- Email: [email protected]
Absolute measurement of aspheric lens with electrically tunable lens in digital holography
Optics and Lasers in Engineering 88 (2017) 313–318
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Absolute measurement of aspheric lens with electrically tunable lens in digital holography
crossmark
⁎
Zhaomin Wanga, , Weijuan Qub, Fang Yangb, Ailing Tianc, Anand Asundia a b c
Centre for Optical and Laser Engineering, Nanyang Technological University, Singapore Ngee Ann Polytechnic, Singapore Shannxi Key Lab of Thin Films Technology and Optical Test, Xi’an Technological University, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Digital holography Aspheric lens Electrically tunable lens Phase measurement
A novel method for testing aspheric lenses using digital holography with an electrically tunable lens (ETL) is proposed and experimentally verified. The ETL generates a tunable deformed wavefront which helps to decrease the high gradient of aspheric lenses. By decomposing the aspheric surface into two resolvable ones, its absolute phase can be determined using a double-exposure measurement. In this method, the wavefront generated by the ETL need not be identical to the aspheric surface as in conventional null interferometer system, but just sufficient to resolve the high gradient surface. On the other hand, the tunability of the ETL allows generation of wavefronts which can be used to test different aspheric lens. Furthermore, advantages of the ETL such as low cost, fast response, and compact configuration make the proposed method a promising technique for aspheric surface measurement.
1. Introduction The performance of a high precision optical system using spherical optics is reduced by optical aberrations, and thus more optical elements are required. Using aspheric lens, these optical aberrations can be significantly reduced or even eliminated with a reduced (even single) number of optical elements [1–3]. However, testing of aspheric lenses is difficult due to high gradients on the aspheric surface resulting in high fringe densities which cannot be resolved and measured. Therefore, there is an urgent need to develop suitable test methods [4]. Many methods have been developed to measure aspheric lens. The stylus instrument is by far the most reliable tool owing to its high accuracy [5–7]. It doesn’t need any null optics, nor is it influenced by the optical properties of the sample. However, being a contact method, the stylus tip could damage the optical surface. Also, the stylus tip radius smoothens the measurement like a low-pass spatial filter. Moreover, being a point-wise measurement method, it is time-consuming. The Shack-Hartmann wavefront sensor is an economical choice for the measurement of aspheric lens form [8–11]. However, the shape and pupil adaptation of the optical system is needed, and the lateral resolution of this technique is limited. In recent years, deflectometry has attracted attention as it is capable of measuring aspheric surface with simple configuration [12,13]. The main drawback of this method is that its resolution is physically limited by the projector or the grating. Interferometers with a computer generated hologram (CGH) to
⁎
generate the desired wavefront have also been used [14–17]. This method has many advantages since the test and reference beams follow almost identical paths. But there is a need to redesign and manufacture new CGH for different asphere which is complicated, expensive, and time-consuming. Another interferometric method uses phase shift interferometry and heterodyne displacement interferometry to measure aspheric lens [18]. It has nanoscale vertical resolution as well as large measurement volume. However, the tested aspheric lens must be axially symmetric, have a measurable apex and defined aspheric equation, and cannot have a reversing curvature. Recently, interferometry with point source array based on structured illumination has also been proposed [19]. It evaluates the aspheric surface by fitting the spherical surface in advance. As this method does not model highfrequency components, it leads to final result containing high order systematic aberrations. Meanwhile, the calibration and alignment also greatly affect the final result. Digital holography (DH) is a well-established high precision 3D imaging technique which is capable of numerically reconstructing the optical field of a sample at any propagation distance [20,21]. It is able to access to the intensity and phase of the sample simultaneously. Along with fast reconstruction speed, high axial resolution, and low cost, it has become a powerful tool in quantitative phase imaging. However, this technique also encounters the same high gradient problem when measuring an aspheric lens. In this paper, a new adaptive optical element, ETL [22,23], is added
Corresponding author. E-mail address: [email protected] (Z. Wang).
http://dx.doi.org/10.1016/j.optlaseng.2016.09.002 Received 26 April 2016; Received in revised form 7 September 2016; Accepted 8 September 2016 Available online 0143-8166/ © 2016 Elsevier Ltd. All rights reserved.
Optics and Lasers in Engineering 88 (2017) 313–318
Z. Wang et al.
Fig. 1. Schematic of experimental configuration.
Fig. 2. ETL focal length variation versus driven current.
Ltd (www.doptron.com). The optical field at a propagation distance d is expressed as
to a DH system to overcome the above problem. The electrical tuning power allows ETL to generate a suitable wavefront that decreases the high gradients of the aspheric surface. A Mach-Zehnder configuration DH system with the ETL is developed and demonstrated for the aspheric lens measurement. The experimental results are in good agreement with factory data for two sample aspheric lenses.
