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Experimental demonstration of scanning phase retrieval by a noniterative method with a Gaussian-amplitude beam
Optics Communications 382 (2017) 428–436
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Experimental demonstration of scanning phase retrieval by a noniterative method with a Gaussian-amplitude beam Nobuharu Nakajima n, Masayuki Yoshino Department of Engineering, Graduate School of Integrated Science and Technology, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu 432-8561, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 30 June 2016 Received in revised form 10 August 2016 Accepted 13 August 2016
We present a proof-of-principle experiment of an analytic (noniterative) phase-retrieval method for coherent imaging systems under scanning illumination of a probe beam. This method allows to reconstruct the amplitude and phase distribution of a semi-transparent object over a wide area from intensities measured at three points in the Fourier plane of the object under scanning illumination of a known Gaussian-amplitude beam in the object plane. The present measurement system is very simple in contrast to ones of interferometric techniques, and also the speed of the calculation of phase retrieval in this method is faster than that in iterative algorithms since this method is based on an analytic solution to the phase retrieval. The effectiveness of this method is shown in experimental examples of the object reconstructions of a converging lens and a plastic plate for scratch standards. & 2016 Elsevier B.V. All rights reserved.
Keywords: Phase retrieval Scanning systems Coherent imaging
1. Introduction In the areas of optical, electron and x-ray microscopy, the imaging system with scanning illumination is utilized in many cases for obtaining the aspect of a large area of a specimen. As a probe that is scanned across the specimen, a coherent beam is frequently used with interferometric measurement systems, because the important information on the specimen is contained in the phase of its transmitted (or reflected) beam. For high-frequency waves such as electrons and x rays, however, the phase measurement by use of interferometric techniques is difficult because, in such waves, a reference wave that is coherent over a wide area is barely obtained and some severe conditions on the system are required when measuring interference fringes. In the past two decades, noninterferometric methods [1–6], by which, from intensities of a wave field, its phase distribution is retrieved, have attracted considerable attention in the fields of coherent x-ray and electron imaging, and have been applied to imaging experiments for a lot of biological and material samples with a x-ray or electron wave. For example, there are the reconstructions of material [7,8] and biological [9,10] specimens in x-ray experiments and of a carbon nanotube [11] in electron-beam experiments by use of the iterative phase retrieval method [12], and the method based on the transport-of-intensity equation [13] has been applied to x-ray imaging for biological objects [14] and electron imaging for magnetic materials [15]. Recently, in n
Corresponding author. E-mail address: [email protected] (N. Nakajima).
http://dx.doi.org/10.1016/j.optcom.2016.08.033 0030-4018/& 2016 Elsevier B.V. All rights reserved.
particular, ptychography [16,17] has been used explosively in the field of the coherent imaging with scanning illumination of x-rays [18,19] and electron waves [20]. In the ptychography method, the modulus and the phase of the transmitted wave through a specimen are reconstructed by using an iterative algorithm from a series of diffraction patterns measured under illumination of multiple overlapping regions of a specimen with a probe beam. On the other hand, we have also proposed twenty years ago another phase-retrieval method with scanning illumination of a known Gaussian-amplitude beam [21,22], by which a complexvalued object with the amplitude and phase distribution can be reconstructed analytically (noniteratively) from intensity data measured at some points in the Fourier plane of the object. So far the effectiveness of this method has been verified only by a computer simulation in one dimension [22]. In this paper, we present a proof-of-principle experiment of our method, by which a two-dimensional complex-valued object function can be reconstructed from intensity data obtained at three points in the Fourier plane of the object with raster scanning of a known Gaussian-amplitude probe beam. The present system of the intensity measurements is very simple in contrast to interferometric techniques, and also the speed of the calculation of phase retrieval is faster than that in iterative algorithms because our method employs an analytic solution to the phase retrieval [23–26]. Two types of objects are reconstructed here: one is a converging lens, and the other is a plastic plate for scratch standards on the market. The experimental results show that the profiles of the reconstructed phase of the lens are in good agreement with theoretical curves and that the reconstructed phase of the plate is in close agreement with that in another experiment by using the
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Fourier transform holography. In Section 2, we formulate the object reconstruction in two dimensions by using the scanning phase-retrieval method with a known Gaussian-amplitude probe beam. In Section 3, we present experimental examples at optical wavelengths, and assess these performances. Concluding remarks are given in Section 4.
2. Formulation of the scanning phase-retrieval In this section, we present an extension of the previous onedimensional (1-D) scanning phase-retrieval method [22] for two dimensions. Fig. 1 shows a schematic diagram of the two-dimensional (2-D) scanning system. A Gaussian amplitude filter G(x, y ) is illuminated by a coherent monochromatic plane wave of unit amplitude with a wavelength λ . Then the amplitude of this filter is assumed to be expressed in the form
⎛ x 2 + y2 ⎞ ⎟, G(x, y) = A exp⎜ − ⎝ 2W 2 ⎠
(1)
where A is the central value and W is the 1/e-width of its intensity distribution, which is assumed to be a known constant. The transmitted light through the filter is Fourier transformed by use of lens L1 of focal length f1, and then the resultant amplitude distribution g (u, v ) is utilized as a probe beam, which is given from Eq. (1) as
⎡
∞
g (u, v) =
⎤
∫ ∫−∞ G(x, y)exp⎢⎢⎣ − 2πi(uxλf + vy) ⎥⎥⎦ dxdy 1
⎡ u2 + v2 ⎢ = 2πAW 2exp⎢ − λf1 / 2 πW ⎢⎣
(
⎤ ⎥ , 2⎥ ⎥⎦
)
(2)
where unimportant multiplicative constants associated with the diffraction integrals are ignored and u and v are coordinates in the object plane. A thin object of complex-amplitude transmittance f (u, v ) is illuminated by the probe beam. Then the object plane is defined as the plane immediately behind the object perpendicular to the optical axis. We here refer to the complex amplitude distribution f (u, v ) in the object plane as the object function to be reconstructed. In the present system we scan the object with respect to the probe and measure intensities in the Fourier plane transformed by a lens L2 of focal length f2. In the Fourier plane the measureable intensity distribution is given by
h(x′, y′; uo , vo)
429
2
⎡
2
⎤
∫ ∫−∞ g (u, v)f (u− uo, v − vo)exp⎢⎢⎣ − 2πi(uxλ′f + vy′) ⎥⎥⎦ dudv ∞
=
, (3)
2
where uo and vo denote the positions of the object with respect to the probe in the directions of u and v axes, respectively. Substituting Eq. (2) into Eq. (3) and combining the Gaussian function and the Fourier transform kernel into one exponential function, we obtain
h(x′ , y′ ; uo, vo )
2
⎡ 2 2 2 ⎢ β x′ + y′ = α 2exp⎢ − 2 2γ ⎢⎣
(
) ⎤⎥ ⎥ ⎥⎦
∞
×
∫ ∫−∞ g ( u + iβ 2x′/2γ, v + iβ 2y′/2γ) f (u− uo, v − vo) dudv
2
,
(4)
where α = 2πAW2, β = λf1 / 2 πW , and γ = λf2 /2π . In the present system we reconstruct the object function from three series of intensity data measured at the points of P0 , P1, and P2 in the Fourier plane by scanning the object across the Gaussian probe. The coordinates of the points of P0 , P1, and P2 are assumed to be ( 0, 0), ( c, 0), and ( 0, c ), respectively, where c is a known constant. The procedure of the object reconstruction consists of four steps: (1) Calculate 1-D phases of h(0, 0; uo , vo) along lines parallel to the u axis from two series of intensity data at the points of P0 and P1 in the scanning of u direction ; (2) calculate a 1-D phase along a line parallel to the v axis from two series of intensity data at the points of P0 and P2 in the scanning of v direction; (3) retrieve a 2-D phase of the function h(0, 0; uo , vo) by adding the calculated 1-D phase in the v direction to the calculated 1-D phases in the u direction as constant phase differences among those 1-D phases; (4) eliminate the effect of the Gaussian probe function from an estimated convolution consisting of the measured modulus (i.e., square root of |h(0, 0; uo , vo)|2) and the retrieved 2-D phase of h(0, 0; uo , vo), which yields f (uo , vo), an estimate of the object. Details of the four steps are as follows. First, to calculate 1-D phases of h(0, 0; uo , vo), we utilize two series of intensities measured at two points P0 and P1 as a function of object position (uo , vo), which are written from Eq. (4) as 2
h(0, 0; uo , vo) = α 2
∞
∫ ∫−∞ g ( u, v) f (u− uo, v − vo) dudv
2
,
(5)
and
Fig. 1. Schematic diagram of the geometry of the scanning phase-retrieval system with a Gaussian-amplitude beam. The object function is reconstructed from intensities measured at three coordinates in the Fourier plane as a function of the object position.
