Dualwavelength holography has a better axial range than singlewavelength holography, allowing unambiguous phase imaging. Partial coherence sources reduce coherent noise, resulting in improved reconstructed images. We measured a ballgrid array using dualwavelength holography with partial coherence sources. This holography method is useful for measurement samples that exhibit coherence noise and have a step height larger than the single wavelength used in holography.
I. INTRODUCTION
Holography records the phase modulation of light reflected or transmitted from a projected object onto a photosensitive plate in the form of interference. A reference and an object beam are required, and an interference pattern is generated as a result of the combination of the two beams. Previously, interference patterns were recorded on film plates. However, chargecoupled device (CCD) and complementary metaloxidesemiconductor (CMOS) technology are now widely used to capture images, and the holograms are then reconstructed on computers. Yaroslavskii and other researchers first proposed hologram reconstruction on the basis of numerical values in the 1970s
[1

5]
. Ounral and Scott used numerical reconstruction to measure the size of a particle after improving the reconstruction algorithm
[6]
. This method of digital recording and reconstruction of a numerical hologram is known as digital holography
[7

10]
. Digital holography has many advantages; for example, it does not require any chemical processing because the reconstructed image can be observed easily on a computer monitor, and numerical data can be obtained for threedimensional (3D) objects
[11
,
12]
.
However, the phase information becomes ambiguous and causes phase wrapping when the phase change exceeds 2π. Dualwavelength holography has been suggested as a way of overcoming this ambiguity
[13

16]
. Dualwavelength holography is a technique that extends profile measurements deeper than singlewave holography. Digital holography usually uses a laser, a highcoherence source. The coherent beams are very sensitive to any defect in the optical paths and are affected by coherence noise, which severely reduces the optical quality of the resulting reconstructed images. To eliminate the coherence noise problem, partial coherence sources are used in digital holography. The partial coherence sources are obtained from filtered lightemitting diode (LED) sources or from a laser source with deliberately decreased coherence. These partial coherence sources improved the reconstructed image and reduced coherence noise in digital holography
[17
,
18]
.
In this study, we measured a ballgrid array (BGA) using a dualwavelength digital holography technique. We chose to use partial coherence sources because the surface of the BGA was rough and its height was larger than the source wavelengths. We also used a reference conjugated hologram (RCH) for a phasecompensated phase map in dualwavelength digital holography.
II. MODEL
 1. Hologram Recording and Reconstruction
In the hologram recording process, a plane reference wave (
R
) and a diffusively reflected 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]
. The digital holographic image can be recorded using a blackandwhite CCD camera. The digital hologram
I_{H}
^{(k. l) }
is an
N×N
array resulting from the twodimensional (2D) sampling of
I_{H}
^{(x, y)}
by the CCD camera. It is given by
where
k
and
_{l}
are integers,
L
×
L
denotes the area of the CCD chip, and Δ
x
and Δ
y
indicate the pixel size of the CCD.
In classical optical holography, the object wave can be reconstructed by illuminating the processed hologram with a plane wave similar to that used in the recording process. Looking through the hologram, one observes a virtual image. If a screen is placed at a distance
d
behind the hologram, a real image is formed on it. Mathematically, the amplitude and phase distributions in the plane of the real image can be found using the FresnelKirchhoff integral
[1
,
2]
. If a plane wave illuminates the hologram with an amplitude transmittance of
I_{H}
^{(x, y)}
, the FresnelKirchhoff integral yields a complex amplitude, Ψ(
ζ, η
), in the real image plane
where λ is the wavelength, and
d
is the reconstruction distance. (
ξ, η
) are the coordinates of reconstruction plane, and pixel sizes in the reconstruction plane are
and
From Eq. (3), the FresnelKirchhoff integral can be considered 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 phasecontrast image is obtained by calculating the argument
The real 3D information is acquired by phase unwrapping with the phasecontrast image. Usually the lasers are used in DHM and are sensitive to any defect in the optical paths, and are affected by coherent noise. These coherent noises degrade the reconstructed image quality. To eliminate the coherent noise, partial coherent sources are suggested in DHM. The partial coherent sources are obtained by laser and RGG
[18
,
22]
.
 2. Dualwavelength Holography and Phase Aberration Compensation
Consider two singlewavelength phase maps,
Φ
_{1}
and
Φ
_{2}
, with an optical path difference, OPD. The beat wavelength λ
_{12}
for λ
_{1}
and λ
_{2}
is given by
The value of λ
_{12}
can be increased by selecting values of λ
_{2}
and λ
_{2}
. that are closer to each other. The phase map for λ
_{12}
is obtained by subtracting one singlewavelength phase map from the other. This map is called a “coarse map”
[14]
. To make a corrected
Φ
_{12}
phase map, the aberrations of each phase map,
Φ
_{1}
and
Φ
_{2}
, should be compensated. We use reference wave (R) and object wave (O) in DHM. To compensate the phase aberrations we could introduce another wave O
_{o}
, corresponding to the object wave without sample
[21]
. If there are phase aberrations in reference wave and object wave, we could introduce them in waves like Eq. (7).
Where k
_{x}
, k
_{y}
define the wave propagation direction and W
_{R}
, Wo
_{o}
are the phase difference between perfect reference wave and object wave. The reconstructed wave front is given by Eq. (8).
Here
and
is the filtered hologram, and W
_{O}
_{o}
–W
_{R}
describing the aberration. We could compensate the aberration, W
_{O}
o
–WR, by introducing the reference conjugated hologram (RCH). To compensate the aberration we introduce the filtered reference hologram (I↓ H↑(R, F) = R↑*O↓o), which is the blank hologram. We define
as the conjugate phase of Eq. (8),
The multiplication of
with filtered hologram results in a suppression of the aberration terms
[21]
,
Eq. (10) shows that the total aberration can be compensated
[19

