Phase noise analysis of two wavelength coherent imaging system Benjamin R. Dapore,1 David J. Rabb,2 and Joseph W. Haus1,* 1
LADAR and Optical Communications Institute, Electro-Optics Program, University of Dayton, 300 College Park, Dayton, OH 45469-2951, USA Air Force Research Laboratory – Sensors Directorate, AFRL/RYMM, 3109 Hobson Way, Wright Patterson Air Force Base, Ohio 45433-7700, USA *[email protected]
2
Abstract: Two wavelength coherent imaging is a digital holographic technique that offers several advantages over conventional coherent imaging. One of the most significant advantages is the ability to extract 3D target information from the phase contrast image at a known difference frequency. However, phase noise detracts from the accuracy at which the target can be faithfully identified. We therefore describe a method for relating phase noise to the correlation of the image planes corresponding to each wavelength, among other parameters. The prediction of the phase noise spectrum of a scene will aid in determining our ability to reconstruct the target. ©2013 Optical Society of America OCIS codes: (090.1995) Digital holography; (100.3175) Interferometric imaging; (110.4280) Noise in imaging systems; (110.6150) Speckle imaging; (070.7345) Wave propagation.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
U. Schnars and W. Jueptner, Digital Holography, (Springer, Berlin, 2005). M. Lee, O. Yaglidere, and A. Ozcan, “Field-portable reflection and transmission microscopy based on lensless holography,” Biomed. Opt. Express 2(9), 2721–2730 (2011). J. C. Marron and R. L. Kendrick, “Distributed aperture active Imaging,” Proc. SPIE 6550, 65500A, 65500A-7 (2007). D. J. Rabb, D. F. Jameson, A. J. Stokes, and J. W. Stafford, “Distributed aperture synthesis,” Opt. Express 18(10), 10334–10342 (2010). J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Höft, “Atmospheric turbulence correction using digital holographic detection: experimental results,” Opt. Express 17(14), 11638–11651 (2009). N. J. Miller, J. W. Haus, P. McManamon, and D. Shemano, “Multi-aperture coherent imaging,” Proc. SPIE 8052, 805207 (2011). G. Nehmetallah, P. Banerjee and N. Kukhtarev, “Single-beam holographic tomography creates images in three dimensions,” SPIE Newsroom (2011). doi: 10.1117/ 2.1201102.003474. M. Yonemura, T. Nishisaka, and H. Machida, “Endoscopic hologram interferometry using fiber optics,” Appl. Opt. 20(9), 1664–1667 (1981). P. P. Banerjee, G. Nehmetallah, N. Kukhtarev, and S. C. Praharaj, “Dynamic holographic interferometry of diffuse objects and its application to determination of airplane attitudes,” Appl. Opt. 47, 3877–3885 (2008). J. W. Goodman, Speckle Phenomena in Optics (Theory and Applications) (Roberts & Company, Englewood, Colorado, 2007). J. C. Marron, “Wavelength decorrelation of laser speckle from three-dimensional diffuse objects,” Opt. Commun. 88(4-6), 305–308 (1992). J. R. Fienup and A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7(3), 450–458 (1990). M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33(2), 156–158 (2008). T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, and C. Depeursinge, “Polarization Imaging by Use of Digital Holography,” Appl. Opt. 41(1), 27–37 (2002). J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15(12), 7231–7242 (2007).
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30642
16. J. W. Haus, B. Dapore, N. Miller, P. Banerjee, G. Nehmetallah, P. Powers, and P. McManamon, “Instantaneously captured images using multi-wavelength digital holography,” Proc. SPIE 8493, 84930W, 84930W-7 (2012).
1. Introduction Digital holography (DH) is a powerful tool that has a wide range of applications. Conventional off-axis DH offers a convenient way of reconstructing the 3D object using numerical reconstruction techniques [1]. Along with DH interferometry and DH microscopy, DH has been used for various metrological applications, viz., deformation detection, visualization and quantization, and 3D surface profilometry. An important issue in DH is the recovery of the phase of the object which relates to its depth information. Applications include microscopy [1,2], laser radar (LADAR) [3–6], tomography [7], medical diagnosis [8], and non-destructive testing [9]. Two-wavelength digital holography offers many advantages over single wavelength holography. Using a single wavelength the height of the object is ambiguous when it is larger than the wavelength of incident light and a surface relief cannot be reconstructed. Two wavelength DH mitigates the surface height ambiguity by constructing a DH with a synthetic wavelength that is greater than each wavelength and therefore the unambiguous range is greatly increased from microns to millimeter lengths or longer. It is able to create a three dimensional topographic image of light scattered from the target using the frequency difference between them to create a synthetic wavelength. The “beat frequency” between the two signals, has a synthetic wavelength, λS, defined as λs =
λ1λ2 λ1 − λ2
(1)
,
where λ1 and λ2 are the two laser wavelengths. The wavelengths are close to one another so in our discussion below we will simply denote them by λ. Creating a synthetic wavelength image at a known difference frequency enables the extraction of the target profile and surface relief data. However, there is an inherent issue in the reconstruction of synthetic wavelength hologram, namely phase noise in the data. Phase noise degrades the precision at which surface relief data can be extracted from a given synthetic wavelength image. Significant work has been undertaken in the past to describe the phase noise between coherent images for a two wavelength system. Much of this work has focused on relating the correlation between the two separate wavelength images, defined as the cross-spectral correlation coefficient, which is used to predict how speckle will impact the phase measurements [10,11]. The definition of the correlation coefficient is taken from Goodman [10]. The correlation coefficient is defined as the correlation between the complex image fields, U1 and U2, corresponding to each wavelength,
μ=
U U* 1 2 2 2 U U 1 2
(2)
.
