- Email: [email protected]
Real-time dual-wavelength digital holographic microscopy based on polarizing separation
Optics Communications 285 (2012) 233–237
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Real-time dual-wavelength digital holographic microscopy based on polarizing separation D.G. Abdelsalam a, b, Daesuk Kim a,⁎ a b
Division of Mechanical System Engineering, Chonbuk National University, Jeonju 561–756, Republic of Korea Engineering and surface metrology lab., National Institute of Standards, Tersa St., El haram, El Giza, Egypt
a r t i c l e
i n f o
Article history: Received 31 August 2011 Accepted 21 September 2011 Available online 5 October 2011 Keywords: Digital holography Phase retrieval Interference microscopy
a b s t r a c t We report on a manifold advanced dual-wavelength digital holographic microscopy (DHM) configuration with a real-time measurement capability. The proposed configuration based on a polarizing separation scheme can be used for microscopic imaging polarimetry as well as dual wavelength digital holographic microscopy. In this paper, we show the feasibility of the proposed scheme by conducting the dual wavelength DHM experiments on a sample with a step height of 1.34 μm nominally. An averaging technique is treated and three-dimensional (3D) topographic measurements are presented. The results obtained by the proposed polarization separation based single shot DHM approach shows it can provide a real time solution for measuring 3D profile information of small objects with excellent accuracy. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Digital holography has been heavily developed over recent years because of newly-available high-resolution CCD cameras and advances in digital and automated image processing techniques [1]. Performing Fourier transforms and spectral filtering without the need for additional optical components and the simplicity by which reconstruction data may be interpreted quantitatively has given advantages of digital holography over conventional holography. Great advantages of off-axis digital holography over in-line holography include the separation of amplitude and phase information [2] and noise reduction [3]. Optical digital holography applications are manifold and include biological microscopy [4–9], metrology of microstructures [10–12], and deformation and vibration analysis [13–15]. By improving nanotechnology, the dimension of structures and functional devices is decreasing in size. Digital holographic microscopy (DHM) allows to investigate the shape and the displacements of objects [16–17] in the nanoscale with high resolution and it highlights its capability as a promising 3D imaging technique. In this paper, an averaging technique is treated and three-dimensional (3D) topographic measurements are presented, obtained by a dualwavelength DHM setup on a 1.34 μm nominal step height.
⁎ Corresponding author at: Division of Mechanical System Engineering, Chonbuk National University, Jeonju 561-756, Republic of Korea. Tel.: + 82 63 270 4632; fax: + 82 63 270 2388. E-mail address: [email protected] (D. Kim). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.09.044
The presented configuration could be called a digital holographic microscope and consists of a microscope objective (MO) that produces a magnified image of the sample that is used as an object for the hologram creation. The principle of the setup is to separate each wavelength beam pair in different reference arms, while combining them in an object arm. The main great advantage of the proposed configuration is that it can be used for single-shot dual-wavelength off-axis interferometry, phase shifting interferometry, and imaging polarimetry without any major modification. Furthermore, it is more compact compared to the previous systems [18]. In this study, we show the ability of the proposed configuration to investigate very small objects by using MO that produces a magnified image (a phase aberration associated with the use of a MO is corrected by a digital method [3]). The investigated object is a sample of a 1.34 μm nominal step height. In the proposed configuration, two HWPs are used to change the polarization direction of light beam coming from one arm to match the polarization direction of light beam coming from the other arm for both wavelengths. Such polarization direction matching for each wavelength is necessary for producing holograms for both wavelengths. The independent phase maps are subtracted and a final phase map for the beat-wavelength is obtained and converted to height map. The harmful noises associated with the height map are treated with an averaging technique. The results obtained by the proposed scheme have been in excellent agreement with the nominal value of the used sample. The main great advantages of the proposed configuration over the previous study are mentioned in details in reference [19]. As far as we know, this is the first time that real-time high resolution DHM using a new configuration shown in Fig. 1 is used to investigate very small objects with excellent accuracy.
234
D.G. Abdelsalam, D. Kim / Optics Communications 285 (2012) 233–237
Fig. 1. Schematic diagram of the proposed single shot dual-wavelength digital holographic microscopy (DHM) system based on polarizing separation.
