Conventional optical refractometry methods are often limited by a narrow measurement range, complex hardware, or relatively high cost. Here, we present a novel refractometry method to measure the bulk refractive index (RI) of materials (including solids and liquids) using lensless holographic on-chip imaging and autofocusing, which is simple, cost-effective, and has a large RI measurement range. As a proof of concept, two compact prototypes were built to measure the RIs of solid materials and liquids, respectively, and they were tested by measuring the RIs of a ZnSe plate and a microscopy immersion oil. Experimental results show that our devices have an average accuracy of ~3 × 10−4 RI unit (RIU) with an estimated precision of ~3 × 10−3 RIU for solids; and an average accuracy of ~1 × 10−4 RIU with an estimated precision of ~3 × 10−4 RIU for liquids. We believe that this cost-effective and portable RI measurement platform holds promise to be used in laboratory and industrial settings.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Refractive index (RI) is a fundamental physical property of materials. The measurement of RI, i.e., refractometry, has broad applications in chemical sensing [1–3], biology [4–8], the food industry [9–11], optical fabrication [12–15], among others. In the past decades, several methods for bulk RI measurement have been developed based primarily on geometric optics [16–20] or wave optics [21–28]. On the basis of geometric optics, the Snell’s law and total internal reflection (TIR) are commonly used principles. The minimum deviation method , in the light of Snell’s law, is widely used for RI measurement of solid transparent materials such as glasses, which provides an accuracy of ~10−5 refractive index unit (RIU) . However, as a disadvantage of this method, one must prepare prism-shaped samples, making the routine measurements cumbersome. As an alternative, RI can also be determined by measuring the critical angle of TIR that occurs at the interface between the sample and a high-RI prism [18,19]. The Abbé refractometer , as one of the most famous refractometry instruments employing the TIR principle, has been widely used to measure the RI of solids with flat surfaces and liquids. Although the TIR-based methods provide a very good accuracy (e.g., ~10−3-10−5 RIU) for transparent liquids and solids, their measurement range is typically limited by the RI of the reference prism , which poses a practical limitation.
On the basis of wave optics, interferometric methods were also developed to detect a relative phase difference with high sensitivity, which can be used to measure the RI of the sample [21–25]. One advantage of interferometric methods is that the RI measurement range is not theoretically limited, which is far wider than TIR-based methods. As an example, a standard Michelson interferometer (MI) can be used to measure the RI of transparent solid plates , where a transparent sample plate is rotated in the sample arm continuously to change the optical path difference, and fringe patterns at different sample orientations are measured. By analyzing the fringe patterns, the RI can be estimated with an accuracy of ~10−3 RIU given that the thickness of the sample is known. However, since the Michelson interferometer is sensitive to air flow in the light path and the vibration of each component in the setup, the accuracy of the measurement is easily influenced by the environment. The Fabry-Perot (FP) interferometer  is another method used to measure the RI of a sample, where the fringe pattern is determined by the phase difference between the directly transmitted light wave and the multiply reflected waves between the reflectors. Thus, the FP arrangement is more robust to environmental perturbations than MI, and the accuracy can be as good as ~10−4 RIU. However, interferometers are in general limited by their relatively high cost, complex optical setup and sensitivity to alignment. Diffraction-based methods to measure RI have been demonstrated with simplified configurations [26–28] achieving an accuracy of e.g., ~10−4 RIU, but portable devices have not been reported. Besides these techniques, various other refractometry methods were proposed, including ellipsometers , optoﬂuidics-based methods , and surface plasmon resonance (SPR) sensors . They have several advantages, such as a low sample volume requirement and high accuracy, but they entail complex configurations and require precise and expensive optical components.