The overbar denotes an ensemble average over the complex field amplitudes. The parameter, μ, determines the probability density function 1 − μ 2 (1 − β 2 )1/2 + βπ − β cos −1 β p ( Δθ ) = 2π (1 − β 2 )3/2
,
(3)
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30643
which describes the overall phase noise where Δθ is the random variable for the system. Equation (3) is a function of the phase difference, Δθ, which appear in the variable β = μ cos(Δθ ) . The phase difference is the deviation of the phase from the average phase of the object for the synthetic wavelength. The theoretic curves which are generated by Eq. (3) are shown for various correlation values in Fig. 1.
Fig. 1. Theoretical probability curves for various correlation values. The parameter μ is varied from 0 to 0.9.
Marron [11] similarly calculated a cross-spectral correlation coefficient to generate a range profile. The range profile is defined as
σ ( z)
2
=
σ ( x, y , z )
2
dxdy ,
(4)
where σ ( x, y, z ) is the complex reflectivity, z is the source to target distance and (x,y) are the cross-range coordinates. The function in Eq. (4) is related to the time-domain reflected intensity of the target projected into x,y,z space. The brackets denote an ensemble average over many target random surface shapes and the function is integrated over the crossrange coordinates. For a target whose only contributions come from roughness, the range profile can be given physical parameters. Assuming a Gaussian functional dependence Eq. (4) becomes
σ ( z)
2 0
1 z 2 = exp − , 2 σ R
(5)
where σR is parameter quantifying the surface roughness of the target. The subscript zero on the brackets denotes a flat surface profile. However this form assumes a globally flat target with random roughness. In practice, a target will be composed of many facets with different tilts. We therefore will extend the expression beyond the flat target case, and explore a relationship that links the phase noise spectrum to a cross-spectral correlation coefficient for a finite aperture size. We begin by taking a process similar to that in Eq. (4) and Eq. (5). First, we assume a diffraction limited rectangular aperture, and generate a range profile. The range profile is given by
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30644
σ ( z) where for simplicity
2
θx
= T
δ ( z − ( tan θ x ) x ) sin c
Dx Dy sin c λL λL
2 dxdy,
(6)
is the tilt angle of the target with respect to the x axis, L = 8.1 m is
source to target distance, and D = 7.7 mm is the receiver aperture width and height. The subscript T on the left hand side denotes a tilted target contribution. The delta function denotes the reflection only takes place at the target surface, that is where z = (tan θ x )x. Since it is assumed that the target is only tilted in x, the y terms integrate out and Eq. (6) can be written as
σ ( z)
2
Dz sin c λ ( tan θ x ) L
= T
2 .
(7)
Beyond the flat case, the range profile becomes a convolution of the flat case’s range profile and the range profile attributed to the PSF,
σ ( z)
2
= σ ( z)
2 0
⊗ σ ( z)
2 T
2 Dz 1 z 2 = exp − . (8) ⊗ sin c σ 2 tan θxλ L R
The range profile is related to the cross-spectral correlation coefficient by [11]
μ ( Δυ ) = σ ( z )
2
−4iπ z ( Δυ ) exp dz , c
(9)
where z is source to target distance, Δν is frequency separation between two sources, and c is the speed of light. Equation (9) is a Fourier Transform of the mean-square surface profile defined by Eq. (8). This leads to the analytic result for the correlation coefficient in the tilted, finite-aperture case, 8π 2σ R2 Δν 2 f x λ L Λ , c2 D
μ = exp −
(10)
where Λ denotes a triangle function as defined by Goodman [10] and the spatial frequency is defined as fx =
2 tan θ x Δυ 2 tan θ x = . c λs
(11)
We define the cutoff angle (μ = 0) as
λs D . 2λ L
θ c = tan −1
(12)
The cutoff spatial frequency, fc, is determined from Eq. (11) at the cutoff angle. 2. Simulation results
We performed a two wavelength imaging simulation that allowed for variable tuning of the two wavelengths. The parameters were chosen such that they matched the experimental setup. A square target with a varying surface roughness of approximately 0.1mm was illuminated with a Gaussian beam. The beam was chosen so as to fully illuminate the 20cm-by-20cm target, which is a distance of 8.1 meters from the sources. The two-wavelength data is
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30645
captured in the pupil plane, to be consistent with the laboratory experiment the pupil plane data was divided into 16 separately processed 64x64 pixel sections. The reduced pupil size allows for a corresponding reduction in the cutoff angle, while the 16 separate sections serve to improve phase difference statistics. Each data section is then zero padded to a size of 256x256 pixels. This allows for the image of the target to appear in a 128x128 quadrant of choice, based on local oscillator or reference beam tilt. At each wavelength the data is Fourier transformed to recover the complex image plane data. An example image of the target after propagation, assuming a far-field approximation is shown below in Fig. 2.
20
Pixels
40 60 80 100 120 20
40
60 Pixels
80
100
120
Fig. 2. Image of target corresponding to one wavelength.
To create a synthetic wavelength image, the complex amplitude field corresponding to the first wavelength is multiplied with the complex conjugate of the field corresponding to the second wavelength. Looking at the angular spectrum of this image provides what is called a synthetic angle image. Figure 3 shows the synthetic angle image before and after processing to remove DC contributions [12,13]. In processing Fig. 3(a), a flat plane is fitted to the synthetic image. Any of the fluctuations that remain after fitting to this tilted plane are assumed to be phase noise. In this way, we can remove the contributions from tilt and investigate only the phase noise of the system.
Fig. 3. (a) Synthetic angle image before processing; (b) Synthetic angle image after processing.
Creating a histogram of the data in Fig. 3(b) gives a PDF that can be compared to the PDFs generated by Eq. (3). By calculating a correlation coefficient value for this figure, a
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30646
PDF can also be generated. These PDFs for the flat target case are compared with the theoretical PDF all the way to the cutoff (tilt) angle. Our correlation coefficient accurately describes the phase noise in the system over the entire range of angles. Figure 4 displays the PDF results for two different tilt cases. Only the positive phase difference values are shown, since the PDF is symmetric around zero.