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Fig. 2. Schematic arrangement of the experimental system. The light beam of a diode laser is collimated by a beam expander with a pinhole. The strength of the light is controlled by a polarizer. A Gaussian-amplitude beam is produced by the Gaussian filter in the back focal plane of lens L1. The object is mounted on a translation stage to implement the scanning process. The complex amplitude of the light transmitted through the object is Fourier transformed by a lens L2 and then its intensities are measured by a CCD detector as a function of the object position.
Fig. 3. Measured intensity data when scanning the converging lens in the object plane by a raster scan process of 32 × 32 positions with a step size of 50 μm: (a), (b), and (c) show the intensity distributions obtained at three points of coordinates ( 0, 0) , ( c, 0) , and ( 0, c ) in the Fourier plane, where c corresponds to the position of the eighth CCD’s pixel from the origin. The color bar in (a) is shared by (b) and (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
⎛ β 2c 2 ⎞ 2 h(c , 0; uo , vo) = α 2exp⎜ − 2 ⎟ ⎝ 2γ ⎠ ∞
∫ ∫−∞ (
and
⎛ 2 d2 ⎞ h(c , 0; uo , vo) = α 2exp⎜ −2 2 ⎟ ⎝ β ⎠
2
)
g u + iβ 2c /2γ , v f (u− uo , v − vo) dudv . (6)
Changing the integral variables u and v into u′ = u − uo and v′ = v − vo , respectively, Eqs. (5) and (6) can be rewritten as 2
h(0, 0; uo , vo) = α 2
∞
∫ ∫−∞ g ( u′ + uo, v′ + vo) f (u′, v′) du′dv′
2
,
(7)
∞
∫ ∫−∞ g ( u′ + uo + id, v′ + vo) f (u′, v′) du′dv′
2
,
(8)
where d = β 2c/2γ . Replacing uo with uo + id in Eq. (7) and substituting the resultant function into the integral on the right hand side of Eq. (8), we obtain the relationship
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431
Fig. 4. Reconstruction of the converging lens from the intensities shown in Fig. 3: (a) and (b) are the modulus and the phase of the reconstructed object, respectively. The blue and the red curves in (c) [or (d)] show the cross-sectional profiles of the modulus and the phase along the dashed lines parallel to the u0 [or v0 ] axis in (a) and (b), respectively. The dotted curves in (c) and (d) show the theoretical phases calculated from the focal length of the lens. The dig at the lower left in (a) is a defect in the surface of the lens. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
⎛ d2 ⎞ 2 2 h(c , 0; uo , vo) = exp⎜ −2 2 ⎟ h(0, 0; uo + id, vo) . ⎝ β ⎠
(9)
This relationship indicates that the measured intensity function 2
h(c , 0; uo , vo) at the coordinate (c, 0) correspond to the product of
(
)
the constant value exp −2d2/β 2 and the intensity of the function
h(0, 0; uo + id, vo) that is shifted the constant value d along the imaginary axis in the complex plane from the function h(0, 0; uo , vo) on the real axis uo . As we shall see, this shift renders the modulus of the function
h(c , 0; uo , vo)
2
dependent on the
phase of the correlation integral h(0, 0; uo , vo) in Eq. (5). In the following procedure, we can retrieve the phase of the correlation integral along a line parallel to the u axis (i.e., where the coordinate vo is assumed to be a constant value). Let m(uo , vo) and ϕ(uo , vo) be the modulus and the phase of h(0, 0; uo , vo);
h(0, 0; uo , vo) = m(uo , vo)exp⎡⎣ i ϕ(uo , vo)⎤⎦.
(10)
Extending the real valuable uo into the complex one uo + id in Eq. (10) and substituting the resultant function into Eq. (9), we can rewrite Eq. (9) as
ln
h(c , 0; uo , vo) m(uo + id, vo)
+
d2 = β2
− ϕI (uo , vo),
(11)
where ϕI (uo , vo) denotes the imaginary part of the extended complex function ϕ(uo + id, vo) = ϕR(uo , vo) + i ϕI (uo , vo). On the left hand side of Eq. (11), the second term is given by the known constants, and the first term can be calculated from the observed modulus h(c , 0; uo , vo) and the function m(uo + id, vo) evaluated by Fourier transforming the product of the inverse Fourier transform of the observed modulus m(uo , vo) = h(0, 0; uo , vo) and an exponential function exp( 2πdx ):
m(uo + id, vo) ∞⎡ = ⎢ −∞ ⎣
∫
⎤
∞
∫−∞ m(uo′, vo) exp( 2πiuo′x) duo′⎦⎥
exp( 2πdx)exp( −2πiuox) dx.
