## Abstract

A non-iterative design and precise fabrication method of diffractive optical elements (DOEs) on multiple freeform surfaces is proposed and investigated in this paper. Complex amplitude modulation (CAM) technology is applied to design complicated DOEs. The wave-front for desired DOEs fabrication is interfered with a plane wave and then be encoded to a pure phase hologram. Simulations for different DOEs (binary and gray scales) on freeform surfaces are performed and the relative errors are 0.56% and 0.78%, respectively. Since the reconstructed optical fields generated by spatial light modulator (SLM) can be recorded into light-sensitive materials (photopolymer), the DOEs fabrication is realized by optical exposure. The results show that the proposed method can design and fabricate DOEs on multi-freeform surfaces at one time with high quality. Since the CAM method ensures precise reconstruction without iterations, the fabrication is accurate as well as the design is fast. It is expected that the proposed method could be applied in the precise 3D optical fabrication and processing in the future.

© 2017 Optical Society of America

## 1. Introduction

A variety of applications such as thin-film transistors [1], electronic eyes cameras [2] and artificial compound eyes [3] employ the curved diffractive optical elements (DOEs). There are several fabrication techniques, including laser direct writing lithography technique [4–9], the ion-beam proximity lithography [10], the soft lithography [11], electron beam lithography [12], nano-imprint lithography [13] and so on. However, considering the complicated processing, it is time-costing and difficult to achieve a nano-structure accuracy in a large size photosensitive material. For interference lithography, a nano-structure precision processing can be achieved high-efficiency in a large arbitrary area [14–20]. The patterning resolution can reach sub-wavelength of the incident light. Baker et al. [14] made sub-wavelength periodic patterns on curved surface by interference exposure method. After that, Mizutani et al. [16] developed a two-spherical wave ultraviolet interferometer to fabricate patterns. Compared with the two-plane-wave interferometer, the most distinguish advantage is that the variation of fringe period on curved surface can be highly restrained. Shi et al. proposed a method to design and fabricate an arbitrary planar DOE with good quality by interference [17]. Then, Zhao et al. improved this approach to design and fabricate the DOE on curved surface. However, these two methods both require the precise alignment of two phase-only spatial light modulators (SLMs) in micrometer level, which produces difficulties in the manufacturing process [18]. Wang et al. designed a method to fabricate DOEs on curved surface utilizing the computer generated hologram in which the depth information of original object is recorded, and there is no need for precise alignment [19]. But several exposures and processing are required in the fabrication of multi-DOEs by the above approaches, which is time-consuming and has low manufacture efficiency. In order to improve the manufacturing efficiency, Tian et al. ameliorated the algorithm, where the algorithm is applied for modulating the complicated three-dimensional (3D) intensity distribution on multiple curved surfaces simultaneously [20]. However, for one thing, algorithm will cost a large amount of time to approach the desired results. And there is no choice but to increase the number of iterations to ensure a good image quality. Moreover, for the sake of iteration algorithms limitation, the rate of error convergence is not fast and easily falls into local minimal value. That is to say, the reconstruction errors can hardly be further reduced. For another, the depths between the multi freeform surfaces and the phase distributions of DOEs cannot be modulated accurately, which have a great impact on fabricated precision.

As we know, the CAM can realize modulations of both amplitude and phase information simultaneously and independently to reconstruct the 3D intensity distribution, in which the image quality could be guaranteed. Besides, since it is free of iterations, so it can also provide the real-time modulation ability [21–26]. In this paper, we would also use interference lithography technique [14–20] to fabricate our desired DOEs on multi-freeform surfaces. And we propose a design method based on CAM to precisely reconstruct wavefront for fabricating DOEs. 3D binary patterns and 3D gray level patterns are recorded on multiple curved surfaces by designed and fabricated DOEs, and the reconstruction results of numerical simulation and the optical experiments both are successful, it indicates that they are in nice agreement. The efficiency and the quality of the fabrication are obviously improved.

## 2. System and principle

Figure 1 shows the schematic of the basic principle. Based on Huygens diffraction theory, the light propagates to the holographic plane$H$from the three surfaces${S}_{1}$,${S}_{2}$,$15\mu m$independently and overlaps with the parallel reference light, and which obtains a desired hologram.

