- Email: [email protected]
Carrier removal method based on the principle of shearing
Optik 126 (2015) 5154–5157
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Carrier removal method based on the principle of shearing Fan Rao ∗ School of Mathematics and Computer Science, Jianghan University, Wuhan, Hubei 430056, PR China
a r t i c l e
i n f o
Article history: Received 9 October 2014 Accepted 7 September 2015 Keywords: Digital holography Carrier removal Shearing principle Phase reconstruction
a b s t r a c t In the method of holographic interferometry and three dimensional profilometry of optic projection grating, the reconstruction of real phase is always affected by the carrier frequency, so removing carrier is a necessary step to obtain the accurate real phase. A novel method is proposed by introducing the shearing principle to calculate the carrier phase. Firstly, the distribution of phase gradient is obtained by shearing the optical field. Then, the mean value is subtracted from the obtained phase gradient to achieve carrier removal. That is, the proposed method is finished before phase unwrapping. The simulation calculation and experiment results show that this method is benefit to reduce the difficulty of phase unwrapping, to remove the carrier frequency effectively, and to make the reconstructed phase more closer to the measured true value. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction The phase distribution of object can be obtained with digital holographic reconstructed optical field, as a means of optical measurement it has been widely used in many areas [1–3] because of its advantages, like non-contact, high sensitivity, real-time and full-court, and so on. Off-axis holography is usually used to overcome the influence of conjugate image and zero-order diffraction image appearing in the process of digital holographic reconstruction. Therefore, the phase of reconstructed field contains a carrier phase which will directly affect the true phase measurement, so it must be removed from the reconstructed phase. Three dimensional profilometry of optic projection grating [4–8] is a method that through projecting fringe onto a 3D object surface and using the deformed fringe which modulated by 3D object contour to get the wrapped phase information, then through phase unwrapping to obtain 3D object shape. In the method, the projection grating is usually made with a high spatial frequency (carrier phase) to improve the accuracy, so the reconstructed phase also contains a carrier phase in general unless frequency shift is done in the frequency domain. Hence, it also must be removed to reconstruct the true three dimensional morphology of object. There are many methods to remove the carrier phase, such as the method in Ref. [9] which through solving the first derivative
∗ Tel.: +86 18086491699. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.09.050 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
of phase unwrapping and regarding it as the approximate value of carrier phase, then subtracting the carrier phase from the total phase to eliminate the influence. The method in Ref. [10] uses frequency shift to filter out the carrier phase, and it is equivalent to subtract a certain phase in spatial domain, thereby removes the carrier phase. Ref. [11] uses Zernike polynomial fitting to eliminate the carrier phase. Although these methods have achieved well results, at the same time, all of them have certain limitation and application premise. For example, the method in Ref. [9] is conducted after completing phase unwrapping, but the carrier phase will lead to a high spatial frequency to the wrapped phase, and this brings difficulty and error to phase unwrapping. The method proposed in Ref. [10] needs to exactly know the carrier phase to build the phase needed to be subtracted, but it is not easy to accurately calculate the carrier phase in practice. In the method of Ref. [11], the item numbers of Zernike polynomial are usually selected empirically and its programming is relatively complex. The method proposed in this paper is based on the shearing principle to remove the carrier phase. Ref. [12] has described the implementation of shearing that using a single digital hologram to realize shearing interferometry. By introducing the shearing principle to the calculation of carrier removal which is completed before phase unwrapping, the method can well eliminate the difficulty of phase unwrapping which caused by carrier phase, meanwhile, it can also avoid error caused by finding carrier phase. Based on theoretical analysis, the specific algorithm is proposed in this paper, its simulation and experiment validation are also given. The results indicate that the method can well reduce the influence of measurement caused by carrier phase.