(12)
As shown in previous papers [23–26], a 1-D phase ϕ(uo , vo) along a line parallel to the u axis can be calculated from Eq. (11) by representing the phase in terms of a Fourier series basis: N
ϕ(uo , vo) ≅
⎛
∑ ⎜⎝ a n cos
n= 1
nπ nπ ⎞⎟ uo + bn sin uo , ⎠ l l
(13)
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Fig. 5. Same as in Fig. 3 except that the scratch plate of 160 μm was used as an object.
where an and bn are unknown coefficients to be solved, the ob2
servational region of h(0, 0; uo , vo) in Eq. (5) is designated −l ≤ uo ≤ l , and N is set to be sufficiently large. Then the imaginary part of ϕ(uo + id, vo) is given from relation (13) as N
ϕI (uo , vo) ≅
⎛
∑ ⎜⎝ −a n sin
n= 1
⎛ nπ ⎞ nπ nπ ⎞⎟ uo + bn cos uo sinh ⎜ d⎟ . ⎠ ⎝ l ⎠ l l
Finally we eliminate the effect of the Gaussian probe function from an estimated correlation function consisting of the measured 2
modulus (i.e., square root of h(0, 0; uo , vo) ) and the retrieved 2-D phase of h(0, 0; uo , vo). Because some noise in the data is inevitable, a Wiener-type filter is used here. The inverse Fourier transform H (0, 0; x, y ) of h(0, 0; uo , vo) is given by
(14)
2
2
intensity data h(0, 0; uo , vo) and h(0, c ; uo , vo) at the points of P0 and P2, respectively, by use of the same procedure as stated above. Since the 1-D phase distribution along the v axis can be regarded as the constant phase differences, the 2-D phase of the correlation function in Eq. (7) can be determined by adding the 1-D phase distribution along the v axis to the 1-D phases along lines parallel to the u axis as the constant phase differences.
H (0, 0; x, y) =
⎡
⎤
∫ ∫−∞ h(0, 0; uo, vo)exp⎢⎢⎣ 2πi(uoxλf + vo y) ⎥⎥⎦ duodvo ∞
The unknown coefficients an and bn (n = 1, … , N ) in relation (13) can be determined as a solution of 2N simultaneous equations that are formulated from Eq. (11) with measured data at 2N value of uo and relation (14). The substitution of the resultant coefficients into relation (13) yields a retrieved 1-D phase along a line passing through a constant value of the coordinate vo . Thus 1-D phases ϕ(uo , vo) along lines parallel to the u axis can be retrieved in the same way as described above except to change the value of the coordinate vo . However, there are unknown constant phase differences among those 1-D phases because the phase calculations are independent of one another. To determine the ambiguity, we calculate a 1-D phase along a line on the v axis from two series of
1
2
( )
= λf1 G( x, y)F ( x, y),
(15)
where F ( x, y ) is the inverse Fourier transform of f (uo , vo) . Using a Wiener-type filter, we can obtain
F (x, y) =
H (0, 0; x, y) , G(x, y) + ε
(16)
where ε is some small constant which is depended on a noise level. Consequently, we can obtain an estimate of the object function f (uo , vo) by Fourier transforming Eq. (16). In the present method, the main time of the calculation by a computer for phase retrieval is expended in solving the 2N simultaneous equations in Eq. (11) with relation (14) from measured data at 2N value of uo . A LU-decomposition method, which comprises the lower and the upper triangularization techniques of matrix, was used here for solution of these linear equations. The computational complexity (CC) of the LU-decomposition method
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433
Fig. 6. Reconstruction of the scratch plate of 160 μm from the intensities shown in Fig. 5: (a) and (b) are the modulus and the phase of the reconstructed object, respectively. The blue and the red curves in (c) [or (d)] show the cross-sectional profiles of the modulus and the phase along the dashed lines parallel to the u0 [or v0 ] axis in (a) and (b), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
( )
is O M2 when these trangularization matrixes are obtained beforehand [27]. M ( ≥2N ) is the number of pixels that the intensity measurement fills up. Thus the CC of the 2-D phase calculation from M × M measured data in this method becomes O M2( M + 1) . On the other hand, the main time of the phase
{
}
retrieval by the iterative methods [12,16] is expended in repeating the calculation of the 2-D Fourier transform several tens to thousands times. The CC of the 2-D fast Fourier transform (i.e., FFT) from M × M measured data becomes O M3log2M . The time of the
(
)
2-D phase calculation in this method is similar in orders of magnitude to that of once 2-D FFT, and so the speed of the calculation of phase retrieval in this method is generally faster than that in the iterative algorithms provided that the trangularization matrixes for the simultaneous equations in Eq. (11) with relation (14) are obtained beforehand. It is easy to prepare the trangularization matrixes before an experiment. As seen in Eqs. (15) and (16), the determinant factors of the lateral resolution of a reconstructed object for the present method are a size of scanning step and a width of the probe beam in the object plane. The lateral resolution is limited by a larger one of them. When the step size is larger than the beam width, the twopoint resolution is obtained by two times the step size. If the beam width is larger than the step size, its resolution is given by approximately the full width of the beam.
Next we discuss the limit of the coordinate c at the measurement point of P1 or P2. From Eq. (3), the inverse Fourier transform H (c , 0; x, y ) of h(c , 0; uo , vo) is given by
⎡
⎤
∫ ∫−∞ h(c, 0; uo, vo)exp⎢⎢⎣ 2πi(uoλxf + voy) ⎥⎥⎦ duodvo ∞
H (c , 0; x, y) =
1
2
( ) (
)
= λf1 G x − f1c /f2 , y F ( x, y).
(17)
Then it can be seen from Eq. (17) that the present phase-retrieval method is applicable provided that the extent of the object's frequency spectrum F ( x, y ) is within the bounds of the shifted Gaussian function (i.e., − 2 W + f1c /f2 ≤ x ≤ 2 W + f1c /f2). Since the rough extent of F ( x, y ) can be estimated from the inverse 2
Fourier transform of the measured intensity data h(0, 0; uo , vo) at the origin [i.e., the autocorrelation function of G( x, y )F ( x, y )], the limit of the coordinate c can be determined approximately from that rough extent and the extent of the Gaussian function.