Fig. 4. (a) Comparison of simulation and theoretical PDFs for a target tilt of 10°, which corrseponds to a value of fx that is 19% fc; (b) PDF comparison, for a target tilt of 22° and the value of fx is 44% fc.
As seen in Fig. 4, the correlation coefficients and their corresponding PDFs closely follow across the phase difference range. The dashed (red) lines display the simulation results which tend to be slightly flattened compared to the theoretical PDF. In the following section experimental data will be compared against the theoretical predictions. 3. Experimental results
An experiment was designed to confirm the results of the model simulations and also to validate our theoretical calculations. A schematic of the experimental setup is shown in Fig. 5. A two-wavelength spatial heterodyne imaging system was set up, with both wavelengths being simultaneously captured. Previous work has shown that this can be done without any limitations due to crosstalk [14–16]. We used two Teraxion Pure Spectrum Laser Modules (Teraxion PS-LM) with frequency stabilized center wavelengths 1545 nm and 1545.4 nm tunable to 5 MHz (i.e. a 50 GHz frequency offset or a 6 mm synthetic wavelength). The output power is 80 mW. The co-bored laser light was collimated at the fiber output of the coupler; the Gaussian beam spread over the target. The two local oscillators (LO 1 and LO 2) emerge from their fibers in the target plane and they are placed to displace the holograms into neighboring quadrants when post processing the data into image plane. The target plane was chosen for the LOs since it gives them the same curvature as the target field on the hologram and thus no focus corrections were required to sharpen the holographic images. The images are simultaneously captured in the pupil plane on a lensless InGaAs camera (FLIR SC2500). A metal target was sandblasted in order to give it a roughness. The roughness was determined for the flattened target by determining the corresponding PDF with a best fit parameter μ = 0.973. This value of μ corresponds to a roughness of 72.3 λ. The target was placed a distance of 8.1 m from the sources, and it was simultaneously illuminated by both sources. A rotation stage is used in order to move the target from flat through the cutoff angle.
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30647
Fig. 5. Simplified schematic of experimental setup.
The intensity recorded on the camera, which is in the pupil plane as shown in Fig. 5, is 2
2
* I ( x, y ) = U LO1 ( x, y ) + U s1 ( x, y ) + U LO1 ( x, y ) U*s1 ( x, y ) + U LO 1 ( x, y )U s1 ( x, y )
(13)
2
2
* + U LO 2 ( x, y ) + U s2 ( x, y ) + U LO 2 ( x, y ) U*s2 ( x, y ) + U LO 2 ( x, y )U s2 ( x, y )
,
where ULO denotes a local oscillator (or reference) field at the camera, Us denotes the backscattered target field at the camera, and * denotes a complex conjugate of the complex field. The first line is the digital hologram recorded for λ1 and the second line is the same for λ2. Experimental data are processed to retrieve the synthetic wavelength images, which are shown in Fig. 6. By placing the local oscillator (LO) sources in the target plane its recorded field has the same phase curvature as that of the backscattered target field and the complex conjugate products cancels the field curvatures, the field in the image plane can be found by taking the Fourier transform of the pupil plane intensity,
{ } + ℑ{ U ( x, y ) } + ℑ{U ( x, y ) U ( x, y )} + ℑ{U ( x, y )U ( x , y ) } + ℑ { U ( x , y ) } + ℑ {U ( x , y ) U ( x , y )} + ℑ {U ( x , y )U ( x , y )}
ℑ {I ( x , y )} = ℑ U LO 1 ( x , y )
{
+ℑ U LO 2
2
2
s1
LO 1
2
2
s2
LO 2
*
*
s1
LO 1
*
*
s2
LO 2
s2
s1
}
( x, y )
(14)
.
The first two terms on each line above are autocorrelations of the LO and backscattered fields, respectively. The last two terms on each line are the images and their twin (complex conjugate) image. The local oscillators are positioned so that at the camera they are tilted plane waves with amplitude ALO. This allows for simplification of Eq. (14) as [4] 2 2 ℑ{ I ( x, y )} = ALO 1δ ( f x , f y ) + ALO 2δ ( f x , f y )
+ ℑ{U s1 ( x, y )} + ℑ{U s1 ( x, y )} 2
{
2
}
+ ALO1ℑ U s*1 ( x, y ) * δ ( f x − f x1 , f y − f y1 ) ,
{ ℑ{U ℑ{U
} ( x, y )} * δ ( f ( x, y )} * δ ( f
+ ALO 2 ℑ U s*2 ( x, y ) * δ ( f x − f x 2 , f y − f y 2 ) + ALO1
s1
+ ALO 2
s2
x x
(15)
+ f x1 , f y + f y1 )
+ fx2 , f y + f y2 )
assuming that the tilt angles are small enough such that the interference fringes are accurately captured. fx1, fy1, fx2, fy2 are spatial heterodyne offset frequencies determined by the tilt angles of each LO and given by fxn = θx/λ, and fyn = θy/λ. In Fig. 6(b) the image plane synthetic wavelength data is shown by processing the dual-wavelength pupil plane data, as discussed earlier. The four images are two pairs of images and their twin (complex conjugate) images. The twin images lie in the diagonal quadrant across from the image. Note the target is
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30648
rectangular and the post holding the target is visible in each quadrant. Figure 6(a) is the corresponding pupil plane data at the synthetic wavelength.
Fig. 6. (a) Pupil plane image of the digital holograms at the synthetic wavelength. (b) The image plane data at the synthetic wavelength obtained by Fourier transforming the dualwavelength pupil plane data as shown in Eq. (14) or Eq. (15). The LO and signal autocorrelation contributions are removed.