EO (x, y, d ) = ℑ− ℑ ER* IH exp ⎡⎣ ikd 1 − r 2 ⎤⎦ ⎡⎣ fx, fy ⎤⎦ (x, y)
{ {
}
(2)
where ℑ and represent Fourier transform and its inverse. fx and fy represent spatial frequencies in x and y directions. The spatial filter radius r can be written as
2. Methodology
r=
2.1. Principle of digital holographic system
2
2
( λf ) + ( λf ) x
y
(3)
The phase of the object field is then obtained [26]
Fig. 1 shows the schematic of the Mach-Zehnder interferometer. A collimated laser beam is split by beam splitter 1 (BS1) into two beams. One beam goes through the ETL and is then normally incident on the aspheric lens surface under test. Since the beam passing through the aspheric lens diverges very fast, a triplet is introduced to reshape the beam. The other beam serves as the reference beam and interferes with object beam at BS4 with a small angular offset between them. The small angular offset is often manually adjusted to ensure the first order spectrum can be separated from the zero order spectrum in frequency domain. The image is resized by a relay imaging lens and captured by an imaging sensor which is often a charge-coupled device (CCD) camera. The recorded image, namely, digital hologram IH x, y [24] is expressed as
φ (x, y) = tan−1
Re[Eo (x, y, d )] Im[Eo (x, y, d )]
(4)
2.2. Numerical aperture calibration of ETL The focal length variation of ETL (Optotune EL-10-30-C-LD) is nonlinear versus its driven current as shown in Fig. 2. Therefore, it is necessary to calibrate the ETL before any further application. The calibration is as described in our previous work [27]. Data are recorded at 10 mA steps over the working range of the ETL from 0 mA to 290 mA. The diameter of the ETL aperture is 10 mm and thus the numerical aperture (NA) of the ETL varies from 0.025 to 0.062. This is too small to compensate for the high NA of most aspheric lenses. Therefore, an offset lens (OL) is introduced to increase the NA. The effective focal length of the ETL and offset lens can be obtained from geometric optics for two lenses in close proximity as
( )
IH (x, y) = EO + ER 2 = EO 2 + ER 2 + EO ER exp{−ikx sin θ} + EO* ER exp{ikx sin θ}
}
ℑ−
(1) 2
where EO is the object wave and ER is the reference wave. EO is the intensity of the object wave and ER 2 is the intensity of the reference wave. EO* denotes the complex conjugate of the object wave. k=2π/λ is the wavenumber, and θ is the angle between the reference wave and the normal. λ is the wavelength of the light source. The angular spectrum method [25] is used for numerical reconstruction through a user-friendly software developed by d′Optron Pte
1 1 1 = + f fETL fOL
(5)
where f is the effective focal length of the combination of ETL and OL. fETL and fOL are focal lengths of ETL and OL, respectively. In this paper, the focal length of OL is −150 mm. The focal length of the combination is shown in Fig. 3. After 314
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Fig. 3. Lens combination's focal length performance curve.
calculation, the NA of lens combination varies from 0.0014 to more than 1.0 thus making it suitable for aspheric lens testing. 2.3. Reconstruction of aspheric surface The essence of the proposed method is to decompose the large gradients of the asphere into two resolvable surfaces, and to measure them separately. Beyond that, the system aberrations need to be determined beforehand so as not to influence the final result. A double-exposure approach is proposed for the absolute phase measurement. In the first exposure, the aspheric lens with thin lens combination of ETL and OL is measured as shown in Fig. 1. The ETL drive current is adjusted to minimize the density of the interference fringe in the digital hologram. Once the fringes can be resolved, the digital hologram is recorded and processed to give the reduced phase of the aspheric lens. In the second exposure, the aspheric lens is removed, and a second digital hologram is recorded. The absolute phase of the aspheric lens can thus be retrieved from these two phases.
φAL (x, y) = φreduced _ AL (x, y) + φETL (x, y)
(6) Fig. 4. Schematic of an aspheric surface.
where φAL is the absolute phase of the aspheric lens. φreduced _ AL and φETL are the unwrapped phases of the reduced aspheric lens and the ETL, respectively. Finally, the topography of the aspheric lens can be expressed as
hAL (x, y) =
λ φ (x, y) 2π (nAL − 1) AL
expression is adopted in fitting. Two aspheric lenses were tested and demonstrated below. 3.1. 4.5 mm diameter aspheric lens testing
(7)
where hAL is the height on aspheric lens surface. nAL is the refractive index of aspheric lens.