3. Experimental results A schematic of the optical system used here is shown in Fig. 2. A diode laser beam of wavelength λ = 0.635 μm was collimated by using a pinhole of 0.2 mm diameter and a beam expander. The
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N. Nakajima, M. Yoshino / Optics Communications 382 (2017) 428–436
Fig. 7. Reconstruction of a portion of the scratch plate of 160 μm by the Fourier transform holography with a reference pinhole of 0.1 mm: (a) and (b) are the modulus and the phase of the reconstructed complex amplitude in the object plane, respectively. The blue and the red curves in (c) [or (d)] show the cross-sectional profiles of the modulus and the phase along the horizontal [or vertical] dashed lines in (a) and (b), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
strength of the laser light was controlled by a polarizer placed in front of the laser. A plane wave with Gaussian amplitude was produced by a Gaussian filter placed in the back focal plane of lens L1 of focal length f1 = 40 mm . As the Gaussian filter, we used a Bull's eye type apodizing filter on the market (Edmund Optics Inc.). The Gaussian-amplitude wave was focused through the lens L1 onto an object placed in the front focal plane of the lens L1, where the full width at 1/e maximum of the focused beam measured about 32 μm . The object was mounted on a u − v translation stage to implement the scanning process, where a raster scan of 32 × 32 positions with a step size of 50 μm was implemented. Since the step size is larger than the beam width, the lateral resolution is obtained by two times the step size (i.e., 100 μm) as described in Section 2. Lens L2 of focal length f2 = 200 mm produced a Fourier transform of the complex amplitude of the light transmitted through the object. The 2-D Fourier intensity data at each position of the object were measured by a CCD detector (a Point Grey Grasshopper: 480 × 360, 14 bit, 18.16 μm pixel pitch). The influence of noise on the measured data was reduced by convolving them with a Gaussian window function, and a constant bias component due to the noise of the CCD was subtracted from the
measured data. According to the procedure in Section 2, series of intensity data at three points of coordinates ( 0, 0), ( c, 0), and
( 0, c ) in the Fourier plane were extracted from the 32 × 32 2-D data measured by the scanning process, where c was set to be the position of the eighth CCD's pixel from the origin (i.e., c = 8 × 18.16 μm ). As the examples of the experiments, the reconstructions of two types of objects are shown: (1) a converging lens with focal length f = 250 mm , and (2) a plastic plate for scratch standards (Edmund Optics Inc.). We first show the reconstruction of the former object. Figs. 3 and 4 show the reconstruction of a portion around the center of the converging lens. Figs. 3(a)–(c) are the intensity data obtained at three points of coordinates ( 0, 0), ( c, 0), and ( 0, c ) in the Fourier plane, which imply intensity distributions of truncated correlation functions between the object and the probe beam. The modulus and phase of the reconstructed object from the data in Fig. 3 by the phase retrieval method in Section 2 are shown in Figs. 4(a) and (b), respectively. Note that Fig. 4(b) is represented by an unwrapped one, because the reconstructed phases were beyond plus or minus π limits. In Fig. 4(c) [or (d)], the blue and the
N. Nakajima, M. Yoshino / Optics Communications 382 (2017) 428–436
red curves show the cross-sectional profiles of the reconstructed modulus and phase along the horizontal [or vertical] dashed black lines in Figs. 4(a) and (b), respectively. The dotted curves in Figs. 4 (c) and (d) show the theoretical phase distributions calculated from the focal length f = 250 mm . There are disturbances at both ends of the reconstructed moduli and phases in Figs. 4(c) and (d), which are due to the errors in the phase calculations from the truncated correlation data in Fig. 3. It is appropriate to eliminate the reconstructed data within a somewhat larger extent than the full width of the probe beam from each end of the reconstruction area. Thus, if the data of two sampling points at each end of the reconstructed figures are ignored, it is found that the reconstructed modulus and phase are very close to the ideal form (i.e., a flat shape) and the theoretical curve, respectively. In Figs. 4 (c) and (d), the root-mean-squared errors between the reconstructed phases and the theoretical ones are 0.416 rad and 0.259 rad, respectively, except for two sampling data points at each end. There is a dig at the lower left of Fig. 4(a), which is a defect in the surface of the lens. Next we present the reconstruction of a plastic plate for scratch standards, where a scratch of 160 μm on the plate is used as an object. Figs. 5 and 6 show the reconstruction of a portion around the scratch. Figs. 5(a)–(c) are the intensity distributions obtained at three points of coordinates ( 0, 0), ( c, 0), and ( 0, c ) in the Fourier plane where the same position c as in Fig. 3 was used. Figs. 6(a) and (b) show the modulus and phase of the reconstructed object from the data in Fig. 5. The cross-sectional profiles of the reconstructed modulus and phase along the horizontal [or vertical] dashed black lines in Figs. 6(a) and (b) are shown in Fig. 6(c) [or (d)] by the blue and the red curves, respectively. From the data of Fig. 6(c), the width of the scratch was estimated at about 150 μm, which is close to the catalog one (160 μm) of the product. To confirm the variation of the reconstructed phase, we investigated the phase distribution of light transmitted through the scratch of the plate by using the Fourier transform holography (FTH) [28]. Fig. 7 shows the object reconstruction by the FTH, where we made an object by covering the same plate in a shading film with a hole of 0.1 mm and a window of 1.6 mm × 1.2 mm , and used the object illumination by the same diode laser as in Figs. 2. Figs. 7(a) and (b) show the modulus and phase of the inverse Fourier-transformed function from a measured hologram. In Fig. 7(c) [or (d)], the blue and the red curves show the cross-sectional profiles of the modulus and phase along the horizontal [or vertical] black dashed lines in Figs. 7(a) and (b), respectively, where the data points are 168 for the interval of 1.6 mm. Comparing two phases along the horizontal lines in Figs. 6(c) and 7 (c), the variation of the phase in and around the scratch in Fig. 6(c) is in close agreement with one in Fig. 7(c) except for the difference of data points. Note that the difference between two phases cannot be evaluated quantitatively here, because it is very difficult that the positions of the cross-sections of two phases reconstructed by different experiments of each other are set to be the same. A comparison between two vertical phases in Figs. 6(d) and 7(d) shows that a tilt to the right in Fig. 6(d) is appeared. This is due to a linear phase induced by a slightly vertical inclination of the scratch plate set up at a holder in the experiment. On the other hand, there is the marked difference between two moduli in Figs. 6 and 7. As seen in Fig. 7(a), that disturbance of the reconstructed modulus by the FTH can be considered the fringe noise due to the interference between reflected lights in the plate. Since our method is noninterferometic phase retrieval, such fringe noise does not appear in the object reconstruction.
4. Conclusions We have experimentally demonstrated the object reconstructions using a scanning phase-retrieval method with a known
435
Gaussian-amplitude probe beam, which is a noninterferometric and noniterative method. The experimental results of this method have presented good reconstructions of the complex-valued objects without a reference wave. In this method, the area of the object reconstruction can be easily enlarged by increasing the number of scanning steps. Although a Gaussian filter is needed here to produce a Gaussian-amplitude probe, the measurement system is very simple and free from interferometric noises in contrast to interferometric techniques, and also the speed of the calculation of phase retrieval from measured data in this method is faster than that in iterative algorithms since our method is based on an analytic solution to the phase retrieval. The resolution in the present experiments is limited by a larger one of the two parameters that are a scanning-step size and a probe-beam width in the object plane. However, it is possible that the resolution is improved using higher-frequency intensity data than those at the measurement points of the present experiments, regardless of magnitudes of the two parameters. The details of such an improvement are currently being studied.
Acknowledgments This research was supported by a Grant-in-Aid for Scientific Research (Grant no. 26390081) from the Japan Society for the Promotion of Science.