2. Hologram reconstruction The interference between the object wave O1 and the reference wave R1 for p-polarization generates a hologram at wavelength λ1 = 635 nm. The interference between the object wave O2 and the reference wave R2 for s-polarization creates a hologram at wavelength λ2 = 635 nm. The hologram intensity resulting from both holograms at the same time real-time monitoring (single camera acquisition) for λ1 and λ2, can be written as follows: 2
2
2
2
IH ðx; yÞ ¼ jO1 j þ jR1 j þ jO2 j þ jR2 j |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} zero order
ð1Þ
þ R1 O1 þ R1 O1 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} þ R2 O2 þ R2 O2 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} λ1 λ2 Where IH is the hologram intensity, x and y are integers, and * denote the complex conjugate. In Eq. (1), the first four intensity terms are of zero order, and the last four are the interference terms for λ1 = 635 nm and λ2 = 635 nm , respectively. As in reference [20] the off-axis geometry is considered. For the p-polarization generated at wavelength λ1 = 635 nm, the Mirror 1 is oriented such that the reference wave R1 reaches the CCD camera with a small incidence angle θ1 (see inset in Fig. 1) with respect to the propagation direction of the object wave O1. Likewise, for the s-polarization generated at wavelength λ2 = 635 nm, the Mirror 2 or Mirror 3 is oriented such that the reference wave R2 reaches the CCD camera with a small incidence θ2 with respect to the propagation direction of the object wave O2. A digital hologram is recorded by a black and white CCD camera with 4.65 μm × 4.65 μm pixel size. The digital hologram IH(k,l) is an array of N × N = 500 × 500 results from the two-dimensional sampling of IH(y,x) by the CCD camera: IH ðk; lÞ ¼ IH ðx; yÞrect
x y ; × L L
N=2
∑
N=2
∑ δðx−kΔx; y−lΔyÞ;
intervals in the hologram plane (pixel size) Δx = Δy = L/N. Holographic microscopy has been proposed in various configurations [3]. Fig. 2 shows the optical arrangement configuration of the holographic microscopy. The MO produces a magnified image of the object, and the hologram plane (the CCD plane) is located between the MO and the image plane, at a distance d from the image. The numerical reconstruction process is based on calculating the Fresnel diffraction pattern of the captured hologram. The result of the calculation is an array of complex numbers called reconstructed wave front Ψ, which represents the complex amplitude of the optical field in the observation plane 0ξη. The distance between the hologram plane 0xy and the observation plane is defined by the reconstruction distance d. In holographic microscopy, image focusing occurs when the reconstruction distance is equal to the distance between the CCD and the image during hologram recording (d in Fig. 2). The reconstructed wave front Ψj(m.n) for wavelength λj in the convolution formulation, in the observation 0ξη, was obtained by computing the discrete Fresnel integral of the digitized hologram IH(k,l) and is given as follows: "
# iπ 2 2 2 2 m Δξj þ n Δηj ; Ψj ðm; nÞ ¼ A exp λj dj (
"
× FFT IH ðk; lÞ exp
iπ 2 2 2 2 k Δx þ l Δy λj dj
ð3Þ
#) m;n
ð2Þ
k¼−N=2l¼−N=2
where k,l are integers, L × L = 2.325 mm × 2.325 mm is the region of interest, δ is a Dirac delta function, and Δx, Δy defines the sampling
Fig. 2. Digital holographic microscope, optical configuration.
D.G. Abdelsalam, D. Kim / Optics Communications 285 (2012) 233–237
pffiffiffiffiffiffiffiffi where m,n,k,l are integers (–N/2 ≤ k,l,m,n ≤ N/2) , i ¼ −1, j = 1,2, and FFT is the fast Fourier transform operator. Δx and Δy define the sampling intervals in the hologram plane. The sampling intervals Δξj and Δηj in the observation plane are related to the size of the CCD (L) and to the distance dj by the following relation: Δξj ¼ Δηj ¼ λj dj =L:
ð4Þ
The MO produces a curvature of the wave front in the object arm. This deformation may affect the phase of the object wave. This phase aberration must be corrected to provide accurate results. The phase aberration can be corrected by multiplication of an array of complex numbers called digital phase mask φj(m,n) is calculated as follows: " # −iπ 2 2 2 2 m Δξj þ n Δηj ; λj Dj
ð5Þ
φj ðm; nÞ ¼ exp
where Dj is a parameter that must be adjusted to compensate the wave-front curvature and expressed as follows: 1 1 ¼ Dj dj
1þ
! d0 : dj
ð6Þ
The complete expression of the reconstruction algorithm becomes " Ψj ðm; nÞ ¼ Aφj ðm; nÞ exp 8 <
# iπ 2 2 2 2 m Δξj þ n Δηj λj dj "
i iπ 2 2 2 2 k Δx þ l Δy × FFT RDj ðk; lÞIH ðk; lÞ exp : λj dj
ð7Þ
) : m;n
Since Ψj(m, n) are an array of complex numbers, an amplitudecontrast image and a phase-contrast image can be obtained by using the following intensity [Re(Ψj) 2 + Im(Ψj) 2] and the argument arctan [Re(Ψj)/Im(Ψj)], respectively. Where (RDj(k, l)) is called a digital reference wave. If we assume that a perfect plane wave is used as a reference for hologram recording, the computed replica of the reference wave RDj can be represented as follows: h i RDj ðk; lÞ ¼ ARj exp i 2π=λj kxj kΔx þ kyj lΔy ;
ð8Þ
where, ARj is the amplitude, λj is the wavelength of the laser source, and kxj and kyj are the two components of the wave vector that must be adjusted such that the propagation direction of RDj matches as closely as possible with that of the experimental reference wave. By using this digital reference wave concept, we can obtain an object wave which is reconstructed in the central region of the observation plane for each wavelength separately. In order to overcome the phase ambiguity produced from single wavelength approach, a synthetic beat-wavelength is used and expressed as follows: λ −λ1 x Φ ¼ arg O1 O2 ¼ ϕ1 −ϕ2 ¼ 2πx 2 ¼ 2π ; Λ λ1 λ2
ð9Þ
