In this paper, a method is proposed for simultaneously measuring the front and back surface profiles of transparent microoptical components. The proposed method combines a dualwavelength digital holographic microscope with liquids to record holograms at different wavelengths, and then numerically reconstructs the threedimensional phase information to image the front and back sides of the sample. A theoretical model is proposed to determine the surface information, and imaging of an achromatic lens is demonstrated experimentally. Unlike conventional interferometry, our proposed method supports nondestructive measurement and direct observation of both front and back profiles of microoptical elements.
I. INTRODUCTION
Holography is a technique for recording phase modulation using interference patterns formed when light is reflected or transmitted by an object onto a photoplate. The recording requires object and reference beams, the combination of which generates the interference pattern. In the past, holographic interference patterns were recorded on film plates; presently, however, chargecoupled devices (CCDs) or complementary metal oxide semiconductor (CMOS) cameras are commonly used, and a computer reconstructs the hologram. Yaroslavskii
et al
. proposed a method for numerical hologram reconstruction in the 1970s, and Ounral and Scott used numerical reconstruction to measure the size of a particle, after improving on the reconstruction algorithm
[1

6]
. This method of digitally recording and reconstructing a numerical hologram is known as digital holography
[7

10]
.
There are two types of digital holographic microscopes (DHMs): reflection and transmission. Typical optical components have two surfaces, front and back. However, transmission interferometers or DHMs cannot measure both threedimensional (3D) surface profiles simultaneously, because the output from the optical components includes integrated surface information. In reflection interferometry, the phase information of each surface is measured separately and then numerically reconstructed to provide 3D information about the sample. Thus reflection interferometry is used to measure both 3D surface profiles. However, these approaches rely on movable mechanisms and accurate position determination to obtain all of the phase information. Lowcoherence interferometry (LCI) could measure the surfaces by controlling the reference optical length as in optical coherence tomography, but LCI requires many images to obtain all of the phase information
[11
,
12]
.
In this paper we propose a method for simultaneously measuring the front and back surface profiles of transparent microoptical components, using a dualwavelength DHM
[13

17]
and liquids. We also propose a theoretical model for determining the surface information, and experimentally demonstrate the use of this model for an achromatic lens.
II. THEORETICAL MODEL
 2.1. Hologram Recording and Reconstruction
In the hologram recording process, a plane reference wave
R
and an object wave
O
interfere at the CCD. The hologram intensity is given by
where
R
* and
O
* denote the complex conjugates of the reference and object waves respectively
[1
,
2]
. A digital holographic image can be recorded using a monochromatic CCD camera. The digital hologram
I_{H}
(
k,l
) is an
N
×
N
array resulting from twodimensional sampling of IH (
x,y
) by the CCD camera, expressed as follows:
where
k
and
l
are integers,
L
×
L
denotes the area of the CCD chip, and Δ
x
and Δ
y
indicate the horizontal and vertical pixel sizes of the CCD.
If a screen is placed at a distance
d
behind the hologram, a real image is formed on the screen. Mathematically the amplitude and phase distributions in the plane of the real image can be calculated using the FresnelKirchhoff integral
[1
,
2]
. If a plane wave illuminates the hologram with an amplitude transmittance
I_{H}
(
x,y
), the FresnelKirchhoff integral yields a complex amplitude Ψ(
ξ,η
) in the real image plane as
where
λ
is the wavelength and
d
is the reconstruction distance. From Eq. (3), the FresnelKirchhoff integral can be considered as the Fourier transformation of the function
I_{H}
(
x,y
)exp[
iπ
(
x
^{2}
+
y
^{2}
)/
λd
] at the spatial frequencies
ξ
and
η
. Because Ψ(
ξ,η
) is an array of complex numbers, one can obtain an amplitudecontrast image using the intensity:
The quantitative phase image is obtained by calculating the argument
[18

