Other (2)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York,1998).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Supplementary Material (2)

Media 1: MOV (2469 KB)     
Media 2: MOV (1312 KB)     

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Figures (6)

Fig. 1.
Fig. 1. Diffraction patterns of an absorbing cylindrical object with the size parameter x=180. The real part of its index of refraction is 1.361 while its imaginary part is 0.0022 (T=21%, Qabs =310), 0.0096 (T=0.1%, Qabs =71) or 0.019 (T=10-4%, Qabs =36). Also in animation. [Media 1]
Fig. 2.
Fig. 2. Measurement geometry for boundary profiling. Angle β is the rotation angle of the major axis of the object with respect to the forward direction and the polar angle α corresponds to the border between the shadow and illuminated regions on the object surface. The border angle ϕb measured from the forward direction is related to α by ϕb = αβ. Projected width w is also denoted.
Fig. 3.
Fig. 3. (a) Reconstructed boundary profile (in black) obtained from calculated diffraction patterns for an opaque cylindrical object whose boundary (shown in red) is described by Eq. (13) with η 1=16.00% and η 2=1.02%. (b) Projected width w as a function of the object rotation angle β. The difference between the reconstructed width (in black) and the actual width (in red) is also shown (in blue). By fitting the reconstructed boundary of (a) with Eq. (13), we obtain η 1=15.90(±0.02)% and η 2=0.98(±0.05)%.
Fig. 4.
Fig. 4. (a) Scheme for measuring angle width Δθ for various rotational angle β of a liquid-jet microcavity (oval shape). A laser beam is indicated by a thick arrow. (b) Angle width versus object rotation angle measured for a liquid-jet microcavity of a mean radius of 14 μm with an Ar-ion laser (λ =514 nm). The angle widths were measured around at θ =5° in order to avoid the strong forward intensity peak. (c) Reconstructed boundary profile (black) and a fit (red). By fitting the reconstructed boundary with Eq. (13), we obtain (quadrupole : octapole)=93.2 : 6.8 with a total deformation of 17.7(±0.3)%. (a) and (b) also in animation. [Media 2]
Fig. 5.
Fig. 5. A plane electromagnetic wave with a wave vector k 0 is incident on an opaque object on the xy-plane. The boundary of the object is described by r(ϕ ) with ϕ the polar angle and the unit vector normal to the boundary. The polar angle is measured with respect to the direction of k 0. The major axis of the object is rotated by β from the k 0 direction. The region of the object specified by the condition ϕ b2 < ϕ < ϕ b1 is shadowed. The diffracted wave with a wave vector k in the direction of θ is considered.
Fig. 6.
Fig. 6. Comparison of the simulated angle width Δθ (black dots) and the inverse w −1 of the projected width (red dots) as a function of object rotation angle β for a totally absorbing quadrupole-deformed object whose boundary is given by Eq. (12). The deformation parameter η is varied from 4% to 22% while the size parameter and the absorption Q are respectively fixed at 200 and 100. The angle widths are measured in the direction of θ 0=5° and the projected widths are measured in the direction of θ 0/2. We observe an excellent agreement between them.

Tables (1)

Tables Icon

Table 1. Reconstructed shape parameters of various pure quadrupolar objects (2 ≤ η ≤ 19) with a common size parameter of 200. Reconstructed boundary profiles are intentionally fitted to the quadru-octapole shape of Eq. (13) in order to test the robustness and sensitivity of the present reconstruction technique. The resulting fit parameters are listed.

Equations (51)