where, x is OPD (Optical Path Difference), which means twice of the topography for reflection scheme. ϕj is the reconstructed phase for wavelength λj and Λ is the synthetic beat wavelength defined as follows: λ λ Λ¼ 1 2 : λ2 −λ1
phase obtained by Eq. (10) can provide a solution for much higher pattern structures by overcoming the phase ambiguity. 3. Experimental results The optical arrangement is depicted in Fig. 1, with the laser diodes sources at λ1 = 635 nm and λ2 = 675 nm, yielding a synthetic wavelength Λ = 10.72 μm. The key concept of the proposed scheme is to separate each wave wavelength beam pair in different reference arms, while combining them in an object arm (the theory has been explained in details in reference [19] for macro scale (without MO)). In this study, we show the ability of the proposed configuration to investigate very small objects by using MO that produces a magnified image. After reflection on the sample as shown in Fig. 1, both collinear object wavefronts are collected by the MO (infinitycorrected), and the object images are formed by an imaging lens about 50 mm behind the CCD plane. A microscope objective (MO) of magnification ×10 with a NA of 0.25 and working distance WD of 10.6 mm is used to magnify the image of the sample that is used as an object for the hologram creation. The CCD camera is a standard 8 bits black and white CCD camera with 4.65 μm pixel size. Each reference arm is adjusted carefully to match the optical path length of the corresponding object arm, in order to create an interference pattern on the CCD for both wavelengths. By tilting the mirror 1 for the first wavelength reference beam and the pair of mirror 2 and mirror 3 for the second one, one can find finely tune each k-vector incident upon the CCD camera. On the other hand, each wavelength hologram fringes can be independently tuned both in spatial frequency and orientation. Finally, the CCD camera records the digital hologram that result from the interference between the object wave O1 and the reference wave R1. And also, the interference between the object wave O2 and the reference wave R2 is recorded simultaneously. The two filtered holograms are reconstructed and a final phase map for the beat-wavelength is obtained and converted to height map. Fig. 3 shows the experimental results obtained through the proposed dual-wavelength off-axis digital holographic microscopy scheme. The surface under test is an object with a nominal height 1.34 μm (see the white rectangle in Fig. 3(a)). The general procedure which is needed for measuring the complex object wave in dual-wavelength off-axis digital holographic microscopy is shown in Fig. 3. Both amplitude and phase information of the object for the two wavelengths can be obtained with a single hologram. Fig. 3 represents the experimental results of application of the 2D-FFT based spatial filtering method for each wavelength separately. After the spatial filtering step [21–24], the digital reference wave RDi is used for the centering process. Then, the final reconstructed object is obtained by adjusting the values of kx and ky,dj, and Dj for each wavelength separately. The digital reference wave used in the calculation process should match as close as possible to the experimental reference wave. This has been done in this paper by selecting the appropriate values of the two components of the wave vector kx = 0.1218565 mm− 1 and ky = −0.12487275 mm−1, d0 = 0.99 mm, d = 49 mm, and D = 212 mm for the hologram captured at λ1 = 635 nm and kx = 0.15987375 mm− 1 and ky = −0.1479725 mm− 1, d0 = 0.99 mm, d = 50 mm, and D = 217 mm for the hologram captured at λ2 = 675 nm. Once the two phase maps are obtained for each wavelength, a phase map can be calculated on the synthetic beatwavelength. The three-dimensional (3D) surface profile height h can be calculated directly as follows: h¼
ð10Þ
We can see that the smaller the difference between the two wavelengths, the larger the synthetic wavelength, typically within the range of micrometers to millimeters. The corresponding synthetic
235
Φ Λ: 4π
ð11Þ
Fig. 3(a) shows the investigated sample (see the white rectangle) of a nominal step height of 1.34 μm. The single-shot dual-wavelength off-axis hologram of the investigated sample (white rectangle) is shown in Fig. 3(b). The filtering windows A and B as shown in Fig. 3(c)
236
D.G. Abdelsalam, D. Kim / Optics Communications 285 (2012) 233–237
Fig. 3. Sequential reconstruction steps of the spatial filtering based phase contrast dual- wavelength off-axis holography (a) original phase object (see the white rectangle in Fig. 3(a)), (b) off-axis hologram, (c) Fourier transformed spatial frequency domain data, (d)–(e) reconstructed amplitude and phase map for λ1 = 635 nm, (f)–(g) amplitude and phase map for λ2 = 675 nm, (h) object phase map on the synthetic beat-wavelength, and (i) 3D surface profile of the selected rectangle of (h).
for λ1 = 635 nm and λ2 = 675 nm, respectively, have been chosen carefully. The reconstructed amplitude and phase map for λ1 = 635 nm and for λ2 = 675 nm are shown in Fig. 3(d–e) and Fig. 3(e–f), respectively. The phase map on the synthetic beat-wavelength is shown in Fig. 3(h). The three dimensional (3D) surface profile of the
selected rectangle of Fig. 3(h) is shown in Fig. 3(i) after converting to height by using Eq. (7). Based on the measured height in Fig. 3(h), the average step height has been estimated to be around 1.34 ± 0.9 μm. The uncertainty in measurement is very high due to the non-coherent noise associated with the profile. The noise shown in Fig. 3(h) is non
Fig. 4. Surface height inside the white rectangular (6 pixels profiles) in Fig. 3(h) (a) 2D, (b) 3D of (a), (c) 2D surface profile along the selected line of (a), and (d–f) are the modified of (a–c) after applying the rms technique, respectively.