In experiment, an aspheric lens primarily used for collimating the beam emerging from an optical fiber was tested. The diameter and focal length of the aspheric lens are 4.5 mm and 2.5 mm, respectively. A 660 nm solid state laser was used as light source and the focal length of the triplet and the relay imaging lens was 40 mm and 100 mm, respectively. The digital hologram was recorded by an 8-bit monochromatic CCD camera with 1280×960 pixels with a pixel size of 4.65 µm×4.65 µm. The distance between ETL and OL thin lens combination and the aspheric lens was 305 mm while the distance of the aspheric lens to triplet was 5 mm. The triplet was 75 mm from the relay imaging lens which in turn was 90 mm from the CCD. As the fitting algorithm was developed in polar co-ordinates, all the phases were apodized into a circular aperture for evaluation. Through calibration with a USAF 1951 resolution target, the diametric field of view was 3.033 mm. Fig. 5(a) shows the interference fringes generated by the aspheric lens without any current applied to the ETL. The fringes cannot be resolved by the imaging system. Next current was applied to the ETL to reduce the density of the fringes and for this example a current of 123.2 mA gave resolvable fringes. The wrapped phase of reduced aspheric lens is shown in Fig. 5(b). In second exposure, the phase
3. Experimental testing A typical aspheric surface schematic, comprising of a conic section with a spherical vertex, is shown in Fig. 4. The ISO 10110-Part 12 standard represents the sag z, as r2 R
z = f (r ) = 1+
1 − (1 +
n r κ )( R )2
+
∑ A2i r 2i i =2
(8)
where r is the lateral coordinate, R is the paraxial surface radius and κ is the conic constant. From Eq. (8), it can be seen that the paraxial surface radius R, conic constant κ, and coefficient A2i are parameters that identify an aspheric surface. To evaluate these parameters, a least squares method is often used in fitting. One should note that the aspheric surface expression is non-linear; hence the evaluation procedure needs to either convert the expression into a linear expression by simplification or to use nonlinear fitting method directly [28–32]. In this paper, a simplified linear 315
Optics and Lasers in Engineering 88 (2017) 313–318
Z. Wang et al.
Fig. 5. The wrapped phase of (a) asphere which cannot be resolved, (b) the reduced aspheric lens and (c) the ETL decomposed by (a) which can now be analyzed.
Fig. 6. Testing 1. Unwrapped phases of (a) the ETL, (b) the reduced aspheric lens, and (c) the aspheric lens. (d) 3D rendering for the phase of the aspheric lens. (e) Reconstructed phase by fitted parameters. (f) Residue phase between measured phase (c) and reconstructed phase (e). (g) Profile comparison of 480th line between measured and reconstructed curves.
316
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Table 1 Aspheric lens parameters comparison.
Test 1 Test 2
Aspheric parameters
A04
A06
A08
A10
Radius (mm)
Conic constant
Factory Fitted Factory Fitted
3.255E-3 3.287E-3 1.954E-5 1.999E-5
8.501E-5 8.572E-5 −1.756E-8 −1.872E-8
−2.227E-5 −2.225E-5 2.597E-11 2.874E-11
– – −2.414E-14 −2.933E-14
−2.3289 −2.3289 −22.09 −22.09
−0.8814 −0.8781 −2.271 −2.232
of the residue is 0.042 µm. The variance in the residue is mainly caused by random noise and irregularity of optical surface in the whole system. The standard deviation coincides with our system error level which is ten nanometers. While the extreme small mean value clearly demonstrates the average statistical characteristic of the random distribution of noise. In order to compare the goodness of the fitting, data along the 480th line in Fig. 6(c) and (e) were plotted in Fig. 6(g). It can be noted that two plots also show a good agreement.
without the aspheric lens is measured and displayed in Fig. 5(c). It can be seen that both wrapped phases are resolvable and can be easily unwrapped. Fig. 6(a) and (b) are the unwrapped phases of the reduced aspheric lens and the ETL respectively. The absolute phase aspheric lens can then be recovered as shown in Fig. 6(c). After implementing the fitting procedure, the aspheric parameters were obtained and compared with factory data in Table 1. It can be seen that the parameters are in good agreement. The reconstructed phase based fitted parameters is displayed in Fig. 6(e) with the residue shown in Fig. 6(f). In this case, the mean value of the residue is 1.29e-17 µm, while the standard deviation
Fig. 7. Testing 2. Unwrapped phases of (a) the ETL, (b) the reduced aspheric lens, and (c) the aspheric lens. (d) 3D rendering for the phase of the aspheric lens. (e) Reconstructed phase by fitted parameters. (f) Residue phase between measured phase (c) and reconstructed phase (e). (g) Profile comparison of 480th line between measured and reconstructed curves.
317
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TIF-2-G-012 from the Singapore Ministry of Education and International Cooperation Project 2014KW05 from Science and Technology Department of Shaanxi Province of China.