In this paper, the forward Huygens-Fresnel diffraction [27,28] is defined as the light propagation in scalar diffraction domain from the hologram$H$to the multi- freedom surfaces ${S}_{1}$,${S}_{2}$,$15\mu m$. Under the paraxial approximation, we can use the Kirchhoff diffraction integral formula. The complex wavefront propagates from the hologram$H$to multi- freedom surfaces ${S}_{1}$,${S}_{2}$,${S}_{3}$ when the reference beam *R* illuminates $H$ plane. This process can be expressed as:

*z*axial distance from the point on a curved surface

*S*to the

_{i}*H*. Since the phase-type SLM cannot load the ideal complex amplitude wavefront of the hologram, the amplitude part of the hologram is usually removed, while only the phase part is remained simply in the traditional algorithmic theory (such as Fidoc phase distribution algorithm [28–32]). However, without several iterations, it is usually prone to result in large errors for fabricating a DOE proposed above, and the solution is just a rough approximation. Especially in condition of processing multi- freeform substrates, numerous iterations would cost time and the effect is limited.

In order to accurately modulate CAM of the incident light waves, we will add the amplitude information into the phase by imitating the optical recording process of phase-type hologram. The CAM interfered on the holographic plane can be expressed as:

Here${A}_{O}(x,y)={A}_{O}\mathrm{exp}[j{\phi}_{O}(x,y)]$ denotes the complex amplitude distribution on hologram$H$ propagated by objective beam$O(x,y)$, and ${A}_{R}(x,y)={A}_{R}\mathrm{exp}[j{\phi}_{R}(x,y)]$ is the distribution on hologram propagated by oblique parallel reference ray$R(x,y)$.The recording process of the phase-type hologram turns intensity distribution to phase profiles. Thus, the phase-type hologram can be calculated as:

*m*order of the Bessel function of the first kind. Then, the diffraction beam of −1 order is written by${E}_{-1}(x,y)=-j{\tau}_{0}{J}_{-1}[\alpha {O}_{0}(x,y)]\mathrm{exp}[j{\varphi}_{o}(x,y)]$. When Bessel function is defined by the infinite power series [33], generally just only the first term can be picked up, and owing to the low refractive index modulation coefficient, we can get an approximation

*J*[

_{-1}*αO*(

_{0}*x, y*)]

*≈-αO*(

_{0}*x, y*). Thus, the diffraction beam of −1 order shall be written as:As we can see, the target CAM will be reconstructed. And a planar holographic phase distribution with more accurate modulation of the intensity distribution of multi- freedom surfaces is designed.

From the above Bessel function, the zero order and high orders of diffraction cannot be eliminated completely. However, the optimized coefficient *α* is effective to limit other orders (high orders especially) [23], so we can modulate it to obtain ± 1 order and zero order as main orders of reconstruction distribution (including that of SLM). The −1 order beam can be picked up by using 4-f system and the appropriate band-pass filter (introduced to filter out zero order and + 1 order) when the reference light is tilted. Then 3D high quality intensity distributions can be obtained since the complex amplitude is reconstructed by −1 order modulation. Afterwards, we will load the designed pure phase hologram into the SLM. Through a proper filtering, the 3D intensity distributions of DOEs on multi- freedom surfaces can be acquired. Finally, our desired DOEs can be fabricated via a single exposure.

## 3. Simulation assessment

To evaluate the feasibility of the proposed algorithm for the precise modulation on multi- freedom surfaces, we perform the numerical simulation to demonstrate reconstruction quality on several different face-type substrates (including cylinders, complex freedom surfaces). And the results of the reconstruction quality are compared and analyzed by combining with the traditional phase recovery algorithm (Fidoc algorithm).

At first, we establish two symmetrical cylinders as our target curved surface substrates. The two symmetrical cylinders are shown in Fig. 2, where the subscript${O}_{1}$and${O}_{2}$means the central of two cylindrical surfaces respectively, L is the side length of cylindrical surface, R is the radius of the curvature, the${S}_{2}$demotes the field angle of cylinders, and $d$is the distance between two surfaces on$z$axial.