D.G. Abdelsalam, D. Kim / Optics Communications 285 (2012) 233–237
coherent noise or inevitable noise may come from read out noise (noise that does not vary with signal), fixed pattern noise (pixel to pixel sensitivity variation), and inherent noise due to the spatial filtering window and 2D-FFT processes. The noise vibration shown in Fig. 3(h) is not coherent but varies sinusoidally; thus, it is more logical to express the height error as the root mean square (rms) value of the disturbed height [25]. The rms height can be expressed as [26–28]: hrms
rhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i P 2 ¼ h− h =ðM−1Þ;
ð12Þ P
where h denotes the height distribution, h denotes the mean values, M represents the pixel number of row of calculation region. The proposed rms method can provide a satisfied solution especially for measuring objects having high abrupt height difference. Fig. 4(a) shows a two-dimensional (2D) surface height inside the white rectangular (6 pixels profiles) in Fig. 3(h). The 3D surface height profile of Fig. 4(a) is shown in Fig. 4(b). The two dimensional (2D) surface profile along the selected line of Fig. 4(a) is shown in Fig. 4(c). Fig. 4(d–e) shows the modified height profile of Fig. 4(a–c) after applying the rms technique. Based on the measured height in Fig. 4(f), the average step height has been estimated to be around 1.34 ± 0.02 μm i.e. the uncertainty in measurement is reduced drastically by the order of more than 90%. 4. Conclusion In this paper, a dual-wavelength digital holographic microscopy (DHM) technique with a manifold advanced configuration has been presented. An averaging technique has been treated and 3D topographic measurements have been presented, on a 1.34 μm nominal step height. The experimental results have been in excellent agreement with the nominal value of the used sample. The results obtained by the proposed polarization separation based single shot DHM approach shows it can provide a real time solution for measuring 3D profile information of small objects with excellent accuracy. Although the microscopic imaging polarimetry capability of the proposed scheme has not been touched in this paper in detail, we claim that the proposed scheme can be used for various microscopic imaging polarimetric measurement applications required for liquid crystal display (LCD) and semiconductor industry.
237
Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (2011– 0002487). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
G. Shen, R. Wei, Optics and Lasers in Engineering 43 (10) (2005) 1039. M.K. Kim, Optics Express 7 (9) (2000) 305. E. Cuche, P. Marquet, C. Depeursinge, Applied Optics 38 (34) (1999) 6994. G.L. Chen, C.Y. Lin, M.K. Kuo, C.C. Chang, Applied Physics B 90 (3–4) (2008) 527. B. Kemper, G. von Bally, Applied Optics 47 (4) (2008) A52. W. Xu, M.H. Jericho, I.A. Meinertzhagen, H.J. Kreuzer, Proceedings of the National Academy of Science USA, (PNAS), 2001, p. 11301. G. Popescu, L.P. Deflores, J.C. Vaughan, K. Badizadegan, H. Iwai, R.R. Dasari, M.S. Feld, Optics Letters 29 (21) (2004) 2503. P. Marquet, B. Rappaz, P.J. Magistretti, E. Cuche, Y. Emery, T. Colomb, C. Depeursinge, Optics Letters 30 (5) (2005) 468. D. Carl, B. Kemper, G. Wernicke, G. von Bally, Applied Optics 43 (36) (2004) 6536. X. Song, Z. Tang, H. Wang, Proceedings of IEEE, Control and Decision Conference (IEEE), 2009, p. 4656. L. Xu, X. Peng, J. Miao, A.K. Asundi, Applied Optics 40 (28) (2001) 5046. G. Coppola, S. De Nicola, P. Ferraro, A. Finizio, S. Grilli, M. Iodice, C. Magro, G. Pierattini, Proceedings of SPIE 4945 (2003) 71. Z. Fan, H. Pang, W. Wang, C. Ning, F. Guo, Proceedings of IEEE International Congress on Image and Signal Processing (IEEE), 2009, p. 1. G. Pedrini, H.J. Tiziani, Optics and Laser Technology 29 (5) (1997) 249. P. Picart, J. Leval, D. Mounier, S. Gougeon, Applied Optics 44 (3) (2005) 337. P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, G. Pierattini, Optics Letters 28 (14) (2003) 1257. I. Yamaguchi, T. Ida, M. Yokota, Strain 44 (5) (2008) 349. J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, C. Depeursinge, Optics Express 15 (2007) 7231. D.G. Abdelsalam, Magnusson Robert, Kim Daesuk, Applied Optics 19 (2011) 3360. E. Cuche, F. Bevilacqua, C. Depeursinge, Optics Letters 24 (1999) 291. E. Cuche, P. Marquet, C. Depeursinge, Applied Optics 39 (2000) 4070. D.G. Abdelsalam, B.J. Baek, Y.J. Cho, D. Kim, Journal of the Optical Society of Korea 14 (2010) 409. D.G. Abdelsalam, B.J. Baek, D. Kim, Journal of the Optical Society of Korea 15 (2011) 146. H. Lee, S. Kim, D. Kim, J. Opt. Soc. Korea 14 (2010) 363. Malacara Daniel, Servin Manuel, Malacara Zacarias, Taylor and Francis Group, United States of America, 2005, p. 384. Feng Pan, Wen Xiao, Shuo Liu, Fan Jing Wang, Lu Rong, and Rui Li, Optics Express 1 (9) (2011) 3863. D.G. Abdelsalam, M.S. Shaalan, M.M. Eloker, Optics and Lasers in Engineering 48 (2010) 543. D.G. Abdelsalam, M.S. Shaalan, M.M. Eloker, Kim Daesuk, Optics and Lasers in Engineering 48 (2010) 643.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Real-time dual-wavelength digital holographic microscopy based on polarizing separation D.G. Abdelsalam a, b, Daesuk Kim a,⁎ a b
