Traditional laparoscopic optical systems consisting of about 30 lenses have low optical magnification. To magnify tissue during surgical operations, one must change from one laparoscope to another or use a magnifying adapter between the laparoscope and the sensor. Our work focuses on how to change the sag of a liquid lens while zooming from 1 × zoom, to 2 × , and 4 × in an optical design for a laparoscope. The design includes several lenses and two liquid lenses with variable focal lengths. A pair of laparoscopes for 3-D stereoscopy is placed within a tube 11 mm in diameter. The predicted depth resolution of tissue is 0.5 mm without interpolation at 4 × zoom.
©2013 Optical Society of America
Traditional laparoscopes used in minimally invasive surgery consist of many lenses and have limited visual properties, such as a fixed field of view, low magnification of 1 × –1.5 × , and high F-number at high zoom. High-definition (HD) image quality and 3-D imaging have recently become necessary in laparoscopes. A conventional surgical laparoscope consisting of more than 30 lenses is 30–40 cm long and 5–11 mm in diameter. The magnification is limited in optical systems using a zoom adaptor between the lens and the sensor. A zoom of 4 × was reported in a laparoscope using liquid lenses, called bio-inspired fluidic lenses . In this paper, a laparoscope offering high zoom and 3-D stereoscopy by using a fluidic lens is proposed. The aim of this work is to determine how to change the sag of liquid lenses from 1 × zoom to 2 × and 4 × with no moving parts. It has been shown experimentally that stereoscopes, which provide 3-D vision, improve the precision and speed of endoscopic surgery [2, 3]. The transverse resolution depends mostly on the size of the lens, wavelength of the light, and CCD . The depth of resolution in stereoscopy, δz (also referred to as the axial resolution) is expressed as follows :Eq. (1), the depth of resolution of 3-D stereoscope will be small at long focal lengths. The depth of resolution of 3-D stereoscope in zoom 4 × is 4 times smaller than zoom 1 × .
2. Tunable focal length by varying the curvature of a liquid lens for zooming
A simplified conceptual model of the zoom system consists of two tunable liquid lenses which works together to adjust the total zoom as shown Fig. 1. Variable focal length in a lens can be achieved by changing its shape. Both liquid lenses are enclosed in a thin, transparent elastomer membrane. The curvature of the liquid-filled lenses at either end of the membrane is changed under pneumatic control. The effective focal length f of the two variable liquid lenses can be expressed asEq. (2) can be expressed as follows.
The general lens shape can be expressed by Eq. (4), where c is the vertex curvature of the lens, r is the radius of the lens aperture, k is a conic constant, and A2, A4, etc., are the coefficients of the polynomial.Eq. (5), where k = A2 = A4 = … = 0. The radius of curvature of the lens, R1, is related to the lens sag z(b1) and aperture b1 as shown in Eq. (6).Eqs. (2) and (6). Figure 2 illustrates how the focal length changes with the lens sag. This fluidic lens design has the advantage of being convertible between a convex and a concave shape.
3. Lens design conditions
The effective focal length of a lens system depends on the refractive index and Abbe number of the lenses. The refractive index and Abbe number of liquid lenses can be modified by making a salt solution, as shown in Fig. 3, that can have a refractive index ranging from n = 1.32 to n = 1.46. The refractive index of oil ranges widely from n = 1.3 to n = 2.3. But it is not easy to keep oil from permeating through polydimethylsiloxane (PDMS). It is known that PDMS can block water, which has a surface energy density of 72.8 mJ/m2, and the surface energy of a liquid increases as the salt concentration increases . In this paper, the optical lenses are designed to use a liquid based on deionized water instead of oil. The liquid lens is enclosed by a transparent PDMS membrane (Sylgard 186, Dow Corning). PDMS has several excellent characteristics and is widely used for micro fabrication of various lab-on-a-chip devices because it is inexpensive, biocompatible, self-sealable, and highly elastic; further, it has excellent optical transparency and allows for easy device fabrication [7, 8]. The refractive index and Abbe number of LiCl solution increases 0.002 and 0.1377 times the salt concentration, respectively, as shown in Fig. 3. The design boundary conditions and goals for the 3-D laparoscope are given in Table 1. The maximum lens radius must be less 2.5 mm in order to insert two lens systems.