We apply a 3D gray pattern and a 3D binary pattern on the two cylindrical surfaces respectively to perform the numerical simulation. Top view of 3D original patterns and their results reconstructed by two algorithms are shown in Fig. 3. The original intensity distribution on surfaces${S}_{1}$,${S}_{2}$ are 3D binary pattern and 3D gray pattern respectively. Their resolutions are both 256 × 256 and then expanded into the patterns with pixels of 400 × 400 by zero-padding.

In the process of simulation, the used parameters are set as follows: ${L}_{1,2}=6.5mm$, ${R}_{1,2}=51.852mm$, $d=10mm$,$\lambda =532nm$, the distance of hologram plane to the first concave surface is${d}_{0}=196mm$, and the resolution of CGH is set as 800 × 800 while the pixel pitch is$8\mu m$. In order to better depart $\pm 1$ diffraction order away from the zero order (eliminating the zero noise), the reference beam is tilted with angle${\theta}_{Ry}={0.95}^{\circ},{\theta}_{Rx}={\theta}_{Ry}/2$. In addition, we introduce the relative error (RE) as the evaluation index of the reproduced image quality, and RE is defined as$RE=\frac{{\displaystyle \sum _{\alpha =1}^{N}{\displaystyle \sum _{\beta =1}^{M}[A{(\alpha ,\beta )}^{2}-{A}_{O}{(\alpha ,\beta )}^{2}]}}}{{\displaystyle \sum _{\alpha =1}^{N}{\displaystyle \sum _{\beta =1}^{M}{[{A}_{O}(\alpha ,\beta )]}^{2}}}}\times 100\%$. It means that the result of reconstruction would become worse as the value of$RE$increasing. In the above expression, $A{(\alpha ,\beta )}^{2}$*,${A}_{O}{(\alpha ,\beta )}^{2}$*denotes reconstruct intensity and original intensity of one point on a curved surface respectively. For the sake of fringes of equal thickness interference impacting the quality of 3D reconstruction profiles, the samples on the z-axis direction are given a small random perturbation within nanometer scale rather than equally spaced on cylindrical substrates (as the same as x-axis and y-axis). The 3D reconstruction results provided by proposed method and Fidoc algorithm are shown in Fig. 4 (b) and 4(c), respectively.

After 10 iterations, the values of RE produced by Fidoc algorithm are 2.59%, 2.34% respectively while the values of our proposed method are reduced to 0.56%, 0.78% without iteration. The relationship between RE and iterations of two patterns are shown in Fig. 5. That is to say, we could both shorten the fabrication time greatly and provide a more accurate modulation in DOE manufacturing on two curved surfaces.

To better demonstrate the feasibility of the method, we construct more complicated models which contain three curved surfaces. Among them, *S _{1}* and

*S*are cylindrical surfaces, ${L}_{1,2}=6.5mm$, ${R}_{1,2}=51.852mm$, $d=10mm$, their parameters are similar with the cylindrical surfaces in simulation, and the second surface is a freeform surface, the expression of freeform surface is ${z}_{2}=a\mathrm{sin}(\omega {x}_{2})\mathrm{sin}(\sigma {y}_{2})$, $a,\omega ,\sigma $are parameters of the freeform surface (we setting $a=1$,$\omega =0.4$,$\sigma =0.5$in our simulation). The ideal patterns on surfaces${S}_{1}$and${S}_{3}$are the same as the preceding patterns above, and the ideal pattern on surface${S}_{2}$is a new gray level pattern. Figure 6 shows the three 3D ideal patterns, their reconstructed patterns simulated by two different algorithms and their relative errors graphs.

_{3}To further compare the two methods, we expand all the original patterns with 128 × 128 pixels to 256 × 256 pixels by zero padding. The distance between the hologram plane and the surface${R}_{1,2}=51.68mm$is *d*_{0} = 1015 *mm*, the distance between these three curved surfaces are both$\lambda =532nm$, the pixels of hologram is 768 × 768. Figure 7 shows the top view of 3D original images and their reconstructed profiles. Among them, the patterns reconstructed by our proposed method are shown in Fig. 7 (b), 7(e) and 7(h), with the value of RE decreasing to only 0.33%, 0.35% and 0.28% without iteration. We also conduct the Fidoc algorithm for 10 iterations. Because their RE easily sink into local minimum values, the RE of Fidoc algorithm are down to 2.07%, 2.35%, 2.31% after 7th iteration and then rise slightly till cycles ending. Figure7 (c), 7(f) and 7(i) show the top view of reconstructed results. It is clear that the reconstructed distribution with high quality can be realized by using accurate complex amplitude modulation based on phase-only hologram, which could be employed in DOE fabrication on multiple curved surfaces. Compared with the traditional algorithm, the method is more feasible to be applied to fabrication for their superiority of improving the reconstruction image quality and saving the calculation time.