Division of Mechanical System Engineering, Chonbuk National University, Jeonju 561–756, Republic of Korea Engineering and surface metrology lab., National Institute of Standards, Tersa St., El haram, El Giza, Egypt
a r t i c l e
i n f o
Article history: Received 31 August 2011 Accepted 21 September 2011 Available online 5 October 2011 Keywords: Digital holography Phase retrieval Interference microscopy
a b s t r a c t We report on a manifold advanced dual-wavelength digital holographic microscopy (DHM) configuration with a real-time measurement capability. The proposed configuration based on a polarizing separation scheme can be used for microscopic imaging polarimetry as well as dual wavelength digital holographic microscopy. In this paper, we show the feasibility of the proposed scheme by conducting the dual wavelength DHM experiments on a sample with a step height of 1.34 μm nominally. An averaging technique is treated and three-dimensional (3D) topographic measurements are presented. The results obtained by the proposed polarization separation based single shot DHM approach shows it can provide a real time solution for measuring 3D profile information of small objects with excellent accuracy. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Digital holography has been heavily developed over recent years because of newly-available high-resolution CCD cameras and advances in digital and automated image processing techniques [1]. Performing Fourier transforms and spectral filtering without the need for additional optical components and the simplicity by which reconstruction data may be interpreted quantitatively has given advantages of digital holography over conventional holography. Great advantages of off-axis digital holography over in-line holography include the separation of amplitude and phase information [2] and noise reduction [3]. Optical digital holography applications are manifold and include biological microscopy [4–9], metrology of microstructures [10–12], and deformation and vibration analysis [13–15]. By improving nanotechnology, the dimension of structures and functional devices is decreasing in size. Digital holographic microscopy (DHM) allows to investigate the shape and the displacements of objects [16–17] in the nanoscale with high resolution and it highlights its capability as a promising 3D imaging technique. In this paper, an averaging technique is treated and three-dimensional (3D) topographic measurements are presented, obtained by a dualwavelength DHM setup on a 1.34 μm nominal step height.
⁎ Corresponding author at: Division of Mechanical System Engineering, Chonbuk National University, Jeonju 561-756, Republic of Korea. Tel.: + 82 63 270 4632; fax: + 82 63 270 2388. E-mail address: [email protected] (D. Kim). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.09.044
The presented configuration could be called a digital holographic microscope and consists of a microscope objective (MO) that produces a magnified image of the sample that is used as an object for the hologram creation. The principle of the setup is to separate each wavelength beam pair in different reference arms, while combining them in an object arm. The main great advantage of the proposed configuration is that it can be used for single-shot dual-wavelength off-axis interferometry, phase shifting interferometry, and imaging polarimetry without any major modification. Furthermore, it is more compact compared to the previous systems [18]. In this study, we show the ability of the proposed configuration to investigate very small objects by using MO that produces a magnified image (a phase aberration associated with the use of a MO is corrected by a digital method [3]). The investigated object is a sample of a 1.34 μm nominal step height. In the proposed configuration, two HWPs are used to change the polarization direction of light beam coming from one arm to match the polarization direction of light beam coming from the other arm for both wavelengths. Such polarization direction matching for each wavelength is necessary for producing holograms for both wavelengths. The independent phase maps are subtracted and a final phase map for the beat-wavelength is obtained and converted to height map. The harmful noises associated with the height map are treated with an averaging technique. The results obtained by the proposed scheme have been in excellent agreement with the nominal value of the used sample. The main great advantages of the proposed configuration over the previous study are mentioned in details in reference [19]. As far as we know, this is the first time that real-time high resolution DHM using a new configuration shown in Fig. 1 is used to investigate very small objects with excellent accuracy.
234
D.G. Abdelsalam, D. Kim / Optics Communications 285 (2012) 233–237
Fig. 1. Schematic diagram of the proposed single shot dual-wavelength digital holographic microscopy (DHM) system based on polarizing separation.