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[17] J.M. Rodenburg, Ptychography and related diffractive imaging methods, in: P. W. Hawkes (Ed.), Advances in Imaging and Electron Physics, 150, Academic, New York, 2008, pp. 87–184. [18] A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, Y. Takahashi, high-resolution multislice x-ray ptychography of extended thick objects, Phys. Rev. Lett. 112 (2014) 053903. [19] A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A.G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, A. Menzel, Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography, J. Struct. Biol., 10.1016/j.jsb.2015.10.008. [20] C.T. Putkunz, A.J. D’Alfonso, A.J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R.E. Scholten, K.A. Nugent, L.J. Allen, Atom-scale ptychorgraphic electron diffractive imaging of boron nitride cones, Phys. Rev. Lett. 108 (2012) 073901. [21] N. Nakajima, Scanning phase-retrieval system with an exponential-filtered probe, Opt. Lett. 21 (1996) 1933–1935.
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Experimental demonstration of scanning phase retrieval by a noniterative method with a Gaussian-amplitude beam Nobuharu Nakajima n, Masayuki Yoshino Department of Engineering, Graduate School of Integrated Science and Technology, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu 432-8561, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 30 June 2016 Received in revised form 10 August 2016 Accepted 13 August 2016
We present a proof-of-principle experiment of an analytic (noniterative) phase-retrieval method for coherent imaging systems under scanning illumination of a probe beam. This method allows to reconstruct the amplitude and phase distribution of a semi-transparent object over a wide area from intensities measured at three points in the Fourier plane of the object under scanning illumination of a known Gaussian-amplitude beam in the object plane. The present measurement system is very simple in contrast to ones of interferometric techniques, and also the speed of the calculation of phase retrieval in this method is faster than that in iterative algorithms since this method is based on an analytic solution to the phase retrieval. The effectiveness of this method is shown in experimental examples of the object reconstructions of a converging lens and a plastic plate for scratch standards. & 2016 Elsevier B.V. All rights reserved.
Keywords: Phase retrieval Scanning systems Coherent imaging
1. Introduction In the areas of optical, electron and x-ray microscopy, the imaging system with scanning illumination is utilized in many cases for obtaining the aspect of a large area of a specimen. As a probe that is scanned across the specimen, a coherent beam is frequently used with interferometric measurement systems, because the important information on the specimen is contained in the phase of its transmitted (or reflected) beam. For high-frequency waves such as electrons and x rays, however, the phase measurement by use of interferometric techniques is difficult because, in such waves, a reference wave that is coherent over a wide area is barely obtained and some severe conditions on the system are required when measuring interference fringes. In the past two decades, noninterferometric methods [1–6], by which, from intensities of a wave field, its phase distribution is retrieved, have attracted considerable attention in the fields of coherent x-ray and electron imaging, and have been applied to imaging experiments for a lot of biological and material samples with a x-ray or electron wave. For example, there are the reconstructions of material [7,8] and biological [9,10] specimens in x-ray experiments and of a carbon nanotube [11] in electron-beam experiments by use of the iterative phase retrieval method [12], and the method based on the transport-of-intensity equation [13] has been applied to x-ray imaging for biological objects [14] and electron imaging for magnetic materials [15]. Recently, in n
Corresponding author. E-mail address: [email protected] (N. Nakajima).
http://dx.doi.org/10.1016/j.optcom.2016.08.033 0030-4018/& 2016 Elsevier B.V. All rights reserved.
particular, ptychography [16,17] has been used explosively in the field of the coherent imaging with scanning illumination of x-rays [18,19] and electron waves [20]. In the ptychography method, the modulus and the phase of the transmitted wave through a specimen are reconstructed by using an iterative algorithm from a series of diffraction patterns measured under illumination of multiple overlapping regions of a specimen with a probe beam. On the other hand, we have also proposed twenty years ago another phase-retrieval method with scanning illumination of a known Gaussian-amplitude beam [21,22], by which a complexvalued object with the amplitude and phase distribution can be reconstructed analytically (noniteratively) from intensity data measured at some points in the Fourier plane of the object. So far the effectiveness of this method has been verified only by a computer simulation in one dimension [22]. In this paper, we present a proof-of-principle experiment of our method, by which a two-dimensional complex-valued object function can be reconstructed from intensity data obtained at three points in the Fourier plane of the object with raster scanning of a known Gaussian-amplitude probe beam. The present system of the intensity measurements is very simple in contrast to interferometric techniques, and also the speed of the calculation of phase retrieval is faster than that in iterative algorithms because our method employs an analytic solution to the phase retrieval [23–26]. Two types of objects are reconstructed here: one is a converging lens, and the other is a plastic plate for scratch standards on the market. The experimental results show that the profiles of the reconstructed phase of the lens are in good agreement with theoretical curves and that the reconstructed phase of the plate is in close agreement with that in another experiment by using the
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Fourier transform holography. In Section 2, we formulate the object reconstruction in two dimensions by using the scanning phase-retrieval method with a known Gaussian-amplitude probe beam. In Section 3, we present experimental examples at optical wavelengths, and assess these performances. Concluding remarks are given in Section 4.
2. Formulation of the scanning phase-retrieval In this section, we present an extension of the previous onedimensional (1-D) scanning phase-retrieval method [22] for two dimensions. Fig. 1 shows a schematic diagram of the two-dimensional (2-D) scanning system. A Gaussian amplitude filter G(x, y ) is illuminated by a coherent monochromatic plane wave of unit amplitude with a wavelength λ . Then the amplitude of this filter is assumed to be expressed in the form
⎛ x 2 + y2 ⎞ ⎟, G(x, y) = A exp⎜ − ⎝ 2W 2 ⎠
(1)
where A is the central value and W is the 1/e-width of its intensity distribution, which is assumed to be a known constant. The transmitted light through the filter is Fourier transformed by use of lens L1 of focal length f1, and then the resultant amplitude distribution g (u, v ) is utilized as a probe beam, which is given from Eq. (1) as