Here, we present a novel method to measure the RI of solids and liquids using holographic imaging [32–38] and autofocusing [39–44]. The measurement principle is based on the relation between RI and optical diffraction: in a holographic imaging system, a change in the RI of the medium after the light hits an object will affect the effective distance of optical diffraction, thereby changing the focus distance of the object’s hologram. Therefore, the RI can be indirectly measured by automatically measuring the focus distance of a two-dimensional (2D) object using a holographic autofocusing algorithm. Because this method does not involve TIR, the theoretical measurement range of this method is not limited, and therefore it can be used to measure both low- and high-RI samples. Based on this principle, we built two portable refractometer prototypes to measure RIs of solids and liquids, respectively. A lensless holographic on-chip imaging configuration was adopted due to its simplicity, cost-effectiveness and robustness to alignment errors. A laser diode provides a partially-coherent illumination, and a complementary metal-oxide-semiconductor (CMOS) image sensor array is used to record the diffraction patterns of a 2D object (e.g., a USAF-1951 target). The sample to be measured is inserted between this 2D object and the image sensor plane, and the auto-focus distances of the 2D object before and after the sample insertion are used to calculate the RI of the sample.
We demonstrated the accuracy as well as the wide measurement range of our technique by measuring the RIs of a ZnSe glass sample and a microscopy immersion oil sample. The average accuracy was quantified to be ~3 × 10−4 RIU for solids and ~1 × 10−4 RIU for liquids; and the precision was estimated as ~3 × 10−3 RIU for solids and ~3 × 10−4 RIU for liquids. The cost-effectiveness, wide RI measurement range and the applicability of this measurement approach to different samples (solids and liquids) can make this technique a versatile tool to measure RIs in laboratory as well as industrial settings.
2. Measurement principle
2.1 Scalar wave propagation under Fresnel approximation
As shown in Fig. 1, for a given complex optical field on the plane z = 0, U(x, y; 0), denote the complex optical field after it propagates by a distance of z as U(x, y; z). Let A(fx, fy; z) be the angular spectrum of U(x, y; z), with a Fourier transform relation45]:Eq. (5), the first part of H (fx, fy; z) is a constant phase factor that is independent of fx and fy, which cannot be sensed by a typical CMOS or charge-coupled device (CCD) based image sensor. Thus, it is ignored in our discussion here and only the second part is kept, given by:Eq. (6) that the effect of wave propagation in a medium with RI = n over a distance z is equivalent to propagation in vacuum (where RI = 1) over a distance of z/n. In other words, having a larger RI reduces the amount of effective diffraction. Therefore, we define here a quantity named as “effective diffraction distance” (EDD), denoted by lE. For wave propagation from z = z1 to z = z2 with multiple layers of materials of different RIs in between, lE is given by:
In practice, for a typical lensless holographic imaging system without pixel super-resolution [32,34,35,38], the additional frequency cut-off due to the finite spacing of the image sensor’s pixels ensures that Fresnel approximation is satisfied. For example, assuming a pixel pitch of Δx = Δy = 1.67 μm, according to the Nyquist theorem, |fx, y| ≤ 1/(2Δx) = 0.299 μm−1. Further assuming λ0 = 655 nm and n = 1, |λ0 fx, y /n| ≤ 0.299 × 0.655 = 0.194 ≪ 1, satisfying the Fresnel approximation condition.
2.2 Holographic autofocusing
Holographic autofocusing algorithms are used to automatically determine the wave propagation distance (i.e., the “focus distance”) of a hologram. Various autofocusing algorithms have been previously demonstrated [29–44] and compared [40,42]. In this work, we adopt a recently proposed edge sparsity-based autofocusing criterion named “Gini index of the gradient” (GoG), which is shown to be robust and accurate for a wide range of specimens [39,40]. GoG of a complex refocused (“back-propagated”) optical field using the angular spectrum method detailed in Section 2.1 is given by46], i.e.,
40]. If the RI is assumed to be 1 during autofocusing (that is, the RI related to the back-propagation kernel ℘(G; z) is set as 1), the calculated is an estimation of lE introduced in Eq. (7).
