Design of off-axis reflective projection lens using spherical Fresnel surface

Optik 122 (2011) 145–149

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Design of off-axis reflective projection lens using spherical Fresnel surface Zhenrong Zheng ∗ State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou, 310027, China

a r t i c l e

i n f o

Article history: Received 28 July 2009 Accepted 9 February 2010

Keywords: Optical design Reflective lens Fresnel

a b s t r a c t In this paper, an off-axis reflective projection lens with single Fresnel reflective surface and three aspheric surfaces were designed. The design method of reflective lens using spherical Fresnel surface is discussed. The MTF (Modulation Transfer Function) of the off-axis reflective lens, with optical magnification 100×, F-number 2.5 and field of view 120◦ , is over 40% at 0.6 lp/mm on the image side, the distortion is less than 2%. This design method can provide reference for application of Fresnel surface in wide field of view imaging, and possesses a bright future with the continuous development of fabrication technique. © 2010 Elsevier GmbH. All rights reserved.

1. Introduction Projection lens is one of the most important parts in many optical systems. The trend of projection lens is towards to short-focus length, wide field of view (FOV), decreasing F-number and high definition [1], and then the design and production of projection lens become more and more difficult. Generally, the projection lens is designed by refraction lens, however, when refraction objective lens is used in short-focus and wide field of view, the variety of aberrations is difficult to reduce because of rapid increase of chromatic aberration and axial coma aberration owing to wide field of view and F-number. The refraction objective lens has original blemish because of dispersion [2]. Hence, the design concept of reflective objective lens is accepted by more and more lens designer. Many reflective projection lens design are reported [3–5], Ogawa studied a projection lens layout with four aspheric mirrors [6], the detail design and manufacture performance were put forward. But the manufacture and measurement of large aspheric surfaces are very difficult. In order to simply the structure of objective lens and reduce the cost, Fresnel surface are used by more and more researchers [7–10]. In this paper, a reflective projection lens using spherical Fresnel surface is designed. A reflective projection lens consisted by a Fresnel reflective surface and three aspheric surfaces is designed. Under the condition of magnification 100×, F-number 2.5 and field angle 120◦ , the MTF reaches over 40% at 0.6 lp/mm on the image side, and the distortion is less than 2%. This method can provide reference for application of Fresnel surface in visible light imaging,

and possesses a bright future with the continuous development of fabrication technique of Fresnel devices. 2. The design method of reflective Fresnel surface with spherical shape The design idea is using spherical Fresnel surface, which has aspheric reflective phase, to design the lens. Eq. (1) shows the expression of even aspheric: Z(r) =

1+



cr 2 1 − (1 + k)c 2 r 2

+ a2 r 4 + a3 r 6 + a4 r 8 + a5 r 10 + . . . (1)

where c is the curvature of the reflective surface, k is conic constant, a2 , a3 , a4 , a5 . . . are respectively even aspheric coefficients. Fig. 1 is the schematic diagram that reflective even aspheric surface is replaced by reflective Fresnel zone plate with spherical shape. Fig. 2 is the enlarged view of a single Fresnel surface of Fig. 1. In Fig. 2, ti = R −





R2 − xi2



R2 − xi2 − R2 − (xi + li ) (h + di ) tg i = i li (xi + li /2)  tg2 i =  f − ti − (hi + di )/2 di =

2

(2)

Meanings of characters in Eq. (2) are shown in Fig. 3; Eq. (2) can be solved and obtained: ∗ Fax: +86 571 8795 1758. E-mail address: [email protected]. 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.02.011

(xi − li /2)tg 2

i

+ 2(f − ti )tg

i

− (xi + li /2) = 0

(3)

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Z. Zheng / Optik 122 (2011) 145–149

Fig. 1. Reflective Fresnel zone plate with spherical shape.

Thus, the parameters of single reflective Fresnel surface can be obtained,

 tg

i

=

−2(f − ti ) +

hi = li · tg

i

− di

2

4(f − ti ) + 4xi2 − li2 2xi − li

(4)

Parameters of initial structure are: x1 = 0,

t1 = 0,

xi+1 = xi + li

(5)

If the radius of reference spherical surface is chosen, structure parameters of reflective Fresnel surface can be obtained according to Eqs. (2)–(5) by different li , and the reflective Fresnel surface can be designed for having same focal power with aspheric surface. The selection of R and li should make that the aberration of reflective Fresnel surface is similar with the replaced aspheric surface. The spot diagram of aspheric surface at specified location can be obtained by ray tracing, the spot diagram of reflective Fresnel zone plate with spherical surface can be calculated by amplitude transmittance. According to the OPD (Optical Path Difference) of rays reflected by Fresnel zone plate, the phase shift between pre- and post of reflective Fresnel zone plate can be shown: (x) = 2k[hi (x) + di (x) + ti (x)],

2 where k = 

Fig. 3. Layout of objective lens with four even aspheric reflective surfaces.