⎡
∞
g (u, v) =
⎤
∫ ∫−∞ G(x, y)exp⎢⎢⎣ − 2πi(uxλf + vy) ⎥⎥⎦ dxdy 1
⎡ u2 + v2 ⎢ = 2πAW 2exp⎢ − λf1 / 2 πW ⎢⎣
(
⎤ ⎥ , 2⎥ ⎥⎦
)
(2)
where unimportant multiplicative constants associated with the diffraction integrals are ignored and u and v are coordinates in the object plane. A thin object of complex-amplitude transmittance f (u, v ) is illuminated by the probe beam. Then the object plane is defined as the plane immediately behind the object perpendicular to the optical axis. We here refer to the complex amplitude distribution f (u, v ) in the object plane as the object function to be reconstructed. In the present system we scan the object with respect to the probe and measure intensities in the Fourier plane transformed by a lens L2 of focal length f2. In the Fourier plane the measureable intensity distribution is given by
h(x′, y′; uo , vo)
429
2
⎡
2
⎤
∫ ∫−∞ g (u, v)f (u− uo, v − vo)exp⎢⎢⎣ − 2πi(uxλ′f + vy′) ⎥⎥⎦ dudv ∞
=
, (3)
2
where uo and vo denote the positions of the object with respect to the probe in the directions of u and v axes, respectively. Substituting Eq. (2) into Eq. (3) and combining the Gaussian function and the Fourier transform kernel into one exponential function, we obtain
h(x′ , y′ ; uo, vo )
2
⎡ 2 2 2 ⎢ β x′ + y′ = α 2exp⎢ − 2 2γ ⎢⎣
(
) ⎤⎥ ⎥ ⎥⎦
∞
×
∫ ∫−∞ g ( u + iβ 2x′/2γ, v + iβ 2y′/2γ) f (u− uo, v − vo) dudv
2
,
(4)
where α = 2πAW2, β = λf1 / 2 πW , and γ = λf2 /2π . In the present system we reconstruct the object function from three series of intensity data measured at the points of P0 , P1, and P2 in the Fourier plane by scanning the object across the Gaussian probe. The coordinates of the points of P0 , P1, and P2 are assumed to be ( 0, 0), ( c, 0), and ( 0, c ), respectively, where c is a known constant. The procedure of the object reconstruction consists of four steps: (1) Calculate 1-D phases of h(0, 0; uo , vo) along lines parallel to the u axis from two series of intensity data at the points of P0 and P1 in the scanning of u direction ; (2) calculate a 1-D phase along a line parallel to the v axis from two series of intensity data at the points of P0 and P2 in the scanning of v direction; (3) retrieve a 2-D phase of the function h(0, 0; uo , vo) by adding the calculated 1-D phase in the v direction to the calculated 1-D phases in the u direction as constant phase differences among those 1-D phases; (4) eliminate the effect of the Gaussian probe function from an estimated convolution consisting of the measured modulus (i.e., square root of |h(0, 0; uo , vo)|2) and the retrieved 2-D phase of h(0, 0; uo , vo), which yields f (uo , vo), an estimate of the object. Details of the four steps are as follows. First, to calculate 1-D phases of h(0, 0; uo , vo), we utilize two series of intensities measured at two points P0 and P1 as a function of object position (uo , vo), which are written from Eq. (4) as 2
h(0, 0; uo , vo) = α 2
∞
∫ ∫−∞ g ( u, v) f (u− uo, v − vo) dudv
2
,
(5)
and
Fig. 1. Schematic diagram of the geometry of the scanning phase-retrieval system with a Gaussian-amplitude beam. The object function is reconstructed from intensities measured at three coordinates in the Fourier plane as a function of the object position.
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Fig. 2. Schematic arrangement of the experimental system. The light beam of a diode laser is collimated by a beam expander with a pinhole. The strength of the light is controlled by a polarizer. A Gaussian-amplitude beam is produced by the Gaussian filter in the back focal plane of lens L1. The object is mounted on a translation stage to implement the scanning process. The complex amplitude of the light transmitted through the object is Fourier transformed by a lens L2 and then its intensities are measured by a CCD detector as a function of the object position.
Fig. 3. Measured intensity data when scanning the converging lens in the object plane by a raster scan process of 32 × 32 positions with a step size of 50 μm: (a), (b), and (c) show the intensity distributions obtained at three points of coordinates ( 0, 0) , ( c, 0) , and ( 0, c ) in the Fourier plane, where c corresponds to the position of the eighth CCD’s pixel from the origin. The color bar in (a) is shared by (b) and (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
⎛ β 2c 2 ⎞ 2 h(c , 0; uo , vo) = α 2exp⎜ − 2 ⎟ ⎝ 2γ ⎠ ∞
∫ ∫−∞ (
and
⎛ 2 d2 ⎞ h(c , 0; uo , vo) = α 2exp⎜ −2 2 ⎟ ⎝ β ⎠
2
)
g u + iβ 2c /2γ , v f (u− uo , v − vo) dudv . (6)
Changing the integral variables u and v into u′ = u − uo and v′ = v − vo , respectively, Eqs. (5) and (6) can be rewritten as 2
h(0, 0; uo , vo) = α 2
∞
∫ ∫−∞ g ( u′ + uo, v′ + vo) f (u′, v′) du′dv′
2
,
(7)
∞
∫ ∫−∞ g ( u′ + uo + id, v′ + vo) f (u′, v′) du′dv′
2
,
(8)
where d = β 2c/2γ . Replacing uo with uo + id in Eq. (7) and substituting the resultant function into the integral on the right hand side of Eq. (8), we obtain the relationship
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431
Fig. 4. Reconstruction of the converging lens from the intensities shown in Fig. 3: (a) and (b) are the modulus and the phase of the reconstructed object, respectively. The blue and the red curves in (c) [or (d)] show the cross-sectional profiles of the modulus and the phase along the dashed lines parallel to the u0 [or v0 ] axis in (a) and (b), respectively. The dotted curves in (c) and (d) show the theoretical phases calculated from the focal length of the lens. The dig at the lower left in (a) is a defect in the surface of the lens. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
⎛ d2 ⎞ 2 2 h(c , 0; uo , vo) = exp⎜ −2 2 ⎟ h(0, 0; uo + id, vo) . ⎝ β ⎠
(9)
This relationship indicates that the measured intensity function 2
h(c , 0; uo , vo) at the coordinate (c, 0) correspond to the product of
(
)
the constant value exp −2d2/β 2 and the intensity of the function
h(0, 0; uo + id, vo) that is shifted the constant value d along the imaginary axis in the complex plane from the function h(0, 0; uo , vo) on the real axis uo . As we shall see, this shift renders the modulus of the function
h(c , 0; uo , vo)
2
dependent on the
phase of the correlation integral h(0, 0; uo , vo) in Eq. (5). In the following procedure, we can retrieve the phase of the correlation integral along a line parallel to the u axis (i.e., where the coordinate vo is assumed to be a constant value). Let m(uo , vo) and ϕ(uo , vo) be the modulus and the phase of h(0, 0; uo , vo);
h(0, 0; uo , vo) = m(uo , vo)exp⎡⎣ i ϕ(uo , vo)⎤⎦.
(10)
Extending the real valuable uo into the complex one uo + id in Eq. (10) and substituting the resultant function into Eq. (9), we can rewrite Eq. (9) as
ln
h(c , 0; uo , vo) m(uo + id, vo)
+
d2 = β2
− ϕI (uo , vo),
(11)
where ϕI (uo , vo) denotes the imaginary part of the extended complex function ϕ(uo + id, vo) = ϕR(uo , vo) + i ϕI (uo , vo). On the left hand side of Eq. (11), the second term is given by the known constants, and the first term can be calculated from the observed modulus h(c , 0; uo , vo) and the function m(uo + id, vo) evaluated by Fourier transforming the product of the inverse Fourier transform of the observed modulus m(uo , vo) = h(0, 0; uo , vo) and an exponential function exp( 2πdx ):
m(uo + id, vo) ∞⎡ = ⎢ −∞ ⎣
∫
⎤
∞
∫−∞ m(uo′, vo) exp( 2πiuo′x) duo′⎦⎥
exp( 2πdx)exp( −2πiuox) dx.