21]
The real 3D information is acquired by phaseunwrapping the quantitative phase image.
 2.2. Optical Path Difference in A DualWavelength Digital Holographic Microscope with Liquids
Consider the opticalpath difference of a beam traveling through a transparent sample of refractive index
n_{L}
that is immersed in liquids with refractive indices
n
_{1}
and
n
_{2}
(see
Fig. 1
). The optical path length between planes A and B of the test surface can be described as
[22]
:
Schematic diagram of a doubly concave lens immersed in liquids.
where
Z_{0i}
and
Z_{i}
(
x,y
) represent the vertex and surface height, respectively, and
L_{i}
(
x,y
) is the distance traveled from the surface to the vertex (
i
= 1, 2). OPD(
x,y
) is the path difference between
OPL
and
OPL_{0}
. The planes
A
and
B
are parallel, so the distance between them is constant. This determines the relationship between
Z
and
L
:
Substituting Eq. (7) into Eq. (6) gives
Equation (8) includes the integrated surface information [
L_{1}
(
x,y
) +
L_{2}
(
x,y
)] of the sample, not information about the individual surface profiles. Equation (8), therefore, represents the coupled information of the surface profiles and can be modified to include the effect of different wavelengths as follows:
OPD_{i}
(
x,y
) (
i
= 1, 2) is the optical path difference at wavelength
λ_{i}
, while
n_{1i}
,
n_{2i}
and
n_{Li}
represent the refractive index of liquid and sample at wavelength
λ_{i}
, respectively. From Eq. (9) we find that the sets of information for individual surfaces,
L_{1}
(
x,y
) and
L_{2}
(
x,y
), are decoupled. The surface profile information in
L_{1}
(
x,y
) and
L_{2}
(
x,y
) can also be expressed as in Eq. (10):
The result above is obtained from the expression for the opticalpath difference with two wavelengths and two different liquids. Equation (10) presents the decoupled surface profile information, and shows that both surface profiles of a transmission optical component can be measured simultaneously using a dualwavelength DHM and two different liquids.
III. EXPERIMENT AND RESULTS
Figure 2
shows a schematic diagram of the experimental setup for a dualwavelength transmission holographic microscope with liquids, for measuring both surfaces simultaneously. The basic experimental setup is similar to that of a Mach Zehnder type interferometer. A 10mW HeNe laser (633 nm) and laser diode (532 nm) were used as the light sources. Water and a refractive index liquid (Cargille Laboratories) were used. The refractive index of the liquid was 1.6171 and 1.6317 at 633 and 532 nm, respectively, while that of water was 1.33168 and 1.33372 at 633 and 532 nm, respectively. Different liquids are filled as in
Fig. 3
.
Transmissiontype dualwavelength digital holographic microscope (DHM) (VN: variabledensity filter; BS: beam splitter; M: mirror; O: object wave; R: reference wave; BE: beam expander with spatial filter; ML: microscope objective lens; CCD: chargecoupled device).
Lens holder for filling liquids before and after the lens.
We used a CCD camera (Imperx) to record the holograms. The pixel size and the number of pixels were 7.4 × 7.4 μm and 1024 × 1024, respectively. Achromatic lenses (Edmund Optics) were used as the samples, each with a different radius of curvature (ROC) 26.82 and 61.38 mm; and refractive indices of 1.51508 and 1.51947 at 633 and 532 nm, respectively.
Figure 4
shows the experimental results for the achromatic lens from dualwavelength DHM with liquids.
Figures 4(a)
and
(b)
show the holograms at 532 and 633 nm respectively.
Figures 4(c)
and
(d)
show respectively the reconstructed quantitative phase image of each hologram with 3D graylevel surface profiles of the samples shown in
Figs. 4(e)
and
(f)
for 532 and 633 nm. To acquire the 3D graylevel images, we used Flynn’s unwrapping method
[23]
.
Experimental results from dualwavelength DHM with liquids. Holograms for wavelengths of (a) 532 nm and (b) 633 nm. Reconstructed quantitative phase image for (c) 532 nm and (d) 633 nm. 3D graylevel images for (e) 532 nm and (f) 633 nm.
Figure 5(a)
illustrates the gray levels measured along the dashed lines in
Fig. 4(e)
and
Fig. 4(f)
. The squares and circles correspond to the results for wavelengths of 532 and 633 nm, respectively. The difference between the profiles for each wavelength was very small, as predicted by Eq. (9). The OPD includes information about the lens shape and refractive index; however, the OPD in
Fig. 5(a)
is not the actual height. The OPD difference between 532 and 633 nm is caused by the difference in the refractive indices. We calculated the front and back profiles using Eq. (10) and data from
Figs. 4(e)
and
(f)
. The calculated results for the 3D color image from each surface are shown in
Figs. 5(b)
and
(c)
. Note that the opticalpath difference between the wavelengths, as well as noise, affected the calculated image. In
Fig. 5(d)
we show the spatial profiles indicated by the dashed lines in
Figs. 5(b)
and
(c)
; the squares and circles correspond to the raw data, while the lines represent the fitted data. The fitted data show that the profile of each surface is different, and the surface information is decoupled. Next, we calculated the ROC from the calculated profiles. The ROC can be found from the completely reconstructed surface by fitting to a bestfit sphere using singularvalue decomposition (SVD). Any spherical surface along the zaxis can be expressed as
Surface profiles of the achromatic lens. (a) Opticalpath difference (height) for 532 and 633 nm. (b) and (c) 3D color images of the front and back surface profiles. (d) Calculated and fitted profiles of each surface for the dashed lines in (b) and (c).
where
a
and
b
are the coefficients of the curve. The ROC is given by
. The calculated ROC values for the surfaces were 28.47 ± 1.22 and 60.54 ± 1.46 mm. These values were reasonably close to the design values, indicating that the proposed dualwavelength DHM with liquids can be used for simultaneous measurement of the front and back profiles of a sample. Thus, unlike the many techniques available for measuring the surface profiles
separately
, the method described in this research provides the means to measure these profiles
simultaneously
. In this study we assume that the refractive indices of the studied materials are known; in future work we will investigate the case in which the refractive index of the sample is not known, or has a distribution.
IV. CONCLUSION
Digital holographic microscopy can be used to measure opticalpath differences with a high degree of accuracy. The opticalpath difference depends on the refractive index and morphology of the samples. Normally optical components have a twosided morphology, with front and back surfaces. We cannot determine the individual surface profiles with normal transmission DHM, because the output wave from the object presents integrated information. This study proposed a new experimental method and models for determining the individual surface profiles simultaneously using transmission DHM and liquids. We integrated dualwavelength DHM with liquids and demonstrated a new method for determining the individual surface profiles of an achromatic lens, which has different ROC surfaces. The proposed method involves recording holograms at different wavelengths and then numerically reconstructing the phase information of the front and back 3D information of the sample. The proposed method can be applied to normal transmission interferometry for determining the 3D surface profiles of transmitted optical components.
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