2.3 Optical refractometry using lensless holography and autofocusing
A typical lensless holographic imaging setup is depicted in Fig. 2(a), where a partially coherent quasi-planar light wave illuminates a high-resolution planar target (e.g., a positive USAF-1951 target), and its diffraction pattern is recorded on a CMOS/CCD image sensor. The EDD, Eq. (7), between the target and the image sensor will change if a sample plate with a thickness of d and a RI of nx is inserted between the sensor and target planes. Let lE,0 be the EDD without inserting the sample, and lE,1 with the sample. They are related to each other by:
3. Results and discussion
3.1 Portable holographic device for RI measurement of solid transparent plates
Based on the working principle detailed in Fig. 2 and Section 2.3, we custom-fabricated a holographic device for the RI measurement of solid samples, shown in Fig. 3. We use a laser diode (LD) (LD201710XHXL, QiLin optics, Nantong, China) with a peak wavelength of 655 nm as the illumination source, and a 10-megapixel CMOS image sensor (1.67 μm pixel size, 3840 × 2748 pixels; DMM 27UJ003-ML, ImagingSource Corporation, Germany) for hologram recording. A positive USAF-1951 target (Thorlabs, Inc.) with a 25.4 mm diameter is adopted as the “target object” used for autofocusing. Among various patterns of the USAF-1951 target, the high-resolution part (i.e., the central part) is more important as it contains richer fine features, useful for autofocusing. The housing of the device is made from 3D-printed plastic (printing resolution 0.1 mm; service provided by Zhantong 3D, Shenzhen, China), and steel posts are used to connect the illumination part and the imaging part. In order to demonstrate that the RI measurement range of our presented technique is very wide, we chose a ZnSe plate (GL2017102145R, QiLin optics, Nantong, China) as the sample with n = 2.5778 at λ = 655 nm (data obtained from Zemax software ), which falls outside the measurement range of a typical commercial Abbé refractometer (typically 1.3 < n < 1.8). The thickness d of the ZnSe plate is measured by a micrometer to be 1697.0 μm.
Following the procedure detailed in Section 2.3, a hologram of the USAF-1951 target is captured before and after inserting the ZnSe sample plate, i.e., Fig. 4(a) and (d), respectively, and autofocusing is performed for each case to estimate the EDDs: lE,0 = 2811.4 μm and lE,1 = 1772.1 μm, respectively (Fig. 4(b) and (e)). An area with 1024 × 1024 pixels at the center of the CMOS image sensor is chosen for autofocusing. The wave curvature from the LD source is corrected by multiplying the hologram with a phase compensation term given by Eq. (11) . The RI of air is taken as na = 1.0003  in Eq. (13). Based on the measurements reported in Fig. 4 (i.e., lE,0 and lE,1) and the known quantities (i.e., d and na), the unknown RI of the solid material, nx, can be calculated as 2.5802. Including this one, we repeated the RI measurement of the same ZnSe plate for a total of 10 times (by repeating all the necessary steps), and obtained nx = 2.5781 ± 0.0016, with the individual RI measurement results as 2.5802, 2.5771, 2.5799, 2.5755, 2.5779, 2.5767, 2.5795, 2.5775, 2.5798, and 2.5767 RIU. According to Zemax software, the RI of ZnSe is n = 2.5778 at λ = 655 nm , which confirms that the mean RI accuracy of our measurement is ~3 × 10−4 RIU.
We also provide here a theoretical analysis on our measurement error to understand the effect of each parameter. Our measurement accuracy is mainly limited by the accuracies of the micrometer, the autofocusing algorithm , and the RI of air . The estimated uncertainties of these parameters are listed in Table 1.
According to the Eq. (13), we can estimate the standard error of nx,Eq. (14) are given by
After inserting the known and measured values into Eqs. (15)-(18) (i.e., na = 1.0003, d = 1697.0 μm, lE,0 = 2811.4 μm and lE,1 = 1772.1 μm), we obtain , , , and . Based on these, we can estimate the total standard error of nx as: .This standard error is approximately 4-fold larger than our measurement standard deviation (0.0016 RIU); this might be partially due to that fact that the above analysis assumes that lE,0 and lE,1 are independent from each other. In reality, the autofocusing algorithm can output a z-distance that is consistently offset by a certain amount from the true z-distance, such that δlE,0 and δlE,1 can partially cancel each other when subtracted in Eq. (13). This is confirmed by the relatively small standard deviation (0.0016 RIU) of 10 individual experiments, which would otherwise be significantly larger if the autofocusing algorithm’s error (~1 μm) is entirely random. As a result, we take the experimentally measured 0.0016 RIU as the random error related to uncertainties in δlE,0, δlE,1 and na. Another major source of error comes from the micrometer’s inaccuracy. Here, we used a micrometer with an accuracy of 1 μm, which, by itself, results in a δnx of 0.0024 RIU. Because we used the same d throughout the 10 measurements, this error source is not reflected in the standard deviation of our experiments, i.e., 0.0016 RIU. Considering the error introduced by the micrometer is 0.0024 RIU and the measurement random error is 0.0016 RIU, the total random error (i.e., precision) of the current device for measuring solids can be estimated to be 0.0029 RIU, i.e., approximately 3 × 10−3 RIU.