The image plane complex field from the two quadrants corresponding to each wavelength can be written as
{ } ℑ{U ( x, y )} * δ ( f
U1 = ALO1ℑ U s*1 ( x, y ) * δ ( f x − f x1 , f y − f y1 ) U 2 = ALO 2
* s2
x
− f x2 , f y − f y2 ) .
(16)
The same process is repeated as in the model, creating synthetic angle images which in turn are used to create PDFs. The cutoff angle is 42.6 degrees for the system, where θc is defined in Eq. (12). Two examples of the corresponding PDFs are shown in Fig. 7.
Fig. 7. (a) PDFs corresponding to five degree target tilt, (b) PDFs corresponding to twenty degree target tilt.
The histogram of the data has amplitude fluctuations due to finite sampling. Figure 7(a) is a relatively small tilt angle, but the distribution extends with nonzero amplitude over the entire range of angles. In Fig. 7(b) the angle is closer to the cutoff angle and the PDF is flattened with almost the same probability density at every phase difference value. The comparison between experimental data and the theoretical result for measurements made
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30649
from zero to the cutoff angle is shown in Fig. 8. The agreement between theory and experiment across the entire range is excellent for a simple planar target.
Fig. 8. The μ parameter from theory and experiment as a function of target tilt.
4. Complex target morphology
We have shown that our equation for a theoretical correlation coefficient holds for a planar target that is tilted at any angle below the cutoff angle. However, this is for a square target with a tilt only in one direction. A real-world target scenario would involve a target that has multiple tilts and facets that are contributing to the phase statistics. Thus, the results need to be extended to a more general situation. First, we define our correlation coefficient in terms of the spatial frequency content of the system, 8π 2σ 2 Δυ 2 T Λ λ Lf x 2 D c
μ ( f x , f y ) = exp −
λ Lf y Λ D ,
(17)
where the spatial frequency fx is defined in Eq. (11) and there is an analogous expression for fy. We then developed a complex scene to see if this relationship holds. Figure 9(a) shows a side view of a computer rendered scene that contains elements with many tilts. Within the box there is a truck behind the tree. The phase plot in Fig. 9(b) is a top view of the same scene; the scene has a constant tilt background, as well as leaf-like objects and a truck that possess many different tilts.
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30650
Fig. 9. (a) Side view of a complex rendered scene; (b) MATLAB point cloud image showing a top view of the scene with multiple tilts and facets for a synthetic wavelength of 0.5 m. The truck appears in the top center of the scene in yellow.
However, when determining the PDF of the phase noise of the complex scene to compare to the theoretical PDF, an account of the spatial frequency content in the scene has to be done. The PDF for the system is a weighted sum over a large number of tilts with the weight being given by the complex angular spectrum of the scene. Thus, to generate an overall PDF for the scene we assume that each correlation coefficient value, which is based on the spatial frequency content, is weighted by the spectral content of the scene
Overall _ PDF =
pdf ( μ ( f
fx , f y
fx , f y
x
)
, f y ) spectrum ( f x , f y )
spectrum ( f x , f y )
2
2
,
(18)
where the spectrum is defined as an imaginary exponential of the target heights h(x,y) relative to the synthetic wavelength 4iπ h ( x, y ) spectrum f x , f y = ℑ exp . λs
(
)
(19)
Using Eq. (19) the relationship between our newly formulated theoretical PDF and the phase noise PDF for the simulated scene is shown (red line) in Fig. 10 for a synthetic wavelength of 0.5 m. In Fig. 10(a) the fit of the PDF to a single μ value is shown. It under estimates the simulated PDF at small angular differences and over estimates it at larger angular differences. In Fig. 10(b) the deviation is over estimated at small angular differences, but remains a faithful representation at larger angular differences. Given the complexity of the scene the agreement is reasonably good with some deviations apparent for small angular differences.
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30651
Fig. 10. Simulated PDFs from the scene in Fig. 9 compared to (a) the PDF from a single μ value and (b) the PDF generated by ensemble average using Eq. (18). The synthetic wavelength is 0.5 m.
5. Conclusions
We have formulated an expression that describes the phase noise within a synthetic wavelength imaging system for a complex target. This equation allows for calculation of correlation coefficients at different target tilt angles, which describes distribution of phase fluctuations over the target. We performed both numerical simulations and experiments on two-frequency holography to quantify the results and compare with the theoretical phase angle distribution expression. Our simulations and experiments on simple flat targets with pre-assigned tilts show good agreement between the theoretical correlation coefficients and the data. A complex scene characterized by a target with many facets with different tilts was characterized by a PDF that is weighted according to the Fourier spectrum of the target surface heights. The complexity of the scene is an initial validation of the main result we presented in Eq. (18). Although our results will require further examination in the future it has been shown to work in cases from a simple flat targets with prescribed tilts to a target with many facets and many tilts. Multi-wavelength holography allows for range information to be faithfully reconstructed using the phase information, which is robust against speckle noise in the scene. The reconstructed object intensity from a hologram of a scene with fully developed speckle would have a signal to noise ratio (SNR) of unity, but the synthetic angle image extracted from a single speckle realization can have significantly higher SNR. The application of the work presented here is to better inform how the phase difference statistics (noise in the synthetic angle image) are affected by the target tilts, synthetic wavelength, and the imaging system’s point spread function. Future work will be to further account for discrepancies between the theoretical and simulated phase difference PDFs for complex targets, and to include higher fidelity modeling of the object with reflectivity calculated as a function of material bidirectional reflectance distribution function. Further work could also include a description of the phase difference PDF for an averaged synthetic angle image created from multiple speckle realizations of the object. Acknowledgments
This effort was supported in part by the U.S. Air Force through contract number FA8650-062-1081, and the University of Dayton Ladar and Optical Communications Institute (LOCI). The views expressed in this article are those of the authors and do not reflect on the official policy of the Air Force, Department of Defense or the U.S. Government.