F. Rao / Optik 126 (2015) 5154–5157
2. Off-axis holography reconstructed optical field characteristics of phase
by shifting in the original position (kxi , myi ) ˜ (kxi , myi ) = A(kxi , (m + s)yi ) exp[j (kxi , (m + s)yi )] U i
In digital holography, the hologram recording the surface (x, y) of ˜ ˜ object optical field O(x, y) and off-axis reference optical field R(x, y) are: ˜ O(x, y) = o(x, y) exp[jϕo (x, y)]
(1)
˜ (2) R(x, y) = A exp[j2(x + y)] √ where, j = −1, o(x, y)and ϕo (x, y) are the amplitude and phase of object optical field, respectively, A is the amplitude of reference optical field, and are the reference spatial frequency(carrier frequency) of the reference optical field in the x and y direction. The intensity distribution of off-axis digital hologram as: 2
˜∗
2
(7) When the shearing quantity s is small, amplitude differences at the same point can be ignored. That A(kxi , myi ) = A(kxi , (m + s)yi ) Divide the two fields: ˜ (kxi , myi ) U i ˜ i (kxi , myi ) U
˜ ∗ (x, y) exp[j2(x + y)] +AO
exp[j (kxi , (m + s)yi )] exp[j (kxi , myi )]
= exp[j
(3)
∂ = ∂yi
Digital and conventional holography reconstructed difference are directly available through computer simulation of diffraction of reconstructed wavefield. The hologram is illuminated by a perpendicular incident plane wave C(x,y) = 1, the transmission wavefield can be expressed as:
−
˜ ˜ y) = o2 (x, y) + A2 + AO(x, y) exp[−j2(x + y)] U(x, ˜ ∗ (x, y) exp[j2(x + y)] = U ˜ 1 (x, y) + U ˜ 2 (x, y) + AO ˜ 4 (x, y) ˜ 3 (x, y) + U +U
(4)
˜ ∗ (x, y) exp[j2(x + y)], by diffraction calcula˜ 4 (x, y) = AO where U tions often can be clear the reconstructed real image. On the clear ˜ i4 (xi , yi ), the phase image (xi , yi ) of the reconstructed wavefield U can be calculated with the following: (xi , yi ) = arctan
=
y (kxi , myi )] k,m
If s = 1, ∂ /∂yi is at the point of phase y direction of the gradient:
˜ = o2 (x, y) + A2 + AO(x, y) exp[−j2(x + y)]
˜ i4 (xi , yi )] Im[U ˜ i4 (xi , yi )] Re[U
(8)
(9)
˜∗
˜ ˜ ˜ ˜ I(x, y) = |R(x, y)| + |O(x, y)| + O(x, y)R (x, y) + O (x, y)R(x, y)
5155
(5)
in which Im[] and Re[] are represented the real and imaginary of operation. Calculate the phase (xi ,yi ) is generally wrapped, in order to get real phase also to be the appropriate unwrapping phase for operation. It should be noted that the phase of the reconstructed wavefield must contain the reference beam phase 2(xi + yi ). Because the carrier does not change in the diffraction process, it will affect the phase of objects accurately reconstructed, in order to get accurate phase distribution of object that removal of carrier phase in the reconstructed wavefield, just the phase of plane reference wave as xi and yi linear distribution, Assume the off-axis spherical wave as reference beam, it phase is sum of a linear distribution and secondary distribution phase [13], which the linear phase distribution corresponds to the carrier phase, so we can introduce principle of shearing removal the linear distribution phase. 3. Using shearing principle of removal carrier ˜ i (kxi , myi ) is reconstructed Pixel of K × M complex matrix U wavefield of digital holographic, (1 ≤ k ≤ K, 1 ≤ m ≤ M). xi and yi for each pixel in the x and y direction on the actual size, that is: ˜ i (kxi , myi ) = A(kxi , myi ) exp[j (kxi , myi )] U
(6)
where A and represents the amplitude and phase of field, For ˜ i (kxi , myi ), shearing is made in image plane along one field U direction (for y direction for example). If the shearing quantity are ˜ (kxi , myi ) is formed s pixels (s is an integer), then a new field U i
y (kxi , myi ) k,m
=
(kxi , myi ) along the
[kxi , (m + 1)yi ]
(kxi , myi )
(10)
As the carrier phase in (kxi , myi ) is linear variation, the phase gradient along the y direction is 2, which is a constant. But the gradient of object along y direction changes; its average should tend to zero. Then ∂ /∂yi subtract average of ∂ /∂yi can be calculated along the y direction on the gradient of removal carrier phase:
∂ ∂ = − mean ∂yi ∂yi
∂ ∂yi
(11)
where mean[] is the averaging operation. The phase gradient of removal carrier for x direction can be obtained using the same way;
∂ ∂ = − mean ∂xi ∂xi
∂ ∂xi
(12)