3.2. 25 mm diameter aspheric lens testing The second aspheric lens tested was a high precision aspheric lens (L-BAL35) from Newport, with 25.00 mm diameter and 37.50 mm focal length, respectively. With 25.00 mm clear aperture, its NA is approximately 0.33. The distance of thin lens combination to the aspheric lens was 285 mm and the distance of the aspheric lens to the triplet was 50 mm. The relay imaging was 65 mm from the lens triplet and the distance between the relay lens and CCD camera was 80 mm. The drive current applied to the ETL was 9.3 mA. The field of view was calibrated again, and the normalized aperture diameter was 9.65 mm for this example. Fig. 7(a) and (b) are the reconstructed phases of the reduced aspheric lens and the ETL, respectively. The absolute phase of the aspheric lens is displayed in Fig. 7(c). After fitting, the aspheric parameters were obtained and compared to factory values in Table 1. It can be seen that as the coefficient goes weaker with the term order increasing, the accuracy of fitting also goes lower. For weak aspheric coefficients, many factors such as fabrication, measurement scheme, system noise, fitting algorithm and computing errors would affect the final result. The reconstructed phase by fitted parameters is shown in Fig. 7(e), and the residue is shown in Fig. 7(f). The mean value of the residue is −2.5188e-16 µm, while the standard deviation of the residue is 0.064 µm. Again, data along the 480th line in Fig. 7(c) and (e) are plotted in Fig. 7(g). In the plot, fitting result and the experimental data are in good agreement. However, the irregularity in the plot fluctuates a bit stronger than that in the previous testing due to the larger aperture which introduces more noise.
References [1] Renaud NM, Moore DT. Design and tolerancing of aspherical and gradient-index germanium singlets of equal performance. Appl Opt 1990;29:4030–5. [2] Milojkovic P, Christensen MP, Haney MW. Trade-offs between lens complexity and real estate utilization in a free-space multichip global interconnection module. J Opt Soc Am A 2006;23:1787–95. [3] Oliker V. On design of free-form refractive beam shapers, sensitivity to figure error, and convexity of lenses. J Opt Soc Am A 2008;25:3067–76. [4] Braunecker B, Hentschel R, Tiziani HJ. Advanced optics using aspherical elements [electronic resource]/. In: Bernhard Braunecker, Rüdiger Hentschel, Hans J, Tiziani, editors. Bellingham, Wash: SPIE Press; c2008, 2008. [5] Bennett JM, Dancy JH. Stylus profiling instrument for measuring statistical properties of smooth optical surfaces. Appl Opt 1981;20:1785–802. [6] Song JF, Vorburger TV. Stylus profiling at high resolution and low force. Appl Opt 1991;30:42–50. [7] O’Donnell KA. Effects of finite stylus width in surface contact profilometry. Appl Opt 1993;32:4922–8. [8] Morales A, Malacara D. Geometrical parameters in the Hartmann test of aspherical mirrors. Appl Opt 1983;22:3957–9. [9] Groening S, Sick B, Donner K, Pfund J, Lindlein N, Schwider J. Wave-front reconstruction with a Shack–Hartmann sensor with an iterative spline fitting method. Appl Opt 2000;39:561–7. [10] Pfund J, Lindlein N, Schwider J, Non- testing of aspherical surfaces by using a Shack-Hartmann sensor. In: Optical fabrication and testing. Optical society of america.: Québec City, 2000. p. OTuC5. [11] Su P, Parks RE, Wang L, Angel RP, Burge JH. Software configurable optical test system: a computerized reverse Hartmann test. Appl Opt 2010;49:4404–12. [12] Tang Y, Su X, Liu Y, Jing H. 3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry. Opt Express 2008;16:15090–6. [13] Tang Y, Su X, Wu F, Liu Y. A novel phase measuring deflectometry for aspheric mirror test. Opt Express 2009;17:19778–84. [14] Ono A, Wyant JC. Aspherical mirror testing using a CGH with small errors. Appl Opt 1985;24:560–3. [15] Kim T-H, Choi S-C. Measurement of highly aspherical surface using computer generated holograms. J Opt Soc Korea 2002;6:21–6. [16] Reichelt S, Pruss C, Tiziani HJ. Absolute interferometric test of aspheres by use of twin computer-generated holograms. Appl Opt 2003;42:4468–79. [17] Wang M, Asselin D, Topart P, Gauvin J, Berlioz P, Harnisch B. CGH null-test design and fabrication for off-axis aspherical mirror tests. In: International optical design. Optical society of america.: Vancouver, 2006. p. ThA3. [18] 〈 < http://www.zygo.com/?/met/interferometers/verifire/asphere/ > 〉. [19] Garbusi E, Pruss C, Osten W. Interferometer for precise and flexible asphere testing. Opt Lett 2008;33:2973–5. [20] Kim MK. Digital holographic microscopy: principles, techniques, and applications. New York: Springer; 2013. [21] Asundi, A. Digital holography for MEMS and microsystem metrology. [Electronic resource]. Chichester, Wiley; c2011, 2011. [22] Blum M, Büeler M, Grätzel C, Aschwanden M. Compact optical design solutions using focus tunable lenses. 2011. pp. 81670W–81670W–81679. [23] Zuo C, Chen Q, Asundi A. Phase retrieval and computational imaging with the transport of intensity equation: boundary conditions, fast solution, and discrepancy compensation. In: Classical optics 2014, Optical society of america.: Kohala Coast, Hawaii, 2014. p. CTh3C.3. [24] Cuche E, Marquet P, Depeursinge C. Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel offaxis holograms. Appl Opt 1999;38:6994–7001. [25] Goodman JW. Introduction to Fourier optics. New York: McGraw-Hill; 1996. [26] Jueptner U.S.a.W. Digital holography. 2004. [27] Wang Z, Qu W, Yang F, Asundi AK. Focal length calibration of an electrically tunable lens by digital holography. Appl Opt 2016;55:749–56. [28] Medhat M, Hassan AF, El-Nashar AF, Salman SMM. Determination of the surface parameters for aspheric aphakic lenses. Opt Laser Technol 1991;23:308–13. [29] Chen Z-l, Guo Z-d, Mi Q, Yang Z-q, Bai R-b. Research of fitting algorithm for coefficients of rotational symmetry aspheric lens. 2009. pp. 72834O–72834O– 72834. [30] Sun W, McBride JW, Hill M. A new approach to characterising aspheric surfaces. Precis Eng 2010;34:171–9. [31] El-Hayek N, Nouira H, Anwer N, Gibaru O, Damak M. A new method for aspherical surface fitting with large-volume datasets. Precis Eng 2014;38:935–47. [32] Cheng X, Yang Y, Hao Q. Aspherical surface profile fitting based on the relationship between polynomial and inner products. Opt Eng 2016;55:5, 015105-015105.