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30652
LADAR and Optical Communications Institute, Electro-Optics Program, University of Dayton, 300 College Park, Dayton, OH 45469-2951, USA Air Force Research Laboratory – Sensors Directorate, AFRL/RYMM, 3109 Hobson Way, Wright Patterson Air Force Base, Ohio 45433-7700, USA *[email protected]
2
Abstract: Two wavelength coherent imaging is a digital holographic technique that offers several advantages over conventional coherent imaging. One of the most significant advantages is the ability to extract 3D target information from the phase contrast image at a known difference frequency. However, phase noise detracts from the accuracy at which the target can be faithfully identified. We therefore describe a method for relating phase noise to the correlation of the image planes corresponding to each wavelength, among other parameters. The prediction of the phase noise spectrum of a scene will aid in determining our ability to reconstruct the target. ©2013 Optical Society of America OCIS codes: (090.1995) Digital holography; (100.3175) Interferometric imaging; (110.4280) Noise in imaging systems; (110.6150) Speckle imaging; (070.7345) Wave propagation.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
U. Schnars and W. Jueptner, Digital Holography, (Springer, Berlin, 2005). M. Lee, O. Yaglidere, and A. Ozcan, “Field-portable reflection and transmission microscopy based on lensless holography,” Biomed. Opt. Express 2(9), 2721–2730 (2011). J. C. Marron and R. L. Kendrick, “Distributed aperture active Imaging,” Proc. SPIE 6550, 65500A, 65500A-7 (2007). D. J. Rabb, D. F. Jameson, A. J. Stokes, and J. W. Stafford, “Distributed aperture synthesis,” Opt. Express 18(10), 10334–10342 (2010). J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Höft, “Atmospheric turbulence correction using digital holographic detection: experimental results,” Opt. Express 17(14), 11638–11651 (2009). N. J. Miller, J. W. Haus, P. McManamon, and D. Shemano, “Multi-aperture coherent imaging,” Proc. SPIE 8052, 805207 (2011). G. Nehmetallah, P. Banerjee and N. Kukhtarev, “Single-beam holographic tomography creates images in three dimensions,” SPIE Newsroom (2011). doi: 10.1117/ 2.1201102.003474. M. Yonemura, T. Nishisaka, and H. Machida, “Endoscopic hologram interferometry using fiber optics,” Appl. Opt. 20(9), 1664–1667 (1981). P. P. Banerjee, G. Nehmetallah, N. Kukhtarev, and S. C. Praharaj, “Dynamic holographic interferometry of diffuse objects and its application to determination of airplane attitudes,” Appl. Opt. 47, 3877–3885 (2008). J. W. Goodman, Speckle Phenomena in Optics (Theory and Applications) (Roberts & Company, Englewood, Colorado, 2007). J. C. Marron, “Wavelength decorrelation of laser speckle from three-dimensional diffuse objects,” Opt. Commun. 88(4-6), 305–308 (1992). J. R. Fienup and A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7(3), 450–458 (1990). M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33(2), 156–158 (2008). T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, and C. Depeursinge, “Polarization Imaging by Use of Digital Holography,” Appl. Opt. 41(1), 27–37 (2002). J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15(12), 7231–7242 (2007).
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30642
16. J. W. Haus, B. Dapore, N. Miller, P. Banerjee, G. Nehmetallah, P. Powers, and P. McManamon, “Instantaneously captured images using multi-wavelength digital holography,” Proc. SPIE 8493, 84930W, 84930W-7 (2012).
1. Introduction Digital holography (DH) is a powerful tool that has a wide range of applications. Conventional off-axis DH offers a convenient way of reconstructing the 3D object using numerical reconstruction techniques [1]. Along with DH interferometry and DH microscopy, DH has been used for various metrological applications, viz., deformation detection, visualization and quantization, and 3D surface profilometry. An important issue in DH is the recovery of the phase of the object which relates to its depth information. Applications include microscopy [1,2], laser radar (LADAR) [3–6], tomography [7], medical diagnosis [8], and non-destructive testing [9]. Two-wavelength digital holography offers many advantages over single wavelength holography. Using a single wavelength the height of the object is ambiguous when it is larger than the wavelength of incident light and a surface relief cannot be reconstructed. Two wavelength DH mitigates the surface height ambiguity by constructing a DH with a synthetic wavelength that is greater than each wavelength and therefore the unambiguous range is greatly increased from microns to millimeter lengths or longer. It is able to create a three dimensional topographic image of light scattered from the target using the frequency difference between them to create a synthetic wavelength. The “beat frequency” between the two signals, has a synthetic wavelength, λS, defined as λs =
λ1λ2 λ1 − λ2
(1)
,
where λ1 and λ2 are the two laser wavelengths. The wavelengths are close to one another so in our discussion below we will simply denote them by λ. Creating a synthetic wavelength image at a known difference frequency enables the extraction of the target profile and surface relief data. However, there is an inherent issue in the reconstruction of synthetic wavelength hologram, namely phase noise in the data. Phase noise degrades the precision at which surface relief data can be extracted from a given synthetic wavelength image. Significant work has been undertaken in the past to describe the phase noise between coherent images for a two wavelength system. Much of this work has focused on relating the correlation between the two separate wavelength images, defined as the cross-spectral correlation coefficient, which is used to predict how speckle will impact the phase measurements [10,11]. The definition of the correlation coefficient is taken from Goodman [10]. The correlation coefficient is defined as the correlation between the complex image fields, U1 and U2, corresponding to each wavelength,
μ=
U U* 1 2 2 2 U U 1 2
(2)
.