2. Hologram reconstruction The interference between the object wave O1 and the reference wave R1 for p-polarization generates a hologram at wavelength λ1 = 635 nm. The interference between the object wave O2 and the reference wave R2 for s-polarization creates a hologram at wavelength λ2 = 635 nm. The hologram intensity resulting from both holograms at the same time real-time monitoring (single camera acquisition) for λ1 and λ2, can be written as follows: 2
2
2
2
IH ðx; yÞ ¼ jO1 j þ jR1 j þ jO2 j þ jR2 j |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} zero order
ð1Þ
þ R1 O1 þ R1 O1 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} þ R2 O2 þ R2 O2 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} λ1 λ2 Where IH is the hologram intensity, x and y are integers, and * denote the complex conjugate. In Eq. (1), the first four intensity terms are of zero order, and the last four are the interference terms for λ1 = 635 nm and λ2 = 635 nm , respectively. As in reference [20] the off-axis geometry is considered. For the p-polarization generated at wavelength λ1 = 635 nm, the Mirror 1 is oriented such that the reference wave R1 reaches the CCD camera with a small incidence angle θ1 (see inset in Fig. 1) with respect to the propagation direction of the object wave O1. Likewise, for the s-polarization generated at wavelength λ2 = 635 nm, the Mirror 2 or Mirror 3 is oriented such that the reference wave R2 reaches the CCD camera with a small incidence θ2 with respect to the propagation direction of the object wave O2. A digital hologram is recorded by a black and white CCD camera with 4.65 μm × 4.65 μm pixel size. The digital hologram IH(k,l) is an array of N × N = 500 × 500 results from the two-dimensional sampling of IH(y,x) by the CCD camera: IH ðk; lÞ ¼ IH ðx; yÞrect
x y ; × L L
N=2
∑
N=2
∑ δðx−kΔx; y−lΔyÞ;
intervals in the hologram plane (pixel size) Δx = Δy = L/N. Holographic microscopy has been proposed in various configurations [3]. Fig. 2 shows the optical arrangement configuration of the holographic microscopy. The MO produces a magnified image of the object, and the hologram plane (the CCD plane) is located between the MO and the image plane, at a distance d from the image. The numerical reconstruction process is based on calculating the Fresnel diffraction pattern of the captured hologram. The result of the calculation is an array of complex numbers called reconstructed wave front Ψ, which represents the complex amplitude of the optical field in the observation plane 0ξη. The distance between the hologram plane 0xy and the observation plane is defined by the reconstruction distance d. In holographic microscopy, image focusing occurs when the reconstruction distance is equal to the distance between the CCD and the image during hologram recording (d in Fig. 2). The reconstructed wave front Ψj(m.n) for wavelength λj in the convolution formulation, in the observation 0ξη, was obtained by computing the discrete Fresnel integral of the digitized hologram IH(k,l) and is given as follows: "
# iπ 2 2 2 2 m Δξj þ n Δηj ; Ψj ðm; nÞ ¼ A exp λj dj (
"
× FFT IH ðk; lÞ exp
iπ 2 2 2 2 k Δx þ l Δy λj dj
ð3Þ
#) m;n
ð2Þ
k¼−N=2l¼−N=2
where k,l are integers, L × L = 2.325 mm × 2.325 mm is the region of interest, δ is a Dirac delta function, and Δx, Δy defines the sampling
Fig. 2. Digital holographic microscope, optical configuration.
D.G. Abdelsalam, D. Kim / Optics Communications 285 (2012) 233–237
pffiffiffiffiffiffiffiffi where m,n,k,l are integers (–N/2 ≤ k,l,m,n ≤ N/2) , i ¼ −1, j = 1,2, and FFT is the fast Fourier transform operator. Δx and Δy define the sampling intervals in the hologram plane. The sampling intervals Δξj and Δηj in the observation plane are related to the size of the CCD (L) and to the distance dj by the following relation: Δξj ¼ Δηj ¼ λj dj =L:
ð4Þ
The MO produces a curvature of the wave front in the object arm. This deformation may affect the phase of the object wave. This phase aberration must be corrected to provide accurate results. The phase aberration can be corrected by multiplication of an array of complex numbers called digital phase mask φj(m,n) is calculated as follows: " # −iπ 2 2 2 2 m Δξj þ n Δηj ; λj Dj
ð5Þ
φj ðm; nÞ ¼ exp
where Dj is a parameter that must be adjusted to compensate the wave-front curvature and expressed as follows: 1 1 ¼ Dj dj
1þ
! d0 : dj
ð6Þ
The complete expression of the reconstruction algorithm becomes " Ψj ðm; nÞ ¼ Aφj ðm; nÞ exp 8 <
# iπ 2 2 2 2 m Δξj þ n Δηj λj dj "
i iπ 2 2 2 2 k Δx þ l Δy × FFT RDj ðk; lÞIH ðk; lÞ exp : λj dj
ð7Þ
) : m;n
Since Ψj(m, n) are an array of complex numbers, an amplitudecontrast image and a phase-contrast image can be obtained by using the following intensity [Re(Ψj) 2 + Im(Ψj) 2] and the argument arctan [Re(Ψj)/Im(Ψj)], respectively. Where (RDj(k, l)) is called a digital reference wave. If we assume that a perfect plane wave is used as a reference for hologram recording, the computed replica of the reference wave RDj can be represented as follows: h i RDj ðk; lÞ ¼ ARj exp i 2π=λj kxj kΔx þ kyj lΔy ;
ð8Þ
where, ARj is the amplitude, λj is the wavelength of the laser source, and kxj and kyj are the two components of the wave vector that must be adjusted such that the propagation direction of RDj matches as closely as possible with that of the experimental reference wave. By using this digital reference wave concept, we can obtain an object wave which is reconstructed in the central region of the observation plane for each wavelength separately. In order to overcome the phase ambiguity produced from single wavelength approach, a synthetic beat-wavelength is used and expressed as follows: λ −λ1 x Φ ¼ arg O1 O2 ¼ ϕ1 −ϕ2 ¼ 2πx 2 ¼ 2π ; Λ λ1 λ2
ð9Þ