4. Conclusion In summary, a new aspheric lens measurement method has been presented. The method introduces an adaptive optical component-an electrical tunable lens (ELT) as phase compensator to measure aspheric lens using digital holography. However, the narrow range of ETL's NA limits its flexibility as well as applications. To overcome this drawback, an OL is introduced. The thin lens combination greatly extends the tuning power and thus improves the range of NA. Unlike the CGH approach, the proposed method does not require the NA of the aspheric lens to identically match the NA of the system. The lens combination is used to compensate the spherical sag in the aspheric lens, thus reducing the high gradients on the lens surface. By double-exposure technique, the proposed method permits absolute phase measurement. Furthermore, quantitative characterization is available which is often required in finish evaluation during design or fabrication. To verify the validity of the proposed method, aspheric lenses with different NA have been tested. Through parametric and visual comparisons, the experimental results are in good agreement with factory data. As all of the samples applied in this paper are plano-convex structure, the Mach-Zehnder interferometer was thus employed. For aspheric lens without plano-surface, Tyman-Green or Fizeau interferometer is recommended as they are more suitable for testing reflective surfaces. Of course, the proposed method can also be adapted to reflective surface measurement. Acknowledgment This work was supported by Translational Research & Development Grant MOE2012-TIF-1-T-003 from the Singapore National Research Foundation, Innovation Fund Grant MOE2013-
318
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Absolute measurement of aspheric lens with electrically tunable lens in digital holography
crossmark
⁎
Zhaomin Wanga, , Weijuan Qub, Fang Yangb, Ailing Tianc, Anand Asundia a b c
Centre for Optical and Laser Engineering, Nanyang Technological University, Singapore Ngee Ann Polytechnic, Singapore Shannxi Key Lab of Thin Films Technology and Optical Test, Xi’an Technological University, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Digital holography Aspheric lens Electrically tunable lens Phase measurement
A novel method for testing aspheric lenses using digital holography with an electrically tunable lens (ETL) is proposed and experimentally verified. The ETL generates a tunable deformed wavefront which helps to decrease the high gradient of aspheric lenses. By decomposing the aspheric surface into two resolvable ones, its absolute phase can be determined using a double-exposure measurement. In this method, the wavefront generated by the ETL need not be identical to the aspheric surface as in conventional null interferometer system, but just sufficient to resolve the high gradient surface. On the other hand, the tunability of the ETL allows generation of wavefronts which can be used to test different aspheric lens. Furthermore, advantages of the ETL such as low cost, fast response, and compact configuration make the proposed method a promising technique for aspheric surface measurement.