(12)
As shown in previous papers [23–26], a 1-D phase ϕ(uo , vo) along a line parallel to the u axis can be calculated from Eq. (11) by representing the phase in terms of a Fourier series basis: N
ϕ(uo , vo) ≅
⎛
∑ ⎜⎝ a n cos
n= 1
nπ nπ ⎞⎟ uo + bn sin uo , ⎠ l l
(13)
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Fig. 5. Same as in Fig. 3 except that the scratch plate of 160 μm was used as an object.
where an and bn are unknown coefficients to be solved, the ob2
servational region of h(0, 0; uo , vo) in Eq. (5) is designated −l ≤ uo ≤ l , and N is set to be sufficiently large. Then the imaginary part of ϕ(uo + id, vo) is given from relation (13) as N
ϕI (uo , vo) ≅
⎛
∑ ⎜⎝ −a n sin
n= 1
⎛ nπ ⎞ nπ nπ ⎞⎟ uo + bn cos uo sinh ⎜ d⎟ . ⎠ ⎝ l ⎠ l l
Finally we eliminate the effect of the Gaussian probe function from an estimated correlation function consisting of the measured 2
modulus (i.e., square root of h(0, 0; uo , vo) ) and the retrieved 2-D phase of h(0, 0; uo , vo). Because some noise in the data is inevitable, a Wiener-type filter is used here. The inverse Fourier transform H (0, 0; x, y ) of h(0, 0; uo , vo) is given by
(14)
2
2
intensity data h(0, 0; uo , vo) and h(0, c ; uo , vo) at the points of P0 and P2, respectively, by use of the same procedure as stated above. Since the 1-D phase distribution along the v axis can be regarded as the constant phase differences, the 2-D phase of the correlation function in Eq. (7) can be determined by adding the 1-D phase distribution along the v axis to the 1-D phases along lines parallel to the u axis as the constant phase differences.
H (0, 0; x, y) =
⎡
⎤
∫ ∫−∞ h(0, 0; uo, vo)exp⎢⎢⎣ 2πi(uoxλf + vo y) ⎥⎥⎦ duodvo ∞
The unknown coefficients an and bn (n = 1, … , N ) in relation (13) can be determined as a solution of 2N simultaneous equations that are formulated from Eq. (11) with measured data at 2N value of uo and relation (14). The substitution of the resultant coefficients into relation (13) yields a retrieved 1-D phase along a line passing through a constant value of the coordinate vo . Thus 1-D phases ϕ(uo , vo) along lines parallel to the u axis can be retrieved in the same way as described above except to change the value of the coordinate vo . However, there are unknown constant phase differences among those 1-D phases because the phase calculations are independent of one another. To determine the ambiguity, we calculate a 1-D phase along a line on the v axis from two series of
1
2
( )
= λf1 G( x, y)F ( x, y),
(15)
where F ( x, y ) is the inverse Fourier transform of f (uo , vo) . Using a Wiener-type filter, we can obtain
F (x, y) =
H (0, 0; x, y) , G(x, y) + ε
(16)
where ε is some small constant which is depended on a noise level. Consequently, we can obtain an estimate of the object function f (uo , vo) by Fourier transforming Eq. (16). In the present method, the main time of the calculation by a computer for phase retrieval is expended in solving the 2N simultaneous equations in Eq. (11) with relation (14) from measured data at 2N value of uo . A LU-decomposition method, which comprises the lower and the upper triangularization techniques of matrix, was used here for solution of these linear equations. The computational complexity (CC) of the LU-decomposition method
N. Nakajima, M. Yoshino / Optics Communications 382 (2017) 428–436
433
Fig. 6. Reconstruction of the scratch plate of 160 μm from the intensities shown in Fig. 5: (a) and (b) are the modulus and the phase of the reconstructed object, respectively. The blue and the red curves in (c) [or (d)] show the cross-sectional profiles of the modulus and the phase along the dashed lines parallel to the u0 [or v0 ] axis in (a) and (b), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
( )
is O M2 when these trangularization matrixes are obtained beforehand [27]. M ( ≥2N ) is the number of pixels that the intensity measurement fills up. Thus the CC of the 2-D phase calculation from M × M measured data in this method becomes O M2( M + 1) . On the other hand, the main time of the phase
{
}
retrieval by the iterative methods [12,16] is expended in repeating the calculation of the 2-D Fourier transform several tens to thousands times. The CC of the 2-D fast Fourier transform (i.e., FFT) from M × M measured data becomes O M3log2M . The time of the
(
)
2-D phase calculation in this method is similar in orders of magnitude to that of once 2-D FFT, and so the speed of the calculation of phase retrieval in this method is generally faster than that in the iterative algorithms provided that the trangularization matrixes for the simultaneous equations in Eq. (11) with relation (14) are obtained beforehand. It is easy to prepare the trangularization matrixes before an experiment. As seen in Eqs. (15) and (16), the determinant factors of the lateral resolution of a reconstructed object for the present method are a size of scanning step and a width of the probe beam in the object plane. The lateral resolution is limited by a larger one of them. When the step size is larger than the beam width, the twopoint resolution is obtained by two times the step size. If the beam width is larger than the step size, its resolution is given by approximately the full width of the beam.
Next we discuss the limit of the coordinate c at the measurement point of P1 or P2. From Eq. (3), the inverse Fourier transform H (c , 0; x, y ) of h(c , 0; uo , vo) is given by
⎡
⎤
∫ ∫−∞ h(c, 0; uo, vo)exp⎢⎢⎣ 2πi(uoλxf + voy) ⎥⎥⎦ duodvo ∞
H (c , 0; x, y) =
1
2
( ) (
)
= λf1 G x − f1c /f2 , y F ( x, y).
(17)
Then it can be seen from Eq. (17) that the present phase-retrieval method is applicable provided that the extent of the object's frequency spectrum F ( x, y ) is within the bounds of the shifted Gaussian function (i.e., − 2 W + f1c /f2 ≤ x ≤ 2 W + f1c /f2). Since the rough extent of F ( x, y ) can be estimated from the inverse 2
Fourier transform of the measured intensity data h(0, 0; uo , vo) at the origin [i.e., the autocorrelation function of G( x, y )F ( x, y )], the limit of the coordinate c can be determined approximately from that rough extent and the extent of the Gaussian function.
