## Abstract

A new method improving the design of panoramic lens with a long focal length based on ogive and aspheric surface is proposed. In this design, we use a special conjugation between “annular entrance pupil” and aperture stop to correct the chromatic transverse aberrations. Moreover, we use a new imaging relationship to increase the Effective Focal Length (EFL) of the panoramic lens and the CCD utilization. We realize a panoramic lens with a 360° × (45°~85°)field of view (FOV) and a 10.375mm EFL, which is 1.54 times than the conventional imaging relationship.

© 2015 Optical Society of America

## 1. Introduction

With the coming of Information Age, the imaging lenses are hoped to have a long focal length, wide FOV and large relative aperture. The panoramic lens projects the cylindrically panoramic view of 360° around the optical axis to an annular area on a two-dimensional plane, is simple to fabricate and has a compact structure. These characters make the panoramic lens available in many applications, such as panoramic video camera, surveillance, robotic vision and remote conferencing [1–4 ]. The two catadioptric panoramic lens has a wider FOV, larger relative aperture, longer focal length and higher reliability compared with other panoramic lenses. As shown in Fig. 1 , it is composed of a panoramic annular lens (PAL), an aperture stop and relay lenses. Rays entering inside the panoramic lens would in turn pass the PAL, aperture stop and the relay lenses, and finally be received by the CCD to form an annular image. According to the principle of flat cylinder perspective (FCP), the annular image is consisted of some concentric circles, which can be divided into two parts. The inner part is the Blind Area corresponding to the non-imaging rays and the outer part is the Imaging Area referring to the imaging rays [5].

However, it is difficult to correct the chromatic transverse aberration of the panoramic lens with a long focal length. Almost all the existing panoramic lenses have complex structures and short focal lengths. Powell described the design of the PAL in details in his paper in 1994. He realized a PAL whose focal length is 2.65mm [6]. Shuang Niu *et al.* reported a PAL optical system with 10mm focal length by using cemented lenses to correct the chromatic aberration in 2007 [7].

A new method improving the design of the panoramic lens with simple structure and long focal length is proposed in this paper. We achieve a special conjugation between the “annular entrance pupil” and the aperture stop by using the ogive and aspheric surface in PAL. By this way, we’ve well corrected the chromatic transverse aberration. Moreover, we use a new imaging relationship$y\text{'}={f}_{p}\omega -h$to increase the EFL of the panoramic lens and the CCD utilization.

We’ve realized a panoramic lens composed of a PAL and six pieces relay lenses with a 360° × (45°~85°)annular FOV and a 10.375mm focal length, which is 1.54 times than the conventional imaging relationship. The structure is simple and has sharp imaging quality.

## 2. Design principle

#### 2.1 Chromatic transverse aberration correcting

It’s difficult to correct the chromatic transverse aberration of the panoramic lens because of the wide FOV and complicated structure. Typically, symmetrical system can effectively correct the chromatic transverse aberration. However, if the system is unsymmetrical, we need to individually correct the chromatic transverse aberration of the each unit [8]. If the aperture stop (exit pupil) is located near the rear surface, only the entrance pupil locating near the front surface could make the system quasi-symmetrical and thus benefit the chromatic transverse aberration correcting.

The panoramic lens is composed of PAL block, aperture stop, relay lenses and belongs to unsymmetrical system. Analyzing the structure of the panoramic lens, PAL brings in the most chromatic transverse aberration in the whole panoramic lens. To enhance the imaging quality, the panoramic lens needs a complex relay lenses to correct the chromatic transverse aberration introduced by the PAL and the PAL itself is hoped to be quasi-symmetrical. In this design, we put the aperture stop locate between the second refractive surface and the relay lenses to suppress the stray lights [9], thus the entrance pupil which is the image of the aperture stop after the rays passing through the PAL in the opposite direction should be put near the first refractive surface.

#### 2.2 Special conjugation between “annular entrance pupil” and aperture stop

As shown in Fig. 2 , a PAL is usually composed of two refractive surfaces (1, 4) and two reflecting surfaces (2, 3). Rays enter inside the PAL from the first refractive surface (surface 1) and then would in turn be reflected by the first reflecting surfaces (surface 2) and the second reflecting surface (surface 3). Finally, rays would exit the PAL from the second refractive surface (surface 4). During this process, surface 1 can refract the ${360}^{\circ}\times \alpha $ annular FOV into the PAL and then form a ${360}^{\circ}\times \gamma $ transmission inside the PAL, where$\gamma \approx \mathrm{arc}\mathrm{sin}(\frac{\alpha}{n})<\alpha $. Surface 2 can transform$\beta $into$\delta $and increase the CCD utilization. The surface 3 is used to reflect the rays outside the PAL and make the structure more compact. The Surface 4 can refract the rays outside the PAL. Moreover, $\delta $is hoped to be as small as possible and the imaging area must meet the new imaging relationship to realize a higher CCD utilization.