1. Introduction The performance of a high precision optical system using spherical optics is reduced by optical aberrations, and thus more optical elements are required. Using aspheric lens, these optical aberrations can be significantly reduced or even eliminated with a reduced (even single) number of optical elements [1–3]. However, testing of aspheric lenses is difficult due to high gradients on the aspheric surface resulting in high fringe densities which cannot be resolved and measured. Therefore, there is an urgent need to develop suitable test methods [4]. Many methods have been developed to measure aspheric lens. The stylus instrument is by far the most reliable tool owing to its high accuracy [5–7]. It doesn’t need any null optics, nor is it influenced by the optical properties of the sample. However, being a contact method, the stylus tip could damage the optical surface. Also, the stylus tip radius smoothens the measurement like a low-pass spatial filter. Moreover, being a point-wise measurement method, it is time-consuming. The Shack-Hartmann wavefront sensor is an economical choice for the measurement of aspheric lens form [8–11]. However, the shape and pupil adaptation of the optical system is needed, and the lateral resolution of this technique is limited. In recent years, deflectometry has attracted attention as it is capable of measuring aspheric surface with simple configuration [12,13]. The main drawback of this method is that its resolution is physically limited by the projector or the grating. Interferometers with a computer generated hologram (CGH) to
⁎
generate the desired wavefront have also been used [14–17]. This method has many advantages since the test and reference beams follow almost identical paths. But there is a need to redesign and manufacture new CGH for different asphere which is complicated, expensive, and time-consuming. Another interferometric method uses phase shift interferometry and heterodyne displacement interferometry to measure aspheric lens [18]. It has nanoscale vertical resolution as well as large measurement volume. However, the tested aspheric lens must be axially symmetric, have a measurable apex and defined aspheric equation, and cannot have a reversing curvature. Recently, interferometry with point source array based on structured illumination has also been proposed [19]. It evaluates the aspheric surface by fitting the spherical surface in advance. As this method does not model highfrequency components, it leads to final result containing high order systematic aberrations. Meanwhile, the calibration and alignment also greatly affect the final result. Digital holography (DH) is a well-established high precision 3D imaging technique which is capable of numerically reconstructing the optical field of a sample at any propagation distance [20,21]. It is able to access to the intensity and phase of the sample simultaneously. Along with fast reconstruction speed, high axial resolution, and low cost, it has become a powerful tool in quantitative phase imaging. However, this technique also encounters the same high gradient problem when measuring an aspheric lens. In this paper, a new adaptive optical element, ETL [22,23], is added
Corresponding author. E-mail address: [email protected] (Z. Wang).
http://dx.doi.org/10.1016/j.optlaseng.2016.09.002 Received 26 April 2016; Received in revised form 7 September 2016; Accepted 8 September 2016 Available online 0143-8166/ © 2016 Elsevier Ltd. All rights reserved.
Optics and Lasers in Engineering 88 (2017) 313–318
Z. Wang et al.
Fig. 1. Schematic of experimental configuration.
Fig. 2. ETL focal length variation versus driven current.
Ltd (www.doptron.com). The optical field at a propagation distance d is expressed as
to a DH system to overcome the above problem. The electrical tuning power allows ETL to generate a suitable wavefront that decreases the high gradients of the aspheric surface. A Mach-Zehnder configuration DH system with the ETL is developed and demonstrated for the aspheric lens measurement. The experimental results are in good agreement with factory data for two sample aspheric lenses.
EO (x, y, d ) = ℑ− ℑ ER* IH exp ⎡⎣ ikd 1 − r 2 ⎤⎦ ⎡⎣ fx, fy ⎤⎦ (x, y)
{ {
}
(2)
where ℑ and represent Fourier transform and its inverse. fx and fy represent spatial frequencies in x and y directions. The spatial filter radius r can be written as
2. Methodology
r=
2.1. Principle of digital holographic system
2
2
( λf ) + ( λf ) x
y
(3)
The phase of the object field is then obtained [26]
Fig. 1 shows the schematic of the Mach-Zehnder interferometer. A collimated laser beam is split by beam splitter 1 (BS1) into two beams. One beam goes through the ETL and is then normally incident on the aspheric lens surface under test. Since the beam passing through the aspheric lens diverges very fast, a triplet is introduced to reshape the beam. The other beam serves as the reference beam and interferes with object beam at BS4 with a small angular offset between them. The small angular offset is often manually adjusted to ensure the first order spectrum can be separated from the zero order spectrum in frequency domain. The image is resized by a relay imaging lens and captured by an imaging sensor which is often a charge-coupled device (CCD) camera. The recorded image, namely, digital hologram IH x, y [24] is expressed as
φ (x, y) = tan−1
Re[Eo (x, y, d )] Im[Eo (x, y, d )]
(4)
2.2. Numerical aperture calibration of ETL The focal length variation of ETL (Optotune EL-10-30-C-LD) is nonlinear versus its driven current as shown in Fig. 2. Therefore, it is necessary to calibrate the ETL before any further application. The calibration is as described in our previous work [27]. Data are recorded at 10 mA steps over the working range of the ETL from 0 mA to 290 mA. The diameter of the ETL aperture is 10 mm and thus the numerical aperture (NA) of the ETL varies from 0.025 to 0.062. This is too small to compensate for the high NA of most aspheric lenses. Therefore, an offset lens (OL) is introduced to increase the NA. The effective focal length of the ETL and offset lens can be obtained from geometric optics for two lenses in close proximity as
( )
IH (x, y) = EO + ER 2 = EO 2 + ER 2 + EO ER exp{−ikx sin θ} + EO* ER exp{ikx sin θ}
}
ℑ−
(1) 2
where EO is the object wave and ER is the reference wave. EO is the intensity of the object wave and ER 2 is the intensity of the reference wave. EO* denotes the complex conjugate of the object wave. k=2π/λ is the wavenumber, and θ is the angle between the reference wave and the normal. λ is the wavelength of the light source. The angular spectrum method [25] is used for numerical reconstruction through a user-friendly software developed by d′Optron Pte
1 1 1 = + f fETL fOL
(5)
where f is the effective focal length of the combination of ETL and OL. fETL and fOL are focal lengths of ETL and OL, respectively. In this paper, the focal length of OL is −150 mm. The focal length of the combination is shown in Fig. 3. After 314
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Fig. 3. Lens combination's focal length performance curve.