According to Eqs. (11) and (12) get phase gradient, with phase unwrapping algorithm based on the principle of shearing. Then can be have removal carrier of phase unwrapping. 4. Simulation of carrier removal based on the principle of shearing In order to test the methods, we simulate the process. Assume x = y = 1.000 mm, K = M = 256, a two-dimensional distribution phase phoo of 512 × 512 pixels is established using peaks function of Matlab and superimposed a linear distribution of two-dimensional phase for simulation of carrier phase caused: pho = 2x + 2y, take = 80 m−1 and = 100 m−1 respectively for the carrier phase in the x and y direction, finally concluded that ph = pho + phoo. The results are shown in Fig. 1. Fig. 1(a) is used for simulation phase phoo. Fig. 1(b) is a wrapping of total phase (including the carrier phase), we can see, because of overlay the carrier of the phase pho, the ph appear overlapping in the two areas (arrow point), that bring difficulties for unwrapping. Fig. 1(c) is obtained by using our new phase phoo, Fig. 1(d) is a new method error distribution of phase phoo. Simulation calculated = 79.6997 m−1 and = 99.7040 m−1 , the relative error of 0.375% and 0.296%. Fig. 1(e) is phase unwrapping result by using least squares algorithms of Fig. 1(b), then use Ref. [9] method for removal of carrier, because of the ph appear overlapping in the two areas, result the unwrapping failure. Fig. 1(f) is used method of Ref. [10] to find and removal carrier, after that use least squares
5156
F. Rao / Optik 126 (2015) 5154–5157
Fig. 1. Simulated results.
Fig. 3. Experimental results by new carrier removal methods.
Fig. 2. Experimental setup.
algorithms unwrapping obtain error distribution of phase. As the carrier phase from the spectrum when the spectrum to find the subscript can only take integer, conversion to get = 85.9375 m−1 , = 101.5625 m−1 , the relative error of 7.422% and 1.563%, larger than the new method. Simulation calculations show that primitive phase more accurate for make use of shearing principle.
removing the carrier phase of cosine and 3D distribution figure of reconstructed wavefield unwrapping phase. Fig. 3(d) is the reconstructed wavefield wrapping phase after the candles burning, while Fig. 3(e) and (f) are new ways to calculate after remove carrier phase of cosine and 3D distribution figure of reconstructed wavefield unwrapping phase. Fig. 3(g) is the experimental distribution from interference fringes, Fig. 3(i) is phase three-dimensional distribution by the candle burning before subtract after of removal carrier phase unwrapping, while Fig. 3(h) is the interference fringe phase obtained by taking the cosine distribution. Compare with Fig. 3(g) and (h) we can see that the theoretical and experimental values match well.
5. Experiments 6. Conclusions We measured the candle burning temperature field in the experiments [12] to verify that the use of shear principle for removing the carrier before unwrapping. Fig. 2 is a schematic diagram of the experiment optical path. Laser beam ( = 532 nm) comes from YAG laser which is divided into two optical beam by beamsplitter (BS1 ). One beam passes through the microscope objective (L1 ), and pinhole filter (h1) and collimated lens (L2 ) generate a parallel ray, by candle combustion area reaches beamsplitter (BS2 ), and then after reflection and lens (L5 ) together as object wavefield reaches of by the holograph. The other beam reflected by reflector (M1 ) and (M2 ) then expanded by lens L3 , pinhole filter (h2 ) and collimated lens (L4 ) generate parallel ray, and then after beamsplitter (BS2 ) and lens (L5 ) together as reference wave then recorded on the holography. By the hologram, we used the CMOS to record the hologram that is produced by the interference between the reference wave and object wave, The CMOS used contains 2048 pixel × 1536 pixel, the pixel size of which is 3.2 m × 3.2 m. The hologram may obtain ˜ the reconstruction optical field U(x, y) after for the diffraction calculation. Then use carrier removal method based on the principle of shearing and reconstruction phase of optical field, the result is listed in Fig. 3 In Fig. 3 image sizes are 300 pixel × 300 pixel to all. Fig. 3(a) is the reconstructed wavefield wrapping phase before the candles burning, while Fig. 3(b) and (c) are new ways to calculate after
In holographic interferometry and projecting grating of 3-D profilometry, the carrier phase affects on the true phase of reconstruction, is required to remove carrier to accurately reconstruct the object phase information. Shearing principle was introduced to removal carrier operations in this paper. Firstly, obtain the phase gradient distribution for shearing optical field, and then, subtract from phase gradient the average to removal carrier, that is, the unwrapping completed before the parcel method to removal carrier. Simulations and experiments show that this method to reduce the difficulty of phase unwrapping, can effectively removal carrier. The method is better than that finishes unwrapping operation subtracting its first order derivative of removal carrier, and is better than using the shift frequency filter out carrier phases, the reconstruction of phase values are close to the object of true phase. References [1] Bingheng Xiong, Zhengrong Wang, Yongan Zhang, et al., A novel testing method of transparent objects in real-time holograph, Acta Opt. Sin. 21 (7) (2001) 841–845. [2] Cheng Liu, Liangyu Li, Yinzhu Li, et al., Digital holography free of zeroorder diffraction and conjugate image, Acta Opt. Sin. 22 (4) (2002) 427–431.