where, x is OPD (Optical Path Difference), which means twice of the topography for reflection scheme. ϕj is the reconstructed phase for wavelength λj and Λ is the synthetic beat wavelength defined as follows: λ λ Λ¼ 1 2 : λ2 −λ1
phase obtained by Eq. (10) can provide a solution for much higher pattern structures by overcoming the phase ambiguity. 3. Experimental results The optical arrangement is depicted in Fig. 1, with the laser diodes sources at λ1 = 635 nm and λ2 = 675 nm, yielding a synthetic wavelength Λ = 10.72 μm. The key concept of the proposed scheme is to separate each wave wavelength beam pair in different reference arms, while combining them in an object arm (the theory has been explained in details in reference [19] for macro scale (without MO)). In this study, we show the ability of the proposed configuration to investigate very small objects by using MO that produces a magnified image. After reflection on the sample as shown in Fig. 1, both collinear object wavefronts are collected by the MO (infinitycorrected), and the object images are formed by an imaging lens about 50 mm behind the CCD plane. A microscope objective (MO) of magnification ×10 with a NA of 0.25 and working distance WD of 10.6 mm is used to magnify the image of the sample that is used as an object for the hologram creation. The CCD camera is a standard 8 bits black and white CCD camera with 4.65 μm pixel size. Each reference arm is adjusted carefully to match the optical path length of the corresponding object arm, in order to create an interference pattern on the CCD for both wavelengths. By tilting the mirror 1 for the first wavelength reference beam and the pair of mirror 2 and mirror 3 for the second one, one can find finely tune each k-vector incident upon the CCD camera. On the other hand, each wavelength hologram fringes can be independently tuned both in spatial frequency and orientation. Finally, the CCD camera records the digital hologram that result from the interference between the object wave O1 and the reference wave R1. And also, the interference between the object wave O2 and the reference wave R2 is recorded simultaneously. The two filtered holograms are reconstructed and a final phase map for the beat-wavelength is obtained and converted to height map. Fig. 3 shows the experimental results obtained through the proposed dual-wavelength off-axis digital holographic microscopy scheme. The surface under test is an object with a nominal height 1.34 μm (see the white rectangle in Fig. 3(a)). The general procedure which is needed for measuring the complex object wave in dual-wavelength off-axis digital holographic microscopy is shown in Fig. 3. Both amplitude and phase information of the object for the two wavelengths can be obtained with a single hologram. Fig. 3 represents the experimental results of application of the 2D-FFT based spatial filtering method for each wavelength separately. After the spatial filtering step [21–24], the digital reference wave RDi is used for the centering process. Then, the final reconstructed object is obtained by adjusting the values of kx and ky,dj, and Dj for each wavelength separately. The digital reference wave used in the calculation process should match as close as possible to the experimental reference wave. This has been done in this paper by selecting the appropriate values of the two components of the wave vector kx = 0.1218565 mm− 1 and ky = −0.12487275 mm−1, d0 = 0.99 mm, d = 49 mm, and D = 212 mm for the hologram captured at λ1 = 635 nm and kx = 0.15987375 mm− 1 and ky = −0.1479725 mm− 1, d0 = 0.99 mm, d = 50 mm, and D = 217 mm for the hologram captured at λ2 = 675 nm. Once the two phase maps are obtained for each wavelength, a phase map can be calculated on the synthetic beatwavelength. The three-dimensional (3D) surface profile height h can be calculated directly as follows: h¼
ð10Þ
We can see that the smaller the difference between the two wavelengths, the larger the synthetic wavelength, typically within the range of micrometers to millimeters. The corresponding synthetic
235
Φ Λ: 4π
ð11Þ
Fig. 3(a) shows the investigated sample (see the white rectangle) of a nominal step height of 1.34 μm. The single-shot dual-wavelength off-axis hologram of the investigated sample (white rectangle) is shown in Fig. 3(b). The filtering windows A and B as shown in Fig. 3(c)
236
D.G. Abdelsalam, D. Kim / Optics Communications 285 (2012) 233–237
Fig. 3. Sequential reconstruction steps of the spatial filtering based phase contrast dual- wavelength off-axis holography (a) original phase object (see the white rectangle in Fig. 3(a)), (b) off-axis hologram, (c) Fourier transformed spatial frequency domain data, (d)–(e) reconstructed amplitude and phase map for λ1 = 635 nm, (f)–(g) amplitude and phase map for λ2 = 675 nm, (h) object phase map on the synthetic beat-wavelength, and (i) 3D surface profile of the selected rectangle of (h).
for λ1 = 635 nm and λ2 = 675 nm, respectively, have been chosen carefully. The reconstructed amplitude and phase map for λ1 = 635 nm and for λ2 = 675 nm are shown in Fig. 3(d–e) and Fig. 3(e–f), respectively. The phase map on the synthetic beat-wavelength is shown in Fig. 3(h). The three dimensional (3D) surface profile of the
selected rectangle of Fig. 3(h) is shown in Fig. 3(i) after converting to height by using Eq. (7). Based on the measured height in Fig. 3(h), the average step height has been estimated to be around 1.34 ± 0.9 μm. The uncertainty in measurement is very high due to the non-coherent noise associated with the profile. The noise shown in Fig. 3(h) is non
Fig. 4. Surface height inside the white rectangular (6 pixels profiles) in Fig. 3(h) (a) 2D, (b) 3D of (a), (c) 2D surface profile along the selected line of (a), and (d–f) are the modified of (a–c) after applying the rms technique, respectively.