The function of amplitude transmittance for reflective Fresnel surface can be expressed by:

T (x) =

N 

exp{−j2k[hi (x) + di (x) + ti (x)]}

(7)

i=−N

According to Eqs. (2) and (4),

 hi (x) + di (x) =

tg

− xi )

i (x

xi < x ≤ xi + li

0

(8)

other

(6) where xi =

i−1 

lk , Eq. (7) can be expressed by:

k=0

⎧ ⎪ ⎪ ⎪ ⎪ N ⎪ ⎨

⎛

⎜x − ⎜ ⎜ T (x) = rect ⎜ ⎜ ⎪ i=−N ⎪ ⎪ ⎝ ⎪ ⎪ ⎩

i−1 

 exp −j2kli tg

⎞ lk − li /2 ⎟

li

 i

·

⎟ ⎟ ⎟· ⎟ ⎠

k=0

x−

i−1  k=0

Fig. 2. One of the reflective Fresnel surface with spherical shape.

 lk

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

· exp[−j2kti (x)]

⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(9)

Z. Zheng / Optik 122 (2011) 145–149

147

Table 1 The parameter of objective lens with four aspheric mirrors.

M1 M2 M3 M4

1/c

K

a2

a3

a4

a5

83.9 63.585 138.04 27.797

0.589 −10.972 −0.169 −3.423

−1.39e−7 4e−08 −3.03e−7 −8.02e−8

7.17e−10 −3.06e−09 2.17e−11 5.92e−12

−1.82e−12 2.46e−10 −1.384e−15 −2.92e−16

2.13e−015 7.11e−16 4.63e−20 8.94e−21

The light intensity distribution on focus plane can be obtained by Fourier transformation for function of amplitude transmittance.

⎧ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ N ⎨ ⎪ ⎨

⎛

⎜x − ⎜ ⎜ F[T (x)] = F rect ⎜ ⎪ ⎪ ⎜ ⎪ i=−N ⎪ ⎪ ⎪ ⎝ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ N  ⎨

⎛

⎜x − ⎜ ⎜ = F rect ⎜ ⎜ ⎪ i=−N ⎪ ⎪ ⎝ ⎪ ⎪ ⎩

i−1 

lk − li /2 ⎟

 ⎟ ⎟ ⎟ · exp −j2kli tg ⎟ ⎠

k=0

li

i−1 

 i

·

x−

i−1 

 lk

lk − li /2 ⎟

li

 ⎟ ⎟ ⎟ · exp −j2kli tg ⎟ ⎠

 i

· exp[−j2kti (x)]

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎭⎪

k=0

⎞

k=0

⎫⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎬⎪

⎞

·

x−

i−1  k=0

 lk

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

· exp[−j2kti (x)]

⎪ ⎪ ⎪ ⎪ ⎪ ⎭

The distribution on Z offset plane against focus plane can be acquired by Fresnel diffraction: U(Z) =

  1 exp(ikZ)F T (x) exp[jkx2 /2Z] jZ

(10)

The light intensity distribution of spot for ray reflected by Fresnel zone plate with spherical surface can be obtained by Eq. (10), according to known parameters R and li . The reflective Fresnel surface can achieve same focal power with the replaced aspheric surface by continuously correction of parameters R and li , meanwhile, the aberration of reflective Fresnel surface is similar with the replaced aspheric surface. 3. Design and optimization Selecting a suitable initial layout for a reflective lens design is very important. Different representation layouts not only have different impacts on the ray tracing speed but also offer the convergence of optimization. Firstly, We design a reflective projection lens composed of even aspheric reflective surface as a suitable representation layout. Fig. 3 is the layout of the initial projection lens layout with four aspheric surface. Object plane is reflected by the four even aspheric surfaces M1, M2, M3, and M4, imaged on the image plane. In this design, M1 and M3 are concave mirrors; M2 and M4 are convex mirrors. The parameters of four even aspheric surfaces are given in Table 1. Magnification of the design shown in Fig. 1 can reach 100× when objective distance is 247 mm. Table 1 shows the parameter of the short-focus length projection lens. Fig. 3 shows that aspheric mirrors M3 and M4 have large-area aspheric mirrors with high precision, which are hard to be manufactured and measured. The focus length of M3 is −79.8 mm and the dimension is 80 mm × 60 mm. In the reflective projection lens shown in Fig. 3, the major function of aspheric mirror M3 is to correct astigmatism and distortion. This short-focus reflective projection lens is used as initial design layout. And then, a reflective Fresnel surface is adopted to replace the M3 even aspheric surface. According to Eqs. (4) and (5), the structure parameters which satisfy the requirement of focal power can be calculated. The reference pitch of Fresnel surface is chosen by 0.02 mm. Meanwhile, the reflective surface