F. Rao / Optik 126 (2015) 5154–5157 [3] Wei Feng, Enpu Li, Qi Fan, et al., Diagnosing microwave plasma thruster s plume by digital holographic interferometry, Acta Opt. Sin. 34 (12) (2005) 1833–1863. [4] Jindong Tian, Xiang Peng, Approach for three-dimensional digital imaging via position-shift point array projection, Acta Opt. Sin. 25 (10) (2005) 1319–2132. [5] Xianyu Su, Wenjian Chen, Qican Zhang, et al., Dynamic 3-D shape measurement method based on FTP, Opt. Lasers Eng. 36 (3) (2001) 49–64. [6] Weiyi Liu, Zhaoqi Wang, Guoguang Mu, et al., Three-dimensional surface profilometry using color-code projection grating, Acta Opt. Sin. 21 (6) (2001) 687–690. [7] Liudong Xiong, Shuhai Jia, Yanfen Du, A novel algorithm for phase retrieval from a single carrier-frequency fringe pattern, Acta Opt. Sin. 30 (1) (2010) 123–126. [8] V. Srinivasan, H.C. Liu, M. Halioua, Automated phase measuring profilometry: a phase mapping approach, Appl. Opt. 22 (24) (1985) 3977–3982.
5157
[9] Jielin Li, Xianyu Su, Hongjun Su, Removal of carrier frequency in phase-shifting techniques, Opt. Lasers Eng. 30 (1) (1998) 107–115. [10] Takeda Mitsuo, Ina Hideke, Kobayashi Seiji, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, J. Opt. Soc. Am. 72 (1) (1982) 156–160. [11] Zhiyun Wu, Qican Zhang, Carrier removal method in fringe projection profilometry using Zernike polynomials, Acta Opt. Sin. 31 (4) (2011) 04120111–0412011-6. [12] Xiaofan Qian, Zhanliang Wang, Te Hu, et al., Reconstructing the phase of wavefront using digital hologram and the principle of shearing interferometry, Chin. J. Lasers 37 (7) (2010) 1821–1826. [13] P. Hariharan, Optical Holography Principles, Techniques, and Applications, Cambridge University Press, 1996, pp. 20–23.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Carrier removal method based on the principle of shearing Fan Rao ∗ School of Mathematics and Computer Science, Jianghan University, Wuhan, Hubei 430056, PR China
a r t i c l e
i n f o
Article history: Received 9 October 2014 Accepted 7 September 2015 Keywords: Digital holography Carrier removal Shearing principle Phase reconstruction
a b s t r a c t In the method of holographic interferometry and three dimensional profilometry of optic projection grating, the reconstruction of real phase is always affected by the carrier frequency, so removing carrier is a necessary step to obtain the accurate real phase. A novel method is proposed by introducing the shearing principle to calculate the carrier phase. Firstly, the distribution of phase gradient is obtained by shearing the optical field. Then, the mean value is subtracted from the obtained phase gradient to achieve carrier removal. That is, the proposed method is finished before phase unwrapping. The simulation calculation and experiment results show that this method is benefit to reduce the difficulty of phase unwrapping, to remove the carrier frequency effectively, and to make the reconstructed phase more closer to the measured true value. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction The phase distribution of object can be obtained with digital holographic reconstructed optical field, as a means of optical measurement it has been widely used in many areas [1–3] because of its advantages, like non-contact, high sensitivity, real-time and full-court, and so on. Off-axis holography is usually used to overcome the influence of conjugate image and zero-order diffraction image appearing in the process of digital holographic reconstruction. Therefore, the phase of reconstructed field contains a carrier phase which will directly affect the true phase measurement, so it must be removed from the reconstructed phase. Three dimensional profilometry of optic projection grating [4–8] is a method that through projecting fringe onto a 3D object surface and using the deformed fringe which modulated by 3D object contour to get the wrapped phase information, then through phase unwrapping to obtain 3D object shape. In the method, the projection grating is usually made with a high spatial frequency (carrier phase) to improve the accuracy, so the reconstructed phase also contains a carrier phase in general unless frequency shift is done in the frequency domain. Hence, it also must be removed to reconstruct the true three dimensional morphology of object. There are many methods to remove the carrier phase, such as the method in Ref. [9] which through solving the first derivative