3. Experimental results A schematic of the optical system used here is shown in Fig. 2. A diode laser beam of wavelength λ = 0.635 μm was collimated by using a pinhole of 0.2 mm diameter and a beam expander. The
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Fig. 7. Reconstruction of a portion of the scratch plate of 160 μm by the Fourier transform holography with a reference pinhole of 0.1 mm: (a) and (b) are the modulus and the phase of the reconstructed complex amplitude in the object plane, respectively. The blue and the red curves in (c) [or (d)] show the cross-sectional profiles of the modulus and the phase along the horizontal [or vertical] dashed lines in (a) and (b), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
strength of the laser light was controlled by a polarizer placed in front of the laser. A plane wave with Gaussian amplitude was produced by a Gaussian filter placed in the back focal plane of lens L1 of focal length f1 = 40 mm . As the Gaussian filter, we used a Bull's eye type apodizing filter on the market (Edmund Optics Inc.). The Gaussian-amplitude wave was focused through the lens L1 onto an object placed in the front focal plane of the lens L1, where the full width at 1/e maximum of the focused beam measured about 32 μm . The object was mounted on a u − v translation stage to implement the scanning process, where a raster scan of 32 × 32 positions with a step size of 50 μm was implemented. Since the step size is larger than the beam width, the lateral resolution is obtained by two times the step size (i.e., 100 μm) as described in Section 2. Lens L2 of focal length f2 = 200 mm produced a Fourier transform of the complex amplitude of the light transmitted through the object. The 2-D Fourier intensity data at each position of the object were measured by a CCD detector (a Point Grey Grasshopper: 480 × 360, 14 bit, 18.16 μm pixel pitch). The influence of noise on the measured data was reduced by convolving them with a Gaussian window function, and a constant bias component due to the noise of the CCD was subtracted from the
measured data. According to the procedure in Section 2, series of intensity data at three points of coordinates ( 0, 0), ( c, 0), and
( 0, c ) in the Fourier plane were extracted from the 32 × 32 2-D data measured by the scanning process, where c was set to be the position of the eighth CCD's pixel from the origin (i.e., c = 8 × 18.16 μm ). As the examples of the experiments, the reconstructions of two types of objects are shown: (1) a converging lens with focal length f = 250 mm , and (2) a plastic plate for scratch standards (Edmund Optics Inc.). We first show the reconstruction of the former object. Figs. 3 and 4 show the reconstruction of a portion around the center of the converging lens. Figs. 3(a)–(c) are the intensity data obtained at three points of coordinates ( 0, 0), ( c, 0), and ( 0, c ) in the Fourier plane, which imply intensity distributions of truncated correlation functions between the object and the probe beam. The modulus and phase of the reconstructed object from the data in Fig. 3 by the phase retrieval method in Section 2 are shown in Figs. 4(a) and (b), respectively. Note that Fig. 4(b) is represented by an unwrapped one, because the reconstructed phases were beyond plus or minus π limits. In Fig. 4(c) [or (d)], the blue and the
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red curves show the cross-sectional profiles of the reconstructed modulus and phase along the horizontal [or vertical] dashed black lines in Figs. 4(a) and (b), respectively. The dotted curves in Figs. 4 (c) and (d) show the theoretical phase distributions calculated from the focal length f = 250 mm . There are disturbances at both ends of the reconstructed moduli and phases in Figs. 4(c) and (d), which are due to the errors in the phase calculations from the truncated correlation data in Fig. 3. It is appropriate to eliminate the reconstructed data within a somewhat larger extent than the full width of the probe beam from each end of the reconstruction area. Thus, if the data of two sampling points at each end of the reconstructed figures are ignored, it is found that the reconstructed modulus and phase are very close to the ideal form (i.e., a flat shape) and the theoretical curve, respectively. In Figs. 4 (c) and (d), the root-mean-squared errors between the reconstructed phases and the theoretical ones are 0.416 rad and 0.259 rad, respectively, except for two sampling data points at each end. There is a dig at the lower left of Fig. 4(a), which is a defect in the surface of the lens. Next we present the reconstruction of a plastic plate for scratch standards, where a scratch of 160 μm on the plate is used as an object. Figs. 5 and 6 show the reconstruction of a portion around the scratch. Figs. 5(a)–(c) are the intensity distributions obtained at three points of coordinates ( 0, 0), ( c, 0), and ( 0, c ) in the Fourier plane where the same position c as in Fig. 3 was used. Figs. 6(a) and (b) show the modulus and phase of the reconstructed object from the data in Fig. 5. The cross-sectional profiles of the reconstructed modulus and phase along the horizontal [or vertical] dashed black lines in Figs. 6(a) and (b) are shown in Fig. 6(c) [or (d)] by the blue and the red curves, respectively. From the data of Fig. 6(c), the width of the scratch was estimated at about 150 μm, which is close to the catalog one (160 μm) of the product. To confirm the variation of the reconstructed phase, we investigated the phase distribution of light transmitted through the scratch of the plate by using the Fourier transform holography (FTH) [28]. Fig. 7 shows the object reconstruction by the FTH, where we made an object by covering the same plate in a shading film with a hole of 0.1 mm and a window of 1.6 mm × 1.2 mm , and used the object illumination by the same diode laser as in Figs. 2. Figs. 7(a) and (b) show the modulus and phase of the inverse Fourier-transformed function from a measured hologram. In Fig. 7(c) [or (d)], the blue and the red curves show the cross-sectional profiles of the modulus and phase along the horizontal [or vertical] black dashed lines in Figs. 7(a) and (b), respectively, where the data points are 168 for the interval of 1.6 mm. Comparing two phases along the horizontal lines in Figs. 6(c) and 7 (c), the variation of the phase in and around the scratch in Fig. 6(c) is in close agreement with one in Fig. 7(c) except for the difference of data points. Note that the difference between two phases cannot be evaluated quantitatively here, because it is very difficult that the positions of the cross-sections of two phases reconstructed by different experiments of each other are set to be the same. A comparison between two vertical phases in Figs. 6(d) and 7(d) shows that a tilt to the right in Fig. 6(d) is appeared. This is due to a linear phase induced by a slightly vertical inclination of the scratch plate set up at a holder in the experiment. On the other hand, there is the marked difference between two moduli in Figs. 6 and 7. As seen in Fig. 7(a), that disturbance of the reconstructed modulus by the FTH can be considered the fringe noise due to the interference between reflected lights in the plate. Since our method is noninterferometic phase retrieval, such fringe noise does not appear in the object reconstruction.
4. Conclusions We have experimentally demonstrated the object reconstructions using a scanning phase-retrieval method with a known
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Gaussian-amplitude probe beam, which is a noninterferometric and noniterative method. The experimental results of this method have presented good reconstructions of the complex-valued objects without a reference wave. In this method, the area of the object reconstruction can be easily enlarged by increasing the number of scanning steps. Although a Gaussian filter is needed here to produce a Gaussian-amplitude probe, the measurement system is very simple and free from interferometric noises in contrast to interferometric techniques, and also the speed of the calculation of phase retrieval from measured data in this method is faster than that in iterative algorithms since our method is based on an analytic solution to the phase retrieval. The resolution in the present experiments is limited by a larger one of the two parameters that are a scanning-step size and a probe-beam width in the object plane. However, it is possible that the resolution is improved using higher-frequency intensity data than those at the measurement points of the present experiments, regardless of magnitudes of the two parameters. The details of such an improvement are currently being studied.
Acknowledgments This research was supported by a Grant-in-Aid for Scientific Research (Grant no. 26390081) from the Japan Society for the Promotion of Science.
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