The overbar denotes an ensemble average over the complex field amplitudes. The parameter, μ, determines the probability density function 1 − μ 2 (1 − β 2 )1/2 + βπ − β cos −1 β p ( Δθ ) = 2π (1 − β 2 )3/2
,
(3)
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30643
which describes the overall phase noise where Δθ is the random variable for the system. Equation (3) is a function of the phase difference, Δθ, which appear in the variable β = μ cos(Δθ ) . The phase difference is the deviation of the phase from the average phase of the object for the synthetic wavelength. The theoretic curves which are generated by Eq. (3) are shown for various correlation values in Fig. 1.
Fig. 1. Theoretical probability curves for various correlation values. The parameter μ is varied from 0 to 0.9.
Marron [11] similarly calculated a cross-spectral correlation coefficient to generate a range profile. The range profile is defined as
σ ( z)
2
=
σ ( x, y , z )
2
dxdy ,
(4)
where σ ( x, y, z ) is the complex reflectivity, z is the source to target distance and (x,y) are the cross-range coordinates. The function in Eq. (4) is related to the time-domain reflected intensity of the target projected into x,y,z space. The brackets denote an ensemble average over many target random surface shapes and the function is integrated over the crossrange coordinates. For a target whose only contributions come from roughness, the range profile can be given physical parameters. Assuming a Gaussian functional dependence Eq. (4) becomes
σ ( z)
2 0
1 z 2 = exp − , 2 σ R
(5)
where σR is parameter quantifying the surface roughness of the target. The subscript zero on the brackets denotes a flat surface profile. However this form assumes a globally flat target with random roughness. In practice, a target will be composed of many facets with different tilts. We therefore will extend the expression beyond the flat target case, and explore a relationship that links the phase noise spectrum to a cross-spectral correlation coefficient for a finite aperture size. We begin by taking a process similar to that in Eq. (4) and Eq. (5). First, we assume a diffraction limited rectangular aperture, and generate a range profile. The range profile is given by
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30644
σ ( z) where for simplicity
2
θx
= T
δ ( z − ( tan θ x ) x ) sin c
Dx Dy sin c λL λL
2 dxdy,
(6)
is the tilt angle of the target with respect to the x axis, L = 8.1 m is
source to target distance, and D = 7.7 mm is the receiver aperture width and height. The subscript T on the left hand side denotes a tilted target contribution. The delta function denotes the reflection only takes place at the target surface, that is where z = (tan θ x )x. Since it is assumed that the target is only tilted in x, the y terms integrate out and Eq. (6) can be written as
σ ( z)
2
Dz sin c λ ( tan θ x ) L
= T
2 .
(7)
Beyond the flat case, the range profile becomes a convolution of the flat case’s range profile and the range profile attributed to the PSF,
σ ( z)
2
= σ ( z)
2 0
⊗ σ ( z)
2 T
2 Dz 1 z 2 = exp − . (8) ⊗ sin c σ 2 tan θxλ L R
The range profile is related to the cross-spectral correlation coefficient by [11]
μ ( Δυ ) = σ ( z )
2
−4iπ z ( Δυ ) exp dz , c
(9)
where z is source to target distance, Δν is frequency separation between two sources, and c is the speed of light. Equation (9) is a Fourier Transform of the mean-square surface profile defined by Eq. (8). This leads to the analytic result for the correlation coefficient in the tilted, finite-aperture case, 8π 2σ R2 Δν 2 f x λ L Λ , c2 D
μ = exp −
(10)
where Λ denotes a triangle function as defined by Goodman [10] and the spatial frequency is defined as fx =
2 tan θ x Δυ 2 tan θ x = . c λs
(11)
We define the cutoff angle (μ = 0) as
λs D . 2λ L
θ c = tan −1
(12)
The cutoff spatial frequency, fc, is determined from Eq. (11) at the cutoff angle. 2. Simulation results
We performed a two wavelength imaging simulation that allowed for variable tuning of the two wavelengths. The parameters were chosen such that they matched the experimental setup. A square target with a varying surface roughness of approximately 0.1mm was illuminated with a Gaussian beam. The beam was chosen so as to fully illuminate the 20cm-by-20cm target, which is a distance of 8.1 meters from the sources. The two-wavelength data is
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30645
captured in the pupil plane, to be consistent with the laboratory experiment the pupil plane data was divided into 16 separately processed 64x64 pixel sections. The reduced pupil size allows for a corresponding reduction in the cutoff angle, while the 16 separate sections serve to improve phase difference statistics. Each data section is then zero padded to a size of 256x256 pixels. This allows for the image of the target to appear in a 128x128 quadrant of choice, based on local oscillator or reference beam tilt. At each wavelength the data is Fourier transformed to recover the complex image plane data. An example image of the target after propagation, assuming a far-field approximation is shown below in Fig. 2.
20
Pixels
40 60 80 100 120 20
40
60 Pixels
80
100
120
Fig. 2. Image of target corresponding to one wavelength.
To create a synthetic wavelength image, the complex amplitude field corresponding to the first wavelength is multiplied with the complex conjugate of the field corresponding to the second wavelength. Looking at the angular spectrum of this image provides what is called a synthetic angle image. Figure 3 shows the synthetic angle image before and after processing to remove DC contributions [12,13]. In processing Fig. 3(a), a flat plane is fitted to the synthetic image. Any of the fluctuations that remain after fitting to this tilted plane are assumed to be phase noise. In this way, we can remove the contributions from tilt and investigate only the phase noise of the system.
Fig. 3. (a) Synthetic angle image before processing; (b) Synthetic angle image after processing.
Creating a histogram of the data in Fig. 3(b) gives a PDF that can be compared to the PDFs generated by Eq. (3). By calculating a correlation coefficient value for this figure, a
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30646
PDF can also be generated. These PDFs for the flat target case are compared with the theoretical PDF all the way to the cutoff (tilt) angle. Our correlation coefficient accurately describes the phase noise in the system over the entire range of angles. Figure 4 displays the PDF results for two different tilt cases. Only the positive phase difference values are shown, since the PDF is symmetric around zero.