∗ Tel.: +86 18086491699. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.09.050 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
of phase unwrapping and regarding it as the approximate value of carrier phase, then subtracting the carrier phase from the total phase to eliminate the influence. The method in Ref. [10] uses frequency shift to filter out the carrier phase, and it is equivalent to subtract a certain phase in spatial domain, thereby removes the carrier phase. Ref. [11] uses Zernike polynomial fitting to eliminate the carrier phase. Although these methods have achieved well results, at the same time, all of them have certain limitation and application premise. For example, the method in Ref. [9] is conducted after completing phase unwrapping, but the carrier phase will lead to a high spatial frequency to the wrapped phase, and this brings difficulty and error to phase unwrapping. The method proposed in Ref. [10] needs to exactly know the carrier phase to build the phase needed to be subtracted, but it is not easy to accurately calculate the carrier phase in practice. In the method of Ref. [11], the item numbers of Zernike polynomial are usually selected empirically and its programming is relatively complex. The method proposed in this paper is based on the shearing principle to remove the carrier phase. Ref. [12] has described the implementation of shearing that using a single digital hologram to realize shearing interferometry. By introducing the shearing principle to the calculation of carrier removal which is completed before phase unwrapping, the method can well eliminate the difficulty of phase unwrapping which caused by carrier phase, meanwhile, it can also avoid error caused by finding carrier phase. Based on theoretical analysis, the specific algorithm is proposed in this paper, its simulation and experiment validation are also given. The results indicate that the method can well reduce the influence of measurement caused by carrier phase.
F. Rao / Optik 126 (2015) 5154–5157
2. Off-axis holography reconstructed optical field characteristics of phase
by shifting in the original position (kxi , myi ) ˜ (kxi , myi ) = A(kxi , (m + s)yi ) exp[j (kxi , (m + s)yi )] U i
In digital holography, the hologram recording the surface (x, y) of ˜ ˜ object optical field O(x, y) and off-axis reference optical field R(x, y) are: ˜ O(x, y) = o(x, y) exp[jϕo (x, y)]
(1)
˜ (2) R(x, y) = A exp[j2(x + y)] √ where, j = −1, o(x, y)and ϕo (x, y) are the amplitude and phase of object optical field, respectively, A is the amplitude of reference optical field, and are the reference spatial frequency(carrier frequency) of the reference optical field in the x and y direction. The intensity distribution of off-axis digital hologram as: 2
˜∗
2
(7) When the shearing quantity s is small, amplitude differences at the same point can be ignored. That A(kxi , myi ) = A(kxi , (m + s)yi ) Divide the two fields: ˜ (kxi , myi ) U i ˜ i (kxi , myi ) U
˜ ∗ (x, y) exp[j2(x + y)] +AO
exp[j (kxi , (m + s)yi )] exp[j (kxi , myi )]
= exp[j
(3)
∂ = ∂yi
Digital and conventional holography reconstructed difference are directly available through computer simulation of diffraction of reconstructed wavefield. The hologram is illuminated by a perpendicular incident plane wave C(x,y) = 1, the transmission wavefield can be expressed as:
−
˜ ˜ y) = o2 (x, y) + A2 + AO(x, y) exp[−j2(x + y)] U(x, ˜ ∗ (x, y) exp[j2(x + y)] = U ˜ 1 (x, y) + U ˜ 2 (x, y) + AO ˜ 4 (x, y) ˜ 3 (x, y) + U +U
(4)
˜ ∗ (x, y) exp[j2(x + y)], by diffraction calcula˜ 4 (x, y) = AO where U tions often can be clear the reconstructed real image. On the clear ˜ i4 (xi , yi ), the phase image (xi , yi ) of the reconstructed wavefield U can be calculated with the following: (xi , yi ) = arctan