D.G. Abdelsalam, D. Kim / Optics Communications 285 (2012) 233–237
coherent noise or inevitable noise may come from read out noise (noise that does not vary with signal), fixed pattern noise (pixel to pixel sensitivity variation), and inherent noise due to the spatial filtering window and 2D-FFT processes. The noise vibration shown in Fig. 3(h) is not coherent but varies sinusoidally; thus, it is more logical to express the height error as the root mean square (rms) value of the disturbed height [25]. The rms height can be expressed as [26–28]: hrms
rhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i P 2 ¼ h− h =ðM−1Þ;
ð12Þ P
where h denotes the height distribution, h denotes the mean values, M represents the pixel number of row of calculation region. The proposed rms method can provide a satisfied solution especially for measuring objects having high abrupt height difference. Fig. 4(a) shows a two-dimensional (2D) surface height inside the white rectangular (6 pixels profiles) in Fig. 3(h). The 3D surface height profile of Fig. 4(a) is shown in Fig. 4(b). The two dimensional (2D) surface profile along the selected line of Fig. 4(a) is shown in Fig. 4(c). Fig. 4(d–e) shows the modified height profile of Fig. 4(a–c) after applying the rms technique. Based on the measured height in Fig. 4(f), the average step height has been estimated to be around 1.34 ± 0.02 μm i.e. the uncertainty in measurement is reduced drastically by the order of more than 90%. 4. Conclusion In this paper, a dual-wavelength digital holographic microscopy (DHM) technique with a manifold advanced configuration has been presented. An averaging technique has been treated and 3D topographic measurements have been presented, on a 1.34 μm nominal step height. The experimental results have been in excellent agreement with the nominal value of the used sample. The results obtained by the proposed polarization separation based single shot DHM approach shows it can provide a real time solution for measuring 3D profile information of small objects with excellent accuracy. Although the microscopic imaging polarimetry capability of the proposed scheme has not been touched in this paper in detail, we claim that the proposed scheme can be used for various microscopic imaging polarimetric measurement applications required for liquid crystal display (LCD) and semiconductor industry.
237
Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (2011– 0002487). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
G. Shen, R. Wei, Optics and Lasers in Engineering 43 (10) (2005) 1039. M.K. Kim, Optics Express 7 (9) (2000) 305. E. Cuche, P. Marquet, C. Depeursinge, Applied Optics 38 (34) (1999) 6994. G.L. Chen, C.Y. Lin, M.K. Kuo, C.C. Chang, Applied Physics B 90 (3–4) (2008) 527. B. Kemper, G. von Bally, Applied Optics 47 (4) (2008) A52. W. Xu, M.H. Jericho, I.A. Meinertzhagen, H.J. Kreuzer, Proceedings of the National Academy of Science USA, (PNAS), 2001, p. 11301. G. Popescu, L.P. Deflores, J.C. Vaughan, K. Badizadegan, H. Iwai, R.R. Dasari, M.S. Feld, Optics Letters 29 (21) (2004) 2503. P. Marquet, B. Rappaz, P.J. Magistretti, E. Cuche, Y. Emery, T. Colomb, C. Depeursinge, Optics Letters 30 (5) (2005) 468. D. Carl, B. Kemper, G. Wernicke, G. von Bally, Applied Optics 43 (36) (2004) 6536. X. Song, Z. Tang, H. Wang, Proceedings of IEEE, Control and Decision Conference (IEEE), 2009, p. 4656. L. Xu, X. Peng, J. Miao, A.K. Asundi, Applied Optics 40 (28) (2001) 5046. G. Coppola, S. De Nicola, P. Ferraro, A. Finizio, S. Grilli, M. Iodice, C. Magro, G. Pierattini, Proceedings of SPIE 4945 (2003) 71. Z. Fan, H. Pang, W. Wang, C. Ning, F. Guo, Proceedings of IEEE International Congress on Image and Signal Processing (IEEE), 2009, p. 1. G. Pedrini, H.J. Tiziani, Optics and Laser Technology 29 (5) (1997) 249. P. Picart, J. Leval, D. Mounier, S. Gougeon, Applied Optics 44 (3) (2005) 337. P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, G. Pierattini, Optics Letters 28 (14) (2003) 1257. I. Yamaguchi, T. Ida, M. Yokota, Strain 44 (5) (2008) 349. J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, C. Depeursinge, Optics Express 15 (2007) 7231. D.G. Abdelsalam, Magnusson Robert, Kim Daesuk, Applied Optics 19 (2011) 3360. E. Cuche, F. Bevilacqua, C. Depeursinge, Optics Letters 24 (1999) 291. E. Cuche, P. Marquet, C. Depeursinge, Applied Optics 39 (2000) 4070. D.G. Abdelsalam, B.J. Baek, Y.J. Cho, D. Kim, Journal of the Optical Society of Korea 14 (2010) 409. D.G. Abdelsalam, B.J. Baek, D. Kim, Journal of the Optical Society of Korea 15 (2011) 146. H. Lee, S. Kim, D. Kim, J. Opt. Soc. Korea 14 (2010) 363. Malacara Daniel, Servin Manuel, Malacara Zacarias, Taylor and Francis Group, United States of America, 2005, p. 384. Feng Pan, Wen Xiao, Shuo Liu, Fan Jing Wang, Lu Rong, and Rui Li, Optics Express 1 (9) (2011) 3863. D.G. Abdelsalam, M.S. Shaalan, M.M. Eloker, Optics and Lasers in Engineering 48 (2010) 543. D.G. Abdelsalam, M.S. Shaalan, M.M. Eloker, Kim Daesuk, Optics and Lasers in Engineering 48 (2010) 643.