4. Lens design results
The optical lens module consists of six glass lenses, one polymer lens, one iris with diffraction optical element (DOE), and two liquid lenses. The lens module was designed in Code V using a two-mega pixel CMOS sensor having a pixel size of 1.75 μm. Figure 4(a) shows the zoom lens configuration, and Fig. 4(b) shows the structure of iris. Figure 4(c) illustrates how to change the lens curvature for different focal lengths. It was designed using the CODE V software from the Optical Solutions Group at Synopsys. In Fig. 4(a), a liquid iris is adapted because it can be made thin and in pairs. The blocking fluid of the iris is moved by the electrowetting force  and can absorb visible light using black ink . Figure 5 illustrates how the two sags of the lens module vary with the focal length. The relation between the effective focal length and the sag of the second lens can be expressed as y = −0.0937 Sag + 0.7357, but the relationship between the focal length and the sag of the first liquid lens is expressed by a curve marked with diamonds, as shown in Fig. 5. Figure 6 shows that the image resolution is changed by iris aperture due to zoom ratio. The F-number can be fixed by control the iris aperture. There is 4 mm gap from the stop to next lens. We designed the F-number 4.9–6.4 by using the slim liquid iris, 2 mm thickness like Fig. 4(b).
To reduce the chromatic aberration of the optical module, a DOE lens is applied to the iris cover surface using PDMS. The DOE is described in Code V by a continuous profile. The profile can be a function of x and y, φ(x, y), or rotationally symmetric, φ(r). The form of the phase profile is determined by the command HCT (Polynomial Grating Type). In a rotationally symmetric model, φ(r) is described by Eq. (7), where C1 = −0.000183, and C2 = −0.000445. The height and focal length of the DOE can be calculated by Eqs. (8) and (9), respectively, where λ0 is the design wavelength of light; n1(λ0) and n2(λ0) are the refractive index of the DOE lens and the surrounding refractive material, respectively. Figure 7 shows the phase function of the DOE. The number of DOE rings obtained using Eqs. (7) and (8) is 2.
5. Lens design characterizations
The modulation transfer function (MTF) is an important parameter for evaluating the optical performance. Figures 8(a)–8(f) illustrate the MTF value and the optical distortion is shown in Fig. 9 in wide-angle, medium-angle, and tele-angle mode. The optical performances are summarized in Table 2. Figure 10 is the result of 2D image simulation using USAF 1951 chart by CODE V utility.
Using two tunable liquid lenses, several lenses, a DOE, and one iris, we designed a variable zoom system that can achieve 1 × , 2 × , and 4 × zoom for a laparoscope with no moving parts. The trace of the sag of the liquid lens from 1 × zoom to 4 × zoom was optimized in the liquid lens module. The total length of the module is 23.26 mm. Its diameter is less than 4.1 mm, so two modules can be placed in a laparoscope 11 mm in diameter. The effective focal lengths of the wide-angle zoom, medium-angle zoom, and zoom-in modes are 3.24 mm, 6.4 mm, and 12.94 mm, respectively. The MTFs in the zoom-out and zoom-in modes are 35% and 30% at a frequency of 140 lp/mm. The fluidic zoom at lower F-number (4.9–6.04) is brighter than that of a conventional solid laparoscope. The optical distortions in the zoom-out and zoom-in modes are 16% and 3%, respectively The z-direction depth resolutions of this 3-D laparoscope are expected to be <0.5 mm and <2 mm in zoom-in and zoom-out modes, respectively, without interpolation.
References and links
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