The additional error that is related to the tilting of the test sample is not considered in these calculations, but it can be avoided or mitigated through the design and precise fabrication of the device. In our design, we placed the solid sample plate into a slot inside the sample holder, and placed the device in an upright (i.e., vertical) position such that the sample lies flat in the sample holder. This ensures that the sample is parallel to the image sensor and the target, mitigating errors due to sample tilting. The RI error related to thickness inaccuracy can also be partially mitigated by increasing the thickness of the sample. Moreover, because most of the errors are random errors, we believe it is possible to reduce the error significantly by averaging results from multiple experiments. In summary, we estimate that our current prototype achieves a mean accuracy of ~3 × 10−4 RIU with an estimated precision of ~3 × 10−3 RIU for the RI measurement of solids.
3.2 Portable holographic device for RI measurement of liquids
The photograph and design schematic of a portable refractometer for measuring the RIs of liquid samples is presented in Fig. 5, which is very similar to the one shown in Fig. 3. The illumination is similarly provided by a LD. A custom-made liquid chamber, which can be inserted in/out, is placed between the USAF-1951 target and the CMOS image sensor. The front surface (close to the USAF-1951 target) and the back surface (close to the CMOS image sensor) of the liquid chamber are both transparent glass cover slips (18 × 18 × 0.1 mm). The inner thickness of the liquid chamber is roughly 3 mm. Owing to the design of the liquid chamber, the device in Fig. 5 is in a horizontal position during the measurements.
We tested this prototype using a standard immersion oil (Type 300, Cargille Labs, US) under room temperature (T = 25°C). Because the exact thickness of the liquid chamber is unknown, a three-step measurement procedure was developed, which also involves calibration of the chamber itself:
- 2) Next, the chamber is filled with deionized (DI) water (RI: nw = 1.3310 ). A
The RI of the unknown liquid sample that is calculated by this method is given by (see the Appendix for derivation):Eq. (19), nx of this experiment (Fig. 6) is calculated to be 1.5117 (λ = 655 nm). Based on 10 independent measurements, we experimentally obtained nx = 1.5115 ± 0.0003 (with the individual RI measurement results as 1.5117, 1.5114, 1.5114, 1.5117, 1.5108, 1.5117, 1.5115, 1.5112, 1.5116, and 1.5119 RIU), while the RI data provided by Cargille Labs is 1.5116 . Based on these measurements, the mean error of our device (i.e., accuracy) was estimated to be ~1 × 10−4 RIU, with an experimental standard deviation of ~3 × 10−4 RIU.
Similar to the error analysis reported earlier for solid RI measurements, we conducted an error analysis to understand the effect of each parameter. Table 2 lists the estimated uncertainties of the measured parameters. According to Eq. (19), the estimated standard error of the measured liquid RI (nx) is given byEq. (20) are given by:
After inserting the known and measured values into Eq. (21)-(25) (i.e., na = 1.0003, nw = 1.3310, lE,0 = 4539.1 μm, lE,1 = 3831.1 μm and lE,2 = 3575.1 μm), we obtain , , , and . Thus, the estimated standard error of nx is , which is ~5 times larger than the standard deviation that is experimentally measured (i.e., ~3 × 10−4 RIU). However, analogous to the analysis in Sec. 3.1, we also assumed here that δlE,0, δlE,1 and δlE,2 are independent. In fact, δlE,0, δlE,1 and δlE,2 may cancel each other significantly as indicated by the low standard deviation of our RI measurements (~3 × 10−4 RIU), and the other errors due to δna and δnw are only on the order of ~10−4. Therefore, self-calibration of the sample thickness using reference liquids (e.g., DI water) is advantageous compared to direct thickness measurement of the sample using e.g., a micrometer, which improves our precision in liquid measurements compared to solid materials. In summary, we estimate that our current prototype achieves a mean accuracy of ~1 × 10−4 RIU and a precision of ~3 × 10−4 RIU for RI measurements of liquids.