of basic structure is divided into surface unit by n(n = 8–12) and corrected. The intensity of reflective Fresnel surface with spherical shape can be calculated, the reference pitch and angle of reflecting surface can be slight modified in order to make it have similar aberration with aspheric mirror. Although most commercially available optical design software, such as Zemax/CodeV, offers the ability to model Fresnel surfaces in standard surface type, user-defined surface (UDS) is a powerful, flexible, and fast way. In our design process, a Windows Dynamic Link Library (DLL) has been built, the benefit of DLL file is which can calculate and optimize the Fresnel surface in software. Fig. 4 shows a part of reflective Fresnel surface. The optical specifications of reflective projection lens with spherical Fresnel surface are summarized in Table 2. During the optimization process, the major constraints were the effective focal length, overall length, as well as general constraints for height and tilt. All the tilts and decenters were kept in the YZ plane. During the final stage of optimization, the third-order distortion was added to limit the system performance. After many times iterations, the system distortion was reduced to 2% at the 120◦ field. Fig. 5 is the comparison diagram of spot distribution between reflective Fresnel surface and even aspheric reflective surface at the center field of image plane.

Fig. 4. The structure of reflective Fresnel reflective surface.

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Z. Zheng / Optik 122 (2011) 145–149

Table 2 Optical specifications. Parameter

Specifications

Configuration Image size Magnification Projection distance F-number Field of view Distortion Vignetting Image quality

Reflective lens Circle 6 mm 100× <250 mm 1/2.5 120◦ <2.2% <10% over total field >40%@0.6 lp on image

Fig. 6. Structure (a), MTF (b) and of 3-aspheric and 1 reflective Fresnel lens.

F-number 2.5, projection distance 250 mm and field angle 120◦ , the MTF reaches above 40% at 0.6 lp/mm on the magnification side, which shown in Fig. 6(b). MTF of the other side can reach above 60 lp/mm by considering the magnification 100× of objective lens, which can satisfy the definition requirement of objective lens. Fig. 7 shows the field curvature and distortion results. The distortion is less than 2% in totally field. Fig. 5. Spot diagram for aspheric surface and spherical Fresnel reflective surface.

4. Results and conclusions In this paper, a deduced method, using reflective Fresnel surface with spherical shape to replace aspheric mirror, was put forward. Also, a reflective projection lens with a spherical Fresnel reflective surface was designed. Under the properties of magnification 100×, F-number 2.5 and field angle 120◦ , the MTF was over 40% at 0.6 lp/mm on the magnification side, the distortion was less than 2%. Short projection distance less than 250 mm was achieved.

Table 3 is the structure parameters of final design. Compare Table 3 with Table 1, we can see that the residual aberration which induced by spherical Fresnel reflective surface has to be partly corrected by the other aspheric mirrors. Fig. 6 is the layout and MTF results of objective lens. Fig. 6(a) is the layout of structure with three aspheric mirrors and a piece of spheric Fresnel mirror, under the condition of magnification 100×, Table 3 The parameter of objective lens with Fresnel mirror and three aspheric mirrors.

M1 M2 M3 M4

Type

1/c

k

a2

Aspheric Aspheric Fresnel Aspheric

83.902 65.373

0.589 −9.232

27.797

−3.837

−1.39e−07 7.17e−10 4e−08 −1.83e−08 Fresnel mirror (R = 150 mm, li = 0.02 mm) −6.30e−08 7.52e−12

a3

a4

a5

−1.82e−12 1.40e−10

2.13e−015 6.31e−16

−3.55e−16

8.84e−21

Z. Zheng / Optik 122 (2011) 145–149

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Acknowledgement This research is partially supported by the National Basic Research Program of China (973 Program) (No. 2009CB320803). References

Fig. 7. Field curvature and distortion of lens.

Although the Fresnel design provides light weight and low replication cost, the intensity efficiency and stray light of the Fresnel surface need to be carefully considered because of the diffraction. However, the design provides reference for application of Fresnel surface in short-focus and wide field of view projection system, and presumes having a bright future with the continuous development of fabrication technique.

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