## 4. Experimental verification

In order to verify the availability of this method, we employ the holographic projection technique to conduct the experiment. Taking the operability of actual fabrication into account, we adopt cylindrical lens which has the same radius of curvature with the above simulation as curved substrates in the experiment directly. The optical path of the experimental setup is shown in Fig. 8.

The used light source is a green laser with wavelength of *532nm* and the power of *50mw*, *P* is a polarizer, *SF* denotes spatial pinhole filter. The light beam produced by the light source illuminate the SLM (BNS XY series with 8μm pixel pitch and 1920 × 1080 pixels, the active area is *7.68mm × 7.68mm*) after passing collimation system. The SLM loads the designed pure-phase hologram, modulate the incident laser beam into the wavefront distribution we desired subsequently. And then the 4-f system and a band-pass filter are applied to eliminate the strong noises containing in the modulated light from the SLM, because the wavefront distribution remain contains strong zero-order and other higher orders noises. Through these setups, the desired intensity distributions of DOEs are reconstructed and finally recorded with a single exposure on the surfaces of two cylindrical lens at their corresponding position, where the two surfaces are painted with photopolymer. The parameters referred in the experiment are as follows: ${L}_{1,2}=6.5mm$, ${R}_{1,2}=51.852mm$, $\lambda =532nm$, the refractive index modulation of the photo-polymer is$\Delta n=0.04$, and the thickness is$15\mu m$. the original resolution is 256 × 256 and is expanded to 400 × 400 pixels by zero padding. The distance between two cylindrical surfaces *d = 10 mm*, and the diffraction distances from the hologram plane to the nearest cylindrical surface${S}_{1}$is *d _{0} = 196 mm*. The pixel array of the SLM we adopt is 4

*00 × 400*in the middle. The focal length of the Fourier lens in the 4-f system is

*500 mm*, so the total distance from SLM to cylindrical lens in the actual optical system is

*2206mm*.

The reconstructed results are exhibited in Fig. 9, where shows the top view of reconstructed intensity distribution. They are recorded on photo-polymer directly and both captured by 50X Series Digital Microscope Olympus BX51M. Among of these two 3D patterns, the four Chinese characters “曲面加工” with high contrast are used as our binary pattern. The experimental reconstructed image is shown in Fig. 9(a). Another 256-gray-level image “flower” is tested similarly. The reconstructed result is show in Fig. 9(b). It is easily observed that the details of this four letters and that of the petal in the picture are displayed with high image quality. Our proposed method demonstrates the high quality on multiple curved DOEs fabrication without iteration.

The complex amplitude modulation method can be applied to DOEs fabrication on multi- freedom surfaces, during the process of CGH designing, the phase-only hologram added on SLM is encoded without iteration. For the complex hologram hardly losses wavefront information of original 3D intensity distribution, it is feasible employing the proposed method to improve the precision of DOEs fabrication on multi- freedom surfaces.

However, some factors still exist affecting the fabrication precision. One is the band-pass filtering architecture, it is inevitable declining efficiency for energy usage and reduction of the space-bandwidth product (SBP), where the aperture as large as possible can lessen the influence. In addition, through plenty of experimentation,we choose a proper laser power and exposure time, which can guarantees the modulated intensity distribution recorded in photopolymer material completely. Furthermore, the accuracy of experimental system is the other significant factor, because the imprecise planar wave illuminating SLM, inaccurate modulation of SLM, imprecise position of optical elements and other error in the optical system can lead to unexpectedly optical path length, which introduces speckle noise to reduce the quality of the pattern projected on photopolymer. The non-uniformity and the unevenness of the photopolymer also affect the fabrication results. In brief, the more ideal experimental environment we provide the better results produced. It reveals that the method we proposed is an effective method to modulate 3D intensity distribution on multiple curved surfaces.