calculation, the NA of lens combination varies from 0.0014 to more than 1.0 thus making it suitable for aspheric lens testing. 2.3. Reconstruction of aspheric surface The essence of the proposed method is to decompose the large gradients of the asphere into two resolvable surfaces, and to measure them separately. Beyond that, the system aberrations need to be determined beforehand so as not to influence the final result. A double-exposure approach is proposed for the absolute phase measurement. In the first exposure, the aspheric lens with thin lens combination of ETL and OL is measured as shown in Fig. 1. The ETL drive current is adjusted to minimize the density of the interference fringe in the digital hologram. Once the fringes can be resolved, the digital hologram is recorded and processed to give the reduced phase of the aspheric lens. In the second exposure, the aspheric lens is removed, and a second digital hologram is recorded. The absolute phase of the aspheric lens can thus be retrieved from these two phases.
φAL (x, y) = φreduced _ AL (x, y) + φETL (x, y)
(6) Fig. 4. Schematic of an aspheric surface.
where φAL is the absolute phase of the aspheric lens. φreduced _ AL and φETL are the unwrapped phases of the reduced aspheric lens and the ETL, respectively. Finally, the topography of the aspheric lens can be expressed as
hAL (x, y) =
λ φ (x, y) 2π (nAL − 1) AL
expression is adopted in fitting. Two aspheric lenses were tested and demonstrated below. 3.1. 4.5 mm diameter aspheric lens testing
(7)
where hAL is the height on aspheric lens surface. nAL is the refractive index of aspheric lens.
In experiment, an aspheric lens primarily used for collimating the beam emerging from an optical fiber was tested. The diameter and focal length of the aspheric lens are 4.5 mm and 2.5 mm, respectively. A 660 nm solid state laser was used as light source and the focal length of the triplet and the relay imaging lens was 40 mm and 100 mm, respectively. The digital hologram was recorded by an 8-bit monochromatic CCD camera with 1280×960 pixels with a pixel size of 4.65 µm×4.65 µm. The distance between ETL and OL thin lens combination and the aspheric lens was 305 mm while the distance of the aspheric lens to triplet was 5 mm. The triplet was 75 mm from the relay imaging lens which in turn was 90 mm from the CCD. As the fitting algorithm was developed in polar co-ordinates, all the phases were apodized into a circular aperture for evaluation. Through calibration with a USAF 1951 resolution target, the diametric field of view was 3.033 mm. Fig. 5(a) shows the interference fringes generated by the aspheric lens without any current applied to the ETL. The fringes cannot be resolved by the imaging system. Next current was applied to the ETL to reduce the density of the fringes and for this example a current of 123.2 mA gave resolvable fringes. The wrapped phase of reduced aspheric lens is shown in Fig. 5(b). In second exposure, the phase
3. Experimental testing A typical aspheric surface schematic, comprising of a conic section with a spherical vertex, is shown in Fig. 4. The ISO 10110-Part 12 standard represents the sag z, as r2 R
z = f (r ) = 1+
1 − (1 +
n r κ )( R )2
+
∑ A2i r 2i i =2
(8)
where r is the lateral coordinate, R is the paraxial surface radius and κ is the conic constant. From Eq. (8), it can be seen that the paraxial surface radius R, conic constant κ, and coefficient A2i are parameters that identify an aspheric surface. To evaluate these parameters, a least squares method is often used in fitting. One should note that the aspheric surface expression is non-linear; hence the evaluation procedure needs to either convert the expression into a linear expression by simplification or to use nonlinear fitting method directly [28–32]. In this paper, a simplified linear 315
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Fig. 5. The wrapped phase of (a) asphere which cannot be resolved, (b) the reduced aspheric lens and (c) the ETL decomposed by (a) which can now be analyzed.
Fig. 6. Testing 1. Unwrapped phases of (a) the ETL, (b) the reduced aspheric lens, and (c) the aspheric lens. (d) 3D rendering for the phase of the aspheric lens. (e) Reconstructed phase by fitted parameters. (f) Residue phase between measured phase (c) and reconstructed phase (e). (g) Profile comparison of 480th line between measured and reconstructed curves.
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Table 1 Aspheric lens parameters comparison.