21]
.
III. EXPERIMENTAL RESULTS
Figure 1
shows a schematic of a dualwavelength transmission holographic microscope. The basic experimental setup is similar to that of a Michelsontype interferometer. A 668nm laser diode and 677nm laser diode were used as the light sources. These coherent beams were focused on the scattering surface of a rotating ground glass (RGG). The RGG was rotated in its plane by a motor. The light
Schematic experimental setup for reflectiontype dualwavelength digital holographic microscope. LD 1 and LD 2: laser diodes; BS: beam splitter; M: mirror; RM: reference mirror; L1, L2, L3, and L4: lenses; OL1 and OL2: microscope objective lenses; S: sample; RGG: rotating ground glass; CCD: chargecoupled device.
transmitted by the RGG was scattered by its rough surface to create a speckle field that varied with the ground glass rotation. The scattered surface of the RGG was placed in the front focal plane of the lens, L2. Lenses L2 and L3 were used for beam expanding. We used a CCD camera (Sony IPX1M48L) to record the holograms. The pixel size and the number of pixels were 7.4 μm × 7.4 μm and 1024 ×1024, respectively. The CCD was placed in the focal plane of lens 4 (L4). We used a BGA with a height of 15 μm as our sample.
Figure 2
shows the partial coherence effect for the BGA.
Figures 2
(a) and (b) present a coherent source image and hologram, (d) and (e) are partial coherence source image and hologram, respectively.
Figure 2
(c) and (f) are expanded image of dotted square part in
Fig. 2
(b) and (e).
Figures 2
(a) (b) (c) and (d) (e) (f) were obtained without and with the RGG in
Fig. 1
, respectively. We have used the 50X microscope object lens for the image in
Fig. 2
(a) and (d), and
Fig. 2
(b), (e) are from 20X microscope object lens. The surface of the BGA was rough, which caused coherent noise. We can show that the noise degraded the image and the hologram in
Figure 2
(a), (c) and the partial coherence sourceimproved image and hologram in
Figure 2
(b), (d). The partialcoherencesource method using the RGG resulted in a spatial lowpass filtering in digital holography
[18]
.
Figure 3
shows the dualwavelength holography experimental results.
Figure 3
(a) and (d) are the holograms created by laser diode 1 (668 nm) and laser diode 2 (677 nm), respectively.
Figure 3
(b) and (e) show the expanded hologram image of
Fig. 3
(a) and (d).
Figure 3
(c) and (f) show the reconstructed phase map of each hologram. These are not aberration compensated. Each phase map has aberrations, tilt, and centering. We could not extract the right sample phase information without compensating for the tilt aberration. We took the RCH to compensate for the tilt and centering aberration, and detail process are in
Fig. 4
.
Figure 4
(a) is
Image and holograms created using coherent (a), (b), (c) and partial coherence sources (d), (e), (f).
Reconstructed BGA phase image. (a), (d) holograms created by the 668nm source and 677nm source (respectively); (b),(e) expanded hologram image; (c), (f) reconstructed phase images of (a) and (d) without phase compensation.
Reconstructed BGA phase image and RCH. (a) holograms created by the 668nm); (b) FFT of (a); (c) filtered hologram; (d) reference; (e),(f) phase aberration compensated phase maps of Fig. 3. (a) and (d).
(a) reconstructed phase map of equilibrium wavelength; (b) 3D gray image; (c) height measurement.
the hologram, which is same in
Fig. 3
(a) and
Fig. 4
(b) is the FFT image of hologram (a). We can get the filtered hologram
as like
Fig. 4
(c) and
Fig. 4
(d) is the hologram without sample
Using
Fig. 4
(c) and (d) we can get the reconstructed phase map (
Fig. 4
(e)), which is phase aberration compensated.
Figure 4
(f) is another phase aberration compensated phase map using
Fig. 3
(d). We subtracted phase map
Fig. 4
(e) from phase map
Fig. 4
(f) to construct a phase map of the equilibrium wavelength (λ
_{12}
= 50248 nm). The result appears in
Figure 5
(a).
Figure 5
(b) shows the gray level of the 3dimensional in
Fig. 5
(a) and
Fig. 5
(c) is the height profile from the phase measured along the white line.
We measured the BGA height as 15±1.2 μm. The error is relatively large because the error was amplified in the subtraction process between phase maps. This amplification is well known. From these results, we could eliminate the 2π ambiguity using dualwavelength holography and reduce the coherent noise by partial coherence sources. Also, we could use the correct phase information by compensating for the aberration using a RCH.
IV. CONCLUSION
Dualwavelength holography is useful for obtaining profile measurements without including a 2π ambiguity. Using partial coherence sources reduced coherent noise. The BGA had a rough surface, which caused coherent noise. Its step height was larger than the visible wavelength, which resulted in a 2π ambiguity in the digital holography. We used a dualwavelength holography method with partial coherence sources to measure the BGA’s height and shape. From experimental results, we showed that the dualwavelength holography method using partial coherence sources is useful in measurement samples that exhibit coherent noise and whose heights are larger than the single wavelength used in holography.
Acknowledgements
This work was supported by a research grant from Cheju National University in 2010.
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