## 5. Conclusion

We propose a method to design the pure-phase distribution for fabricating DOEs on multi- freeform surfaces, which can modulate the phase and the amplitude of the light wave simultaneously and respectively. Without iteration, 3D intensity distributions are reconstructed successfully without introducing any time consuming process. Both the numerical simulations and the optical experiment are performed, they are in good agreement. We can manufacture any desired complicated stripes on multi- freeform surfaces according to the design and fabrication method we propose. The proposed method is convenient and greatly saving fabrication time. It is believed that the proposed method is an effective method to fabricate high precision complicated DOEs on curved surfaces simultaneously, and it could be useful in various optical fields.

## Funding

Program 863 (2015AA015905); National Natural Science Foundation of China (NSFC) (61575024, 61235002, 61420106014).

## References and links

**1. **G. Yoo, H. Lee, D. Radtke, M. Stumpf, U. Zeitner, and J. Kanicki, “A maskless laser-write lithography processing of thin-film transistors on a hemispherical surface,” Microelectron. Eng. **87**(1), 83–87 (2010).

**2. **H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C.-J. Yu, J. B. Geddes 3rd, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature **454**(7205), 748–753 (2008). [PubMed]

**3. **D. Radtke, J. Duparré, U. D. Zeitner, and A. Tünnermann, “Laser lithographic fabrication and characterization of a spherical artificial compound eye,” Opt. Express **15**(6), 3067–3077 (2007). [PubMed]

**4. **Y. Xie, Z. Lu, F. Li, J. Zhao, and Z. Weng, “Lithographic fabrication of large diffractive optical elements on a concave lens surface,” Opt. Express **10**(20), 1043–1047 (2002). [PubMed]

**5. **Y. Xie, Z. Lu, and F. Li, “Lithographic fabrication of large curved hologram by laser writer,” Opt. Express **12**(9), 1810–1814 (2004). [PubMed]

**6. **D. Radtke and U. D. Zeitner, “Laser-lithography on non-planar surfaces,” Opt. Express **15**(3), 1167–1174 (2007). [PubMed]

**7. **T. Wang, W. Yu, D. Zhang, C. Li, H. Zhang, W. Xu, Z. Xu, H. Liu, Q. Sun, and Z. Lu, “Lithographic fabrication of diffractive optical elements in hybrid sol-gel glass on 3-D curved surfaces,” Opt. Express **18**(24), 25102–25107 (2010). [PubMed]

**8. **M. Mikutis, T. Kudtius, G. Slekys, D. Paipulas, and S. Juodkazis, “High 90% efficiency Bragg gratings formed in fused silica by femtosecond Gauss-Bessel laser beams,” Opt. Mater. Express **3**(11), 1862–1871 (2013).

**9. **L. Yuan, M. L. Ng, and P. R. Herman, “Femtosecond laser writing of phase-tuned volume gratings for symmetry control in 3D photonic crystal holographic lithography,” Opt. Mater. Express **5**(3), 515–529 (2015).

**10. **P. Ruchhoeft, M. Colburn, B. Choi, H. Nounu, S. Johnson, T. Bailey, S. Damle, M. Stewart, J. Ekerdt, S. V. Sreenivasan, J. C. Wolfe, and C. G. Willson, “Patterning curved surfaces: template generation by ion beam proximity lithography and relief transfer by step and flash imprint lithography,” J. Vac. Sci. Technol. B **17**(6), 2965–2969 (1999).

**11. **J. G. Kim, N. Takama, B. J. Kim, and H. Fujita, “Optical-softlithographic technology for patterning on curved surfaces,” J. Micromech. Microeng. **19**(5), 055017 (2009).

**12. **L. Yuan and P. R. Herman, “Layered nano-gratings by electron beam writing to form 3-level diffractive optical elements for 3D phase-offset holographic lithography,” The Royal Society of Chemistry **7**(47), 19905–19913 (2015).