=
y (kxi , myi )] k,m
If s = 1, ∂ /∂yi is at the point of phase y direction of the gradient:
˜ = o2 (x, y) + A2 + AO(x, y) exp[−j2(x + y)]
˜ i4 (xi , yi )] Im[U ˜ i4 (xi , yi )] Re[U
(8)
(9)
˜∗
˜ ˜ ˜ ˜ I(x, y) = |R(x, y)| + |O(x, y)| + O(x, y)R (x, y) + O (x, y)R(x, y)
5155
(5)
in which Im[] and Re[] are represented the real and imaginary of operation. Calculate the phase (xi ,yi ) is generally wrapped, in order to get real phase also to be the appropriate unwrapping phase for operation. It should be noted that the phase of the reconstructed wavefield must contain the reference beam phase 2(xi + yi ). Because the carrier does not change in the diffraction process, it will affect the phase of objects accurately reconstructed, in order to get accurate phase distribution of object that removal of carrier phase in the reconstructed wavefield, just the phase of plane reference wave as xi and yi linear distribution, Assume the off-axis spherical wave as reference beam, it phase is sum of a linear distribution and secondary distribution phase [13], which the linear phase distribution corresponds to the carrier phase, so we can introduce principle of shearing removal the linear distribution phase. 3. Using shearing principle of removal carrier ˜ i (kxi , myi ) is reconstructed Pixel of K × M complex matrix U wavefield of digital holographic, (1 ≤ k ≤ K, 1 ≤ m ≤ M). xi and yi for each pixel in the x and y direction on the actual size, that is: ˜ i (kxi , myi ) = A(kxi , myi ) exp[j (kxi , myi )] U
(6)
where A and represents the amplitude and phase of field, For ˜ i (kxi , myi ), shearing is made in image plane along one field U direction (for y direction for example). If the shearing quantity are ˜ (kxi , myi ) is formed s pixels (s is an integer), then a new field U i
y (kxi , myi ) k,m
=
(kxi , myi ) along the
[kxi , (m + 1)yi ]
(kxi , myi )
(10)
As the carrier phase in (kxi , myi ) is linear variation, the phase gradient along the y direction is 2, which is a constant. But the gradient of object along y direction changes; its average should tend to zero. Then ∂ /∂yi subtract average of ∂ /∂yi can be calculated along the y direction on the gradient of removal carrier phase:
∂ ∂ = − mean ∂yi ∂yi
∂ ∂yi
(11)
where mean[] is the averaging operation. The phase gradient of removal carrier for x direction can be obtained using the same way;
∂ ∂ = − mean ∂xi ∂xi
∂ ∂xi
(12)
According to Eqs. (11) and (12) get phase gradient, with phase unwrapping algorithm based on the principle of shearing. Then can be have removal carrier of phase unwrapping. 4. Simulation of carrier removal based on the principle of shearing In order to test the methods, we simulate the process. Assume x = y = 1.000 mm, K = M = 256, a two-dimensional distribution phase phoo of 512 × 512 pixels is established using peaks function of Matlab and superimposed a linear distribution of two-dimensional phase for simulation of carrier phase caused: pho = 2x + 2y, take = 80 m−1 and = 100 m−1 respectively for the carrier phase in the x and y direction, finally concluded that ph = pho + phoo. The results are shown in Fig. 1. Fig. 1(a) is used for simulation phase phoo. Fig. 1(b) is a wrapping of total phase (including the carrier phase), we can see, because of overlay the carrier of the phase pho, the ph appear overlapping in the two areas (arrow point), that bring difficulties for unwrapping. Fig. 1(c) is obtained by using our new phase phoo, Fig. 1(d) is a new method error distribution of phase phoo. Simulation calculated = 79.6997 m−1 and = 99.7040 m−1 , the relative error of 0.375% and 0.296%. Fig. 1(e) is phase unwrapping result by using least squares algorithms of Fig. 1(b), then use Ref. [9] method for removal of carrier, because of the ph appear overlapping in the two areas, result the unwrapping failure. Fig. 1(f) is used method of Ref. [10] to find and removal carrier, after that use least squares
5156
F. Rao / Optik 126 (2015) 5154–5157
Fig. 1. Simulated results.