Fig. 4. (a) Comparison of simulation and theoretical PDFs for a target tilt of 10°, which corrseponds to a value of fx that is 19% fc; (b) PDF comparison, for a target tilt of 22° and the value of fx is 44% fc.
As seen in Fig. 4, the correlation coefficients and their corresponding PDFs closely follow across the phase difference range. The dashed (red) lines display the simulation results which tend to be slightly flattened compared to the theoretical PDF. In the following section experimental data will be compared against the theoretical predictions. 3. Experimental results
An experiment was designed to confirm the results of the model simulations and also to validate our theoretical calculations. A schematic of the experimental setup is shown in Fig. 5. A two-wavelength spatial heterodyne imaging system was set up, with both wavelengths being simultaneously captured. Previous work has shown that this can be done without any limitations due to crosstalk [14–16]. We used two Teraxion Pure Spectrum Laser Modules (Teraxion PS-LM) with frequency stabilized center wavelengths 1545 nm and 1545.4 nm tunable to 5 MHz (i.e. a 50 GHz frequency offset or a 6 mm synthetic wavelength). The output power is 80 mW. The co-bored laser light was collimated at the fiber output of the coupler; the Gaussian beam spread over the target. The two local oscillators (LO 1 and LO 2) emerge from their fibers in the target plane and they are placed to displace the holograms into neighboring quadrants when post processing the data into image plane. The target plane was chosen for the LOs since it gives them the same curvature as the target field on the hologram and thus no focus corrections were required to sharpen the holographic images. The images are simultaneously captured in the pupil plane on a lensless InGaAs camera (FLIR SC2500). A metal target was sandblasted in order to give it a roughness. The roughness was determined for the flattened target by determining the corresponding PDF with a best fit parameter μ = 0.973. This value of μ corresponds to a roughness of 72.3 λ. The target was placed a distance of 8.1 m from the sources, and it was simultaneously illuminated by both sources. A rotation stage is used in order to move the target from flat through the cutoff angle.
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30647
Fig. 5. Simplified schematic of experimental setup.
The intensity recorded on the camera, which is in the pupil plane as shown in Fig. 5, is 2
2
* I ( x, y ) = U LO1 ( x, y ) + U s1 ( x, y ) + U LO1 ( x, y ) U*s1 ( x, y ) + U LO 1 ( x, y )U s1 ( x, y )
(13)
2
2
* + U LO 2 ( x, y ) + U s2 ( x, y ) + U LO 2 ( x, y ) U*s2 ( x, y ) + U LO 2 ( x, y )U s2 ( x, y )
,
where ULO denotes a local oscillator (or reference) field at the camera, Us denotes the backscattered target field at the camera, and * denotes a complex conjugate of the complex field. The first line is the digital hologram recorded for λ1 and the second line is the same for λ2. Experimental data are processed to retrieve the synthetic wavelength images, which are shown in Fig. 6. By placing the local oscillator (LO) sources in the target plane its recorded field has the same phase curvature as that of the backscattered target field and the complex conjugate products cancels the field curvatures, the field in the image plane can be found by taking the Fourier transform of the pupil plane intensity,
{ } + ℑ{ U ( x, y ) } + ℑ{U ( x, y ) U ( x, y )} + ℑ{U ( x, y )U ( x , y ) } + ℑ { U ( x , y ) } + ℑ {U ( x , y ) U ( x , y )} + ℑ {U ( x , y )U ( x , y )}
ℑ {I ( x , y )} = ℑ U LO 1 ( x , y )
{
+ℑ U LO 2
2
2
s1
LO 1
2
2
s2
LO 2
*
*
s1
LO 1
*
*
s2
LO 2
s2
s1
}
( x, y )
(14)
.
The first two terms on each line above are autocorrelations of the LO and backscattered fields, respectively. The last two terms on each line are the images and their twin (complex conjugate) image. The local oscillators are positioned so that at the camera they are tilted plane waves with amplitude ALO. This allows for simplification of Eq. (14) as [4] 2 2 ℑ{ I ( x, y )} = ALO 1δ ( f x , f y ) + ALO 2δ ( f x , f y )
+ ℑ{U s1 ( x, y )} + ℑ{U s1 ( x, y )} 2
{
2
}
+ ALO1ℑ U s*1 ( x, y ) * δ ( f x − f x1 , f y − f y1 ) ,
{ ℑ{U ℑ{U
} ( x, y )} * δ ( f ( x, y )} * δ ( f
+ ALO 2 ℑ U s*2 ( x, y ) * δ ( f x − f x 2 , f y − f y 2 ) + ALO1
s1
+ ALO 2
s2
x x
(15)
+ f x1 , f y + f y1 )
+ fx2 , f y + f y2 )
assuming that the tilt angles are small enough such that the interference fringes are accurately captured. fx1, fy1, fx2, fy2 are spatial heterodyne offset frequencies determined by the tilt angles of each LO and given by fxn = θx/λ, and fyn = θy/λ. In Fig. 6(b) the image plane synthetic wavelength data is shown by processing the dual-wavelength pupil plane data, as discussed earlier. The four images are two pairs of images and their twin (complex conjugate) images. The twin images lie in the diagonal quadrant across from the image. Note the target is
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30648
rectangular and the post holding the target is visible in each quadrant. Figure 6(a) is the corresponding pupil plane data at the synthetic wavelength.
Fig. 6. (a) Pupil plane image of the digital holograms at the synthetic wavelength. (b) The image plane data at the synthetic wavelength obtained by Fourier transforming the dualwavelength pupil plane data as shown in Eq. (14) or Eq. (15). The LO and signal autocorrelation contributions are removed.
The image plane complex field from the two quadrants corresponding to each wavelength can be written as
{ } ℑ{U ( x, y )} * δ ( f
U1 = ALO1ℑ U s*1 ( x, y ) * δ ( f x − f x1 , f y − f y1 ) U 2 = ALO 2
* s2
x
− f x2 , f y − f y2 ) .