**13. **T. Senn, J. P. Esquivel, N. Sabate, and B. Lochel, “Fabrication of high aspect ratio nanostructures on 3D surfaces,” Microelectron. Eng. **88**(9), 3043–3048 (2011).

**14. **K. M. Baker, “Highly corrected close-packed microlens arrays and moth-eye structuring on curved surfaces,” Appl. Opt. **38**(2), 352–356 (1999). [PubMed]

**15. **J. Nishii, “Glass-imprinting for optical device fabrication,” in Advances in Optical Materials, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AThC1.

**16. **A. Mizutani, S. Takahira, and H. Kikuta, “Two-spherical-wave ultraviolet interferometer for making an antireflective subwavelength periodic pattern on a curved surface,” Appl. Opt. **49**(32), 6268–6275 (2010). [PubMed]

**17. **R. Shi, J. Liu, J. Xu, D. Liu, Y. Pan, J. Xie, and Y. Wang, “Designing and fabricating diffractive optical elements with a complex profile by interference,” Opt. Lett. **36**(20), 4053–4055 (2011). [PubMed]

**18. **H. Zhao, J. Liu, R. Xiao, X. Li, R. Shi, P. Liu, H. Zhong, B. Zou, and Y. Wang, “Modulation of optical intensity on curved surfaces and its application to fabricate DOEs with arbitrary profile by interference,” Opt. Express **21**(4), 5140–5148 (2013). [PubMed]

**19. **X. Wang, J. Liu, J. Han, N. Zhang, X. Li, B. Hu, and Y. Wang, “3D optical intensity modulation on curved surfaces by optimization method and its application to fabricate arbitrary patterns,” Opt. Express **22**(17), 20387–20395 (2014). [PubMed]

**20. **R. Tian, J. Liu, X. Li, X. Wang, and Y. Wang, “Design and fabrication of complicated diffractive optical elements on multiple curved surfaces,” Opt. Express **23**(26), 32917–32925 (2015). [PubMed]

**21. **J. P. Liu, W. Y. Hsieh, T. C. Poon, and P. Tsang, “Complex Fresnel hologram display using a single SLM,” Appl. Opt. **50**(34), H128–H135 (2011). [PubMed]

**22. **Q. Gao, J. Liu, J. Han, and X. Li, “Monocular 3D see-through head-mounted display via complex amplitude modulation,” Opt. Express **24**(15), 17372–17383 (2016). [PubMed]

**23. **X. Li, J. Liu, J. Jia, Y. Pan, and Y. Wang, “3D dynamic holographic display by modulating complex amplitude experimentally,” Opt. Express **21**(18), 20577–20587 (2013). [PubMed]

**24. **Q. Gao, J. Liu, X. Duan, T. Zhao, X. Li, and P. Liu, “Compact see-through 3D head-mounted display based on wavefront modulation with holographic grating filter,” Opt. Express **25**(7), 8412–8424 (2017). [PubMed]

**25. **J. P. Liu, W. Y. Hsieh, T. C. Poon, and P. Tsang, “Complex Fresnel hologram display using a single SLM,” Appl. Opt. **50**(34), H128–H135 (2011). [PubMed]

**26. **H. Song, G. Sung, S. Choi, K. Won, H.-S. Lee, and H. Kim, “Optimal synthesis of double-phase computer generated holograms using a phase-only spatial light modulator with grating filter,” Opt. Express **20**(28), 29844–29853 (2012). [PubMed]

**27. **J. W. Goodman, *Introduction to Fourier Optics* (Roberts & Company Publishers, 2005).

**28. **A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. **10**(3), 035302 (2008).

**29. **H. Akahori, “Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform,” Appl. Opt. **25**(5), 802–811 (1986). [PubMed]

**30. **F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A **7**, 961–969 (1990).

**31. **M. Gruber, “Diffractive optical elements as raster-image generators,” Appl. Opt. **40**(32), 5830–5839 (2001). [PubMed]

**32. **C. Bay, N. Hübner, J. Freeman, and T. Wilkinson, “Maskless photolithography via holographic optical projection,” Opt. Lett. **35**(13), 2230–2232 (2010). [PubMed]

**33. **F. Bowman, Introduction to Bessel functions (Dover Publications, 1958), Chap. 1.