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I ( θ ) [ sin ( πd sin θ / λ ) πd sin θ / λ ] 2 ,
Δ θ = λ d cos θ ,
Q abc = m r 2 m i ,
x = 2 πr λ
Δ θ = λ 2 w ,
w ( β ) = r ( α ) sin ( α β ) .
tan β = d [ r ( α ) sin α ] d [ r ( α ) cos α ] .
w ( β ) = w β + w α α β = r ( α ) cos ( α β ) + { d [ r ( α ) sin α ] cos β d [ r ( α ) cos α ] sin β } .
w ( β ) = r ( α ) cos ( α β ) .
α = β arctan w ( β ) w ( β )
r ( β arctan w ( β ) w ( β ) ) = w ( β ) sin [ arctan w ( β ) w ( β ) ] .
r ( ψ ) = a ( 1 + η cos 2 ψ ) ,
r ( ψ ) = a ( 1 + η 1 cos 2 ψ + η 2 cos 4 ψ ) .
E i 4 π [ ω z ̂ ( n ̂ × B s ) + z ̂ ( k × ( n ̂ × E s ) ] e i k r ( ϕ ) dl ,
E sh = E 0 4 πi z ̂ sh ( n ̂ × ( k 0 × z ̂ ) + k × ( n ̂ × z ̂ ) ) e i ( k 0 k ) r ( ϕ ) dl
= E 0 4 πi z ̂ sh ( k 0 ( n ̂ z ̂ ) z ̂ ( n ̂ k 0 ) + n ̂ ( k ̂ z ̂ ) z ̂ ( k ̂ n ̂ ) ) e i ( k 0 k ) r ( ϕ ) dl
= E 0 4 πi sh ( k + k 0 ) n ̂ e i ( k 0 k ) r ( ϕ ) dl ,
E ill = E 0 4 πi z ̂ ill R ( n ̂ × ( k 0 × z ̂ ) k × ( n ̂ × z ̂ ) ) e i ( k 0 k ) r ( ϕ ) dl
= E 0 4 πi z ̂ ill R ( k 0 ( n ̂ z ̂ ) z ̂ ( n ̂ k 0 ) n ̂ ( k z ̂ ) + z ̂ ( k n ̂ ) ) e i ( k 0 k ) r ( ϕ ) dl
= E 0 4 πi ill R ( k 0 k ) n ̂ e i ( k 0 k ) r ( ϕ ) dl
k = x ̂ k cos θ + y ̂ k sin θ ,
r ( ϕ ) = x ̂ r ( ϕ ) cos ϕ + y ̂ r ( ϕ ) sin ϕ ,
dl = [ d ( r cos ϕ ) ] 2 + [ d ( r sin ϕ ) ] 2 ,
n ̂ = [ x ̂ d ( r sin ϕ ) y ̂ d ( r cos ϕ ) ] dl .
[ ( 1 cos θ ) cos ϕ sin θ sin ϕ ] = sin θ [ sin ( ϕ θ ) + sin ϕ ]
E sh = E 0 k 4 πi ϕ b2 ϕ b1 e ikr ( ϕ ) [ ( 1 cos θ ) cos ϕ sin θ sin ϕ ] d { r ( ϕ ) [ sinϕ ( 1 + cos θ ) cos ϕ sin θ ] }
= E 0 k 4 πi ϕ b2 ϕ b1 e −ikr ( ϕ ) sin θ 1 + cos θ [ sin ( ϕ θ ) + sin ϕ ] d { r ( ϕ ) [ sin ( ϕ θ ) + sin ϕ ] } ,
E ill = E 0 k 4 πi ϕ b1 ϕ b 2 + 2 π R ( ϕ ) e ikr ( ϕ ) ( ( 1 cos θ ) cos ϕ sin θ sin ϕ ) d { r ( ϕ ) [ sin ϕ ( 1 cos θ ) + cos ϕ sin θ ] }
= E 0 k 4 πi ϕ b1 ϕ b 2 + 2 π R ( ϕ ) e ikr ( ϕ ) sin θ 1 + cos θ [ sin ( ϕ θ ) + sin ϕ ] d { r ( ϕ ) [ sin ( ϕ θ ) sin ϕ ] } .
E sh E 0 k 2 πi ϕ b 2 ϕ b 1 e ikθr ( ϕ ) sin ( ϕ θ 2 ) d { r ( ϕ ) sin ( ϕ θ 2 ) } E 0 k 2 πi X ,
E ill θ E 0 k 4 πi ϕ b 1 ϕ b 2 + 2 π e ikθr ( ϕ ) sin ( ϕ θ 2 ) d { r ( ϕ ) cos ϕ } θ E 0 k 4 πi Y .
I = E 2 X 2 + θ 2 Y 2 X 2
E ( θ ) E 0 k 2 πi e ikr ( ϕ b 2 ) θ sin ( ϕ b 2 θ 2 ) e ikr ( ϕ b 1 ) θ sin ( ϕ b 1 θ 2 ) ikθ .
E ( θ ) E 0 πi sin [ kr ( ϕ b ) θ sin ( ϕ b θ 2 ) ] θ .
I ( θ ) sin 2 [ kr ( ϕ b ) θ sin ( ϕ b θ 2 ) ] θ 2 .
e ikθy 2 Δx 2 + e ikθy 1 Δx 1
( e ikθy 1 e ikθy 1 ) Δ x 1 i 2 sin y 1
r ( ϕ b + θ 0 2 ) sin ( ϕ b ) r ( ϕ b ) sin ( ϕ b θ 0 2 ) θ 0 2 2 .
Δθ = π kw
d d ϕ r ( ϕ ) | ϕ b = r ´ ( ϕ b ) sin ϕ b + r ( ϕ b ) cos ϕ b = 0
D [ r ( ϕ b ) + r ´ ( ϕ b ) θ ] sin ϕ b r ( ϕ b ) [ sin ( ϕ b ) cos θ cos ϕ b sin θ ]
= r ( ϕ b ) sin ϕ b ( 1 cos θ ) + r ( ϕ b ) sin ϕ b θ + r ( ϕ b ) cos ϕ b sin θ
r ( ϕ b ) sin ϕ b ( θ 2 2 ) + [ r ( ϕ b ) sin ϕ b + r ( ϕ b ) cos ϕ b ] θ r ( ϕ b ) cos ϕ b ( θ 3 6 )
= r ( ϕ b ) sin ϕ b ( θ 2 2 ) r ( ϕ b ) cos ϕ b ( θ 3 6 )
2 ,
w ( β ) = r ( α ) sin ( α β θ 0 2 ) r ( α + θ 0 2 ) sin ( α β ) ,
w ( θ 0 2 ) = r ( α + θ 0 2 ) sin ( α + θ 0 2 ) = r ( α ) sin α ,
w ( β ) = r ( α ) sin ( α β ) .
w ( β ) = r ( α ) sin ( α β ) .
E ( θ ) E 0 2 πθ [ e ikr ( ϕ b 2 ) θ sin ( ϕ b 2 θ 2 ) e ikr ( ϕ b 1 ) θ sin ( ϕ b 1 θ 2 ) ] ,
I ( θ ) sin 2 [ ( w 1 + w 2 ) 2 ] θ 2

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