The main components in our devices are a LD (0.05 USD), a positive USAF-1951 target (~200 USD) and a CMOS image sensor (~400 USD). For industrial applications, the positive USAF-1951 target can be substituted by other high-resolution planar patterns that may even be printed by a high-resolution printer, which will significantly reduce the cost of our instrument. When custom-designing a target pattern, one should note that the pattern should have various fine spatial features and sharp edges for accurate autofocusing [39,40], but the feature size must be larger than the resolution of the imaging system. Furthermore, various low-cost monochromatic CMOS/CCD image sensors can be used for image acquisition, which might altogether bring down the total cost of the parts of our platform to <100 USD under large volume manufacturing.
In this manuscript, we present a novel bulk refractometry principle based on lensless holographic on-chip imaging and autofocusing, which has the advantages of cost-effectiveness, simplicity, and a wide RI measurement range. As a proof of concept, we designed two compact and portable prototypes to measure the RIs of solids and liquids, respectively, and demonstrated RI measurement of a solid ZnSe plate and a microscopy immersion oil using these prototypes. We show that our devices can achieve an average RI measurement accuracy of ~3 × 10−4 RIU for solid samples, and ~1 × 10−4 RIU for liquid samples. This presented refractometry principle is promising to be widely applicable in laboratory and industrial settings.
Appendix Derivation of Eq. (19)
Let us denote the inner thickness of the liquid chamber as t; denote the EDD outside the liquid chamber (after light hits the 2D object) as lE,S, which should be constant. Thus, lE,0, lE,1 and lE,2 in Sec. 3.2 can be expressed as:
According to Eqs. (26) and (27), the inner thickness of the chamber for liquid, t, is
National Science Foundation; Howard Hughes Medical Institute (HHMI); National Natural Science Foundation of China (61575175, 61427818); National Key Research and Development Program of China (2016YFB1001502); China Scholarship Council (No. 201606320051).
The authors declare that there are no conflicts of interest related to this article.
1. C. R. Parkerson, “Determination of Saturation Temperatures of Inorganic Salt Solutions. Refractive Index Measurements,” Anal. Chem. 23(4), 610–613 (1951). [CrossRef]
2. C. Y. Tan and Y. X. Huang, “Dependence of Refractive Index on Concentration and Temperature in Electrolyte Solution, Polar Solution, Nonpolar Solution, and Protein Solution,” J. Chem. Eng. Data 60(10), 2827–2833 (2015). [CrossRef]
3. M. Yahya and M. Z. Saghir, “Prediction and Experimental Measurement of Refractive Index in Ternary Hydrocarbon Mixtures,” J. Chem. Eng. Data 60(8), 2329–2342 (2015). [CrossRef]
4. W. Lukosz, “Integrated optical chemical and direct biochemical sensors,” Sens. Actuators B Chem. 29(1-3), 37–50 (1995). [CrossRef]
6. T. R. Corner, T. C. Beaman, J. T. Greenamyre, and P. Gerhardt, “Spectrophotometric determination of refractive index increment for bacterial cells,” J. Microbiol. Methods 11(3-4), 255–260 (1990). [CrossRef]
9. F. Rogerson and C. Symington, “A method for the estimation of alcohol in fortified wines using hydrometer Baume and refractometer Brix,” Am. J. Enol. Vitic. 57, 486–490 (2006).
10. M. L. Dongare, P. B. Buchade, and A. D. Shaligram, “Refractive index based optical Brix measurement technique with equilateral angle prism for sugar and Allied Industries,” Optik (Stuttg.) 126(20), 2383–2385 (2015). [CrossRef]
11. A. F. Rawle, “Refractive Index Measurements,” in Encyclopedia of Spectroscopy and Spectrometry (Third Edition), J. C. Lindon, G. E. Tranter, and D. W. Koppenaal, eds. (Academic Press), pp. 936–945.