Fig. 3. Experimental results by new carrier removal methods.
Fig. 2. Experimental setup.
algorithms unwrapping obtain error distribution of phase. As the carrier phase from the spectrum when the spectrum to find the subscript can only take integer, conversion to get = 85.9375 m−1 , = 101.5625 m−1 , the relative error of 7.422% and 1.563%, larger than the new method. Simulation calculations show that primitive phase more accurate for make use of shearing principle.
removing the carrier phase of cosine and 3D distribution figure of reconstructed wavefield unwrapping phase. Fig. 3(d) is the reconstructed wavefield wrapping phase after the candles burning, while Fig. 3(e) and (f) are new ways to calculate after remove carrier phase of cosine and 3D distribution figure of reconstructed wavefield unwrapping phase. Fig. 3(g) is the experimental distribution from interference fringes, Fig. 3(i) is phase three-dimensional distribution by the candle burning before subtract after of removal carrier phase unwrapping, while Fig. 3(h) is the interference fringe phase obtained by taking the cosine distribution. Compare with Fig. 3(g) and (h) we can see that the theoretical and experimental values match well.
5. Experiments 6. Conclusions We measured the candle burning temperature field in the experiments [12] to verify that the use of shear principle for removing the carrier before unwrapping. Fig. 2 is a schematic diagram of the experiment optical path. Laser beam ( = 532 nm) comes from YAG laser which is divided into two optical beam by beamsplitter (BS1 ). One beam passes through the microscope objective (L1 ), and pinhole filter (h1) and collimated lens (L2 ) generate a parallel ray, by candle combustion area reaches beamsplitter (BS2 ), and then after reflection and lens (L5 ) together as object wavefield reaches of by the holograph. The other beam reflected by reflector (M1 ) and (M2 ) then expanded by lens L3 , pinhole filter (h2 ) and collimated lens (L4 ) generate parallel ray, and then after beamsplitter (BS2 ) and lens (L5 ) together as reference wave then recorded on the holography. By the hologram, we used the CMOS to record the hologram that is produced by the interference between the reference wave and object wave, The CMOS used contains 2048 pixel × 1536 pixel, the pixel size of which is 3.2 m × 3.2 m. The hologram may obtain ˜ the reconstruction optical field U(x, y) after for the diffraction calculation. Then use carrier removal method based on the principle of shearing and reconstruction phase of optical field, the result is listed in Fig. 3 In Fig. 3 image sizes are 300 pixel × 300 pixel to all. Fig. 3(a) is the reconstructed wavefield wrapping phase before the candles burning, while Fig. 3(b) and (c) are new ways to calculate after
In holographic interferometry and projecting grating of 3-D profilometry, the carrier phase affects on the true phase of reconstruction, is required to remove carrier to accurately reconstruct the object phase information. Shearing principle was introduced to removal carrier operations in this paper. Firstly, obtain the phase gradient distribution for shearing optical field, and then, subtract from phase gradient the average to removal carrier, that is, the unwrapping completed before the parcel method to removal carrier. Simulations and experiments show that this method to reduce the difficulty of phase unwrapping, can effectively removal carrier. The method is better than that finishes unwrapping operation subtracting its first order derivative of removal carrier, and is better than using the shift frequency filter out carrier phases, the reconstruction of phase values are close to the object of true phase. References [1] Bingheng Xiong, Zhengrong Wang, Yongan Zhang, et al., A novel testing method of transparent objects in real-time holograph, Acta Opt. Sin. 21 (7) (2001) 841–845. [2] Cheng Liu, Liangyu Li, Yinzhu Li, et al., Digital holography free of zeroorder diffraction and conjugate image, Acta Opt. Sin. 22 (4) (2002) 427–431.
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