(16)
The same process is repeated as in the model, creating synthetic angle images which in turn are used to create PDFs. The cutoff angle is 42.6 degrees for the system, where θc is defined in Eq. (12). Two examples of the corresponding PDFs are shown in Fig. 7.
Fig. 7. (a) PDFs corresponding to five degree target tilt, (b) PDFs corresponding to twenty degree target tilt.
The histogram of the data has amplitude fluctuations due to finite sampling. Figure 7(a) is a relatively small tilt angle, but the distribution extends with nonzero amplitude over the entire range of angles. In Fig. 7(b) the angle is closer to the cutoff angle and the PDF is flattened with almost the same probability density at every phase difference value. The comparison between experimental data and the theoretical result for measurements made
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30649
from zero to the cutoff angle is shown in Fig. 8. The agreement between theory and experiment across the entire range is excellent for a simple planar target.
Fig. 8. The μ parameter from theory and experiment as a function of target tilt.
4. Complex target morphology
We have shown that our equation for a theoretical correlation coefficient holds for a planar target that is tilted at any angle below the cutoff angle. However, this is for a square target with a tilt only in one direction. A real-world target scenario would involve a target that has multiple tilts and facets that are contributing to the phase statistics. Thus, the results need to be extended to a more general situation. First, we define our correlation coefficient in terms of the spatial frequency content of the system, 8π 2σ 2 Δυ 2 T Λ λ Lf x 2 D c
μ ( f x , f y ) = exp −
λ Lf y Λ D ,
(17)
where the spatial frequency fx is defined in Eq. (11) and there is an analogous expression for fy. We then developed a complex scene to see if this relationship holds. Figure 9(a) shows a side view of a computer rendered scene that contains elements with many tilts. Within the box there is a truck behind the tree. The phase plot in Fig. 9(b) is a top view of the same scene; the scene has a constant tilt background, as well as leaf-like objects and a truck that possess many different tilts.
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30650
Fig. 9. (a) Side view of a complex rendered scene; (b) MATLAB point cloud image showing a top view of the scene with multiple tilts and facets for a synthetic wavelength of 0.5 m. The truck appears in the top center of the scene in yellow.
However, when determining the PDF of the phase noise of the complex scene to compare to the theoretical PDF, an account of the spatial frequency content in the scene has to be done. The PDF for the system is a weighted sum over a large number of tilts with the weight being given by the complex angular spectrum of the scene. Thus, to generate an overall PDF for the scene we assume that each correlation coefficient value, which is based on the spatial frequency content, is weighted by the spectral content of the scene
Overall _ PDF =
pdf ( μ ( f
fx , f y
fx , f y
x
)
, f y ) spectrum ( f x , f y )
spectrum ( f x , f y )
2
2
,
(18)
where the spectrum is defined as an imaginary exponential of the target heights h(x,y) relative to the synthetic wavelength 4iπ h ( x, y ) spectrum f x , f y = ℑ exp . λs
(
)
(19)
Using Eq. (19) the relationship between our newly formulated theoretical PDF and the phase noise PDF for the simulated scene is shown (red line) in Fig. 10 for a synthetic wavelength of 0.5 m. In Fig. 10(a) the fit of the PDF to a single μ value is shown. It under estimates the simulated PDF at small angular differences and over estimates it at larger angular differences. In Fig. 10(b) the deviation is over estimated at small angular differences, but remains a faithful representation at larger angular differences. Given the complexity of the scene the agreement is reasonably good with some deviations apparent for small angular differences.
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30651
Fig. 10. Simulated PDFs from the scene in Fig. 9 compared to (a) the PDF from a single μ value and (b) the PDF generated by ensemble average using Eq. (18). The synthetic wavelength is 0.5 m.
5. Conclusions
We have formulated an expression that describes the phase noise within a synthetic wavelength imaging system for a complex target. This equation allows for calculation of correlation coefficients at different target tilt angles, which describes distribution of phase fluctuations over the target. We performed both numerical simulations and experiments on two-frequency holography to quantify the results and compare with the theoretical phase angle distribution expression. Our simulations and experiments on simple flat targets with pre-assigned tilts show good agreement between the theoretical correlation coefficients and the data. A complex scene characterized by a target with many facets with different tilts was characterized by a PDF that is weighted according to the Fourier spectrum of the target surface heights. The complexity of the scene is an initial validation of the main result we presented in Eq. (18). Although our results will require further examination in the future it has been shown to work in cases from a simple flat targets with prescribed tilts to a target with many facets and many tilts. Multi-wavelength holography allows for range information to be faithfully reconstructed using the phase information, which is robust against speckle noise in the scene. The reconstructed object intensity from a hologram of a scene with fully developed speckle would have a signal to noise ratio (SNR) of unity, but the synthetic angle image extracted from a single speckle realization can have significantly higher SNR. The application of the work presented here is to better inform how the phase difference statistics (noise in the synthetic angle image) are affected by the target tilts, synthetic wavelength, and the imaging system’s point spread function. Future work will be to further account for discrepancies between the theoretical and simulated phase difference PDFs for complex targets, and to include higher fidelity modeling of the object with reflectivity calculated as a function of material bidirectional reflectance distribution function. Further work could also include a description of the phase difference PDF for an averaged synthetic angle image created from multiple speckle realizations of the object. Acknowledgments
This effort was supported in part by the U.S. Air Force through contract number FA8650-062-1081, and the University of Dayton Ladar and Optical Communications Institute (LOCI). The views expressed in this article are those of the authors and do not reflect on the official policy of the Air Force, Department of Defense or the U.S. Government.
#197854 - $15.00 USD Received 17 Sep 2013; revised 30 Oct 2013; accepted 31 Oct 2013; published 5 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030642 | OPTICS EXPRESS 30652