13. D. Malacara, Optical Shopping Test, 3rd ed. (John Wiley & Sons, Inc.).
14. D. Malacara and Z. Malacara, Handbook of Optical Design, 2nd ed. (Marcel Dekker, Inc.).
17. H. Hattori, H. Yamanaka, H. Kurniawan, S. Yokoi, and K. Kagawa, “Using minimum deviation of a secondary rainbow and its application to water analysis in a high-precision, refractive-index comparator for liquids,” Appl. Opt. 36(22), 5552–5556 (1997). [CrossRef] [PubMed]
19. G. H. Meeten, “Refractive index errors in the critical-angle and the Brewster-angle methods applied to absorbing and heterogeneous materials,” Meas. Sci. Technol. 8(7), 728–733 (1997). [CrossRef]
22. G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. 42(19), 3882–3887 (2003). [CrossRef] [PubMed]
24. G. D. Gillen and S. Guha, “Refractive-index measurements of zinc germanium diphosphide at 300 and 77 K by use of a modified Michelson interferometer,” Appl. Opt. 43(10), 2054–2058 (2004). [CrossRef] [PubMed]
25. G. D. Gillen and S. Guha, “Use of Michelson and Fabry-Perot interferometry for independent determination of the refractive index and physical thickness of wafers,” Appl. Opt. 44(3), 344–347 (2005). [CrossRef] [PubMed]
29. M. Stchakovsky, Y. Battie, and A. E. Naciri, “An original method to determine complex refractive index of liquids by spectroscopic ellipsometry and illustrated applications,” Appl. Surf. Sci. 421, 802–806 (2017). [CrossRef]
30. C. Prokop, N. Irmler, B. Laegel, S. Wolff, A. Mitchell, and C. Karnutsch, “Optofluidic refractive index sensor based on air-suspended SU-8 grating couplers,” Sens. Actuators A Phys. 263, 439–444 (2017). [CrossRef]
31. L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017). [CrossRef]
32. W. Luo, A. Greenbaum, Y. Zhang, and A. Ozcan, “Synthetic aperture-based on-chip microscopy,” Light Sci. Appl. 4(3), e261 (2015). [CrossRef]
33. C. Zuo, J. Sun, J. Zhang, Y. Hu, and Q. Chen, “Lensless phase microscopy and diffraction tomography with multi-angle and multi-wavelength illuminations using a LED matrix,” Opt. Express 23(11), 14314–14328 (2015). [CrossRef] [PubMed]
35. Y. Zhang, S. Y. C. Lee, Y. Zhang, D. Furst, J. Fitzgerald, and A. Ozcan, “Wide-field imaging of birefringent synovial fluid crystals using lens-free polarized microscopy for gout diagnosis,” Sci. Rep. 6(1), 28793 (2016). [CrossRef] [PubMed]
36. Y. Zhang, Y. Wu, Y. Zhang, and A. Ozcan, “Color calibration and fusion of lens-free and mobile-phone microscopy images for high-resolution and accurate color reproduction,” Sci. Rep. 6(1), 27811 (2016). [CrossRef] [PubMed]
38. A. Greenbaum and A. Ozcan, “Maskless imaging of dense samples using pixel super-resolution based multi-height lensfree on-chip microscopy,” Opt. Express 20(3), 3129–3143 (2012). [CrossRef] [PubMed]
39. M. Tamamitsu, Y. Zhang, H. Wang, Y. Wu, and A. Ozcan, “Comparison of Gini index and Tamura coefficient for holographic autofocusing based on the edge sparsity of the complex optical wavefront,” arXiv:1708.08055 [physics.optics] (2017).
41. Z. Ren, Z. Xu, and E. Y. Lam, “Learning-based nonparametric autofocusing for digital holography,” Optica 5(4), 337–344 (2018). [CrossRef]
42. E. S. R. Fonseca, P. T. Fiadeiro, M. Pereira, and A. Pinheiro, “Comparative analysis of autofocus functions in digital in-line phase-shifting holography,” Appl. Opt. 55(27), 7663–7674 (2016). [CrossRef] [PubMed]
44. P. Gao, B. Yao, J. Min, R. Guo, B. Ma, J. Zheng, M. Lei, S. Yan, D. Dan, and T. Ye, “Autofocusing of digital holographic microscopy based on off-axis illuminations,” Opt. Lett. 37(17), 3630–3632 (2012). [CrossRef] [PubMed]
45. J.W.Goodman, Introduction to Fourier Optics (Roberts).