Test 1 Test 2
Aspheric parameters
A04
A06
A08
A10
Radius (mm)
Conic constant
Factory Fitted Factory Fitted
3.255E-3 3.287E-3 1.954E-5 1.999E-5
8.501E-5 8.572E-5 −1.756E-8 −1.872E-8
−2.227E-5 −2.225E-5 2.597E-11 2.874E-11
– – −2.414E-14 −2.933E-14
−2.3289 −2.3289 −22.09 −22.09
−0.8814 −0.8781 −2.271 −2.232
of the residue is 0.042 µm. The variance in the residue is mainly caused by random noise and irregularity of optical surface in the whole system. The standard deviation coincides with our system error level which is ten nanometers. While the extreme small mean value clearly demonstrates the average statistical characteristic of the random distribution of noise. In order to compare the goodness of the fitting, data along the 480th line in Fig. 6(c) and (e) were plotted in Fig. 6(g). It can be noted that two plots also show a good agreement.
without the aspheric lens is measured and displayed in Fig. 5(c). It can be seen that both wrapped phases are resolvable and can be easily unwrapped. Fig. 6(a) and (b) are the unwrapped phases of the reduced aspheric lens and the ETL respectively. The absolute phase aspheric lens can then be recovered as shown in Fig. 6(c). After implementing the fitting procedure, the aspheric parameters were obtained and compared with factory data in Table 1. It can be seen that the parameters are in good agreement. The reconstructed phase based fitted parameters is displayed in Fig. 6(e) with the residue shown in Fig. 6(f). In this case, the mean value of the residue is 1.29e-17 µm, while the standard deviation
Fig. 7. Testing 2. Unwrapped phases of (a) the ETL, (b) the reduced aspheric lens, and (c) the aspheric lens. (d) 3D rendering for the phase of the aspheric lens. (e) Reconstructed phase by fitted parameters. (f) Residue phase between measured phase (c) and reconstructed phase (e). (g) Profile comparison of 480th line between measured and reconstructed curves.
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TIF-2-G-012 from the Singapore Ministry of Education and International Cooperation Project 2014KW05 from Science and Technology Department of Shaanxi Province of China.
3.2. 25 mm diameter aspheric lens testing The second aspheric lens tested was a high precision aspheric lens (L-BAL35) from Newport, with 25.00 mm diameter and 37.50 mm focal length, respectively. With 25.00 mm clear aperture, its NA is approximately 0.33. The distance of thin lens combination to the aspheric lens was 285 mm and the distance of the aspheric lens to the triplet was 50 mm. The relay imaging was 65 mm from the lens triplet and the distance between the relay lens and CCD camera was 80 mm. The drive current applied to the ETL was 9.3 mA. The field of view was calibrated again, and the normalized aperture diameter was 9.65 mm for this example. Fig. 7(a) and (b) are the reconstructed phases of the reduced aspheric lens and the ETL, respectively. The absolute phase of the aspheric lens is displayed in Fig. 7(c). After fitting, the aspheric parameters were obtained and compared to factory values in Table 1. It can be seen that as the coefficient goes weaker with the term order increasing, the accuracy of fitting also goes lower. For weak aspheric coefficients, many factors such as fabrication, measurement scheme, system noise, fitting algorithm and computing errors would affect the final result. The reconstructed phase by fitted parameters is shown in Fig. 7(e), and the residue is shown in Fig. 7(f). The mean value of the residue is −2.5188e-16 µm, while the standard deviation of the residue is 0.064 µm. Again, data along the 480th line in Fig. 7(c) and (e) are plotted in Fig. 7(g). In the plot, fitting result and the experimental data are in good agreement. However, the irregularity in the plot fluctuates a bit stronger than that in the previous testing due to the larger aperture which introduces more noise.
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4. Conclusion In summary, a new aspheric lens measurement method has been presented. The method introduces an adaptive optical component-an electrical tunable lens (ELT) as phase compensator to measure aspheric lens using digital holography. However, the narrow range of ETL's NA limits its flexibility as well as applications. To overcome this drawback, an OL is introduced. The thin lens combination greatly extends the tuning power and thus improves the range of NA. Unlike the CGH approach, the proposed method does not require the NA of the aspheric lens to identically match the NA of the system. The lens combination is used to compensate the spherical sag in the aspheric lens, thus reducing the high gradients on the lens surface. By double-exposure technique, the proposed method permits absolute phase measurement. Furthermore, quantitative characterization is available which is often required in finish evaluation during design or fabrication. To verify the validity of the proposed method, aspheric lenses with different NA have been tested. Through parametric and visual comparisons, the experimental results are in good agreement with factory data. As all of the samples applied in this paper are plano-convex structure, the Mach-Zehnder interferometer was thus employed. For aspheric lens without plano-surface, Tyman-Green or Fizeau interferometer is recommended as they are more suitable for testing reflective surfaces. Of course, the proposed method can also be adapted to reflective surface measurement. Acknowledgment This work was supported by Translational Research & Development Grant MOE2012-TIF-1-T-003 from the Singapore National Research Foundation, Innovation Fund Grant MOE2013-
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