However, the FOV of the PAL is an annular field, which is beyond the limitation of the paraxial optics. The locations of entrance pupil are different under the effect of pupil aberration while the field angle is changing [10]. In our paper, we still use the paraxial optic concepts of aperture stop and entrance pupil where the image of the aperture stop’s center is the entrance pupil’s center after the rays passing the PAL. The center of the entrance pupil is located relatively with the PAL structure and the angle of the chief rays. Figure 2 illustrates that the center point C of aperture stop will image at two symmetrical point A and A’ in meridian plane. The chief rays have negative angle image at point A and the positive angle image at point A’. So, we can know that the center of the entrance pupil is no longer a point but a circle. We can also infer that the entrance pupil in the whole imaging space should be an annular shape, which is called the “annular entrance pupil” in this paper. In order to correct chromatic transverse aberration introduced by the PAL, we need to make the center of the entrance pupil located near the surface 1 and all coincide.

According to the description above, the special conjugation between “annular entrance pupil” and aperture stop can be realized.

#### 2.3 New imaging relationship

The image of the panoramic lens is a ring shape on the imaging plane, as shown in Fig. 3 . Diminish the diameter of inner circle while keep the outer circle unchanged can receive a higher EFL and CCD utilization at the same FOV.

To achieve a larger focal length and a higher CCD utilization, we use a new imaging relationship in our design:

where$h$is the “compression value”, ${f}_{p}$is the EFL of the panoramic lens, $\omega $is the field angle, ${y}^{\prime}$is the image height. Equation (1) helps to compress the image of the blind area and makes the image heights at imaging area proportionate to the FOV when comparing with the conventional imaging relationship.Moreover, from the Eq. (1), we can see that the EFL of the panoramic lens is given by:

Obviously, we can increase the EFL of the panoramic lens by increasing$h$while${y}^{\prime}$is unchanged. We should note that$h$is limited by the PAL. As shown in Fig. 2, surface 2 can converge the off-axis point A at on-axis point B and transform$\beta $into$\delta $. An increase in$h$ means a larger compression value of the field angle, thus will led a decrease in$\delta $. However, $\delta $is limited by the surface 2 and surface 4. Meanwhile, $h$is related with the distortion of the relay lenses.

## 3. Design process

#### 3.1 Ogive surface

An ogive is identical to the standard surface, except the axis of rotation of the surface is offset by an amount${r}_{o}$. The surface sag is given by Eq. (3) [11]:

Here$c$is the curvature (reciprocal of the radius) and$k$is the conic constant.

Figure 4 illustrates that an ogive surface can make the converging points have a lateral deviation at the imaging plane. This imaging property provides a theoretical basis of the annular aperture stop and the “annular entrance pupil”. Moreover, the ogive surface can change the incident angle of the off-axis rays. So it helps to realize the new imaging relationship.

However, the ogive surface increases the optimization difficulty of the optical system due to the vertex of ogive surface is non-differentiable.

#### 3.2 Coefficients solving of the ogive and aspheric surfaces

In order to calculate the aspheric coefficients, we use the standard Even aspheric equation:

Here$c$is the curvature (reciprocal of the radius), $k$is the conic constant, and${\alpha}_{\text{i}}$is aspheric coefficient. As shown in Fig. 5(a) , $d$is the deviation between the origin of the coordinates$O$and the vertex of the aspheric surface$O\text{'}$.

Moreover, before use such a high order, we used a series orders to carry out curve fitting. A high order up to 14 is finally believed to be the best choice while simultaneously considering the cost and the fabrication difficulty. Then, we can see that we have eight coefficients under resolved. Eight coefficients thus must have eight equations to calculate their value. We choose eight points on the surface 2 which meet the equal optical path condition (from point A to point B). However, other points on the surface 2 don’t still satisfy the equal optical path condition except these eight points. Constantly adjust the relative positions of these eight points and make the chief rays of different fields close to converge at point B after reflecting by the aspheric surface 2 until find the aspheric coefficients meet the equal optical path condition. The advantage of this solving process is the adjustment of every point only has a chief effect on the surrounding aspheric surface of this point and has little effect on the surrounding aspheric surface of other points.

Furthermore, we input the aspheric coefficients into Zemax and then optimize surface 1 (ogive surface) coefficients to make the chief rays meet the new imaging relationship${y}^{\prime}={f}_{p}\omega -h$, thus the coefficients of ogive surface could be solved. Figure 5(b) is the layout of the ultimate aspheric surface and ogive surface in the PAL.

#### 3.3 Other surfaces in the PAL

The main function of the surface 3 is to reflect the rays outside the PAL. As shown in Fig. 6 , surface 3 is located at the right side of point B. Rays should have converged at point B now would be reflected outside the PAL by the surface 3 to make the PAL more compact. Then, according to the reflecting imaging formula$\frac{1}{l\text{'}}+\frac{1}{l}=\frac{2}{r}$, the position and the curvature of the surface 3 can be solved. When calculate the parameters, the structure limitation and the stray lights suppression all need to be taken into consideration. As shown in Fig. 6, the surface 3 we solved is a convex spherical surface. Moreover, we select a flat surface as the surface 4 in the design because flat surface can provide a reference for the single point diamond turning (SPDT) and the assembly of the panoramic lens. Furthermore, we put the aperture stop 3mm away from the surface 4.

#### 3.4 Relay lenses matching

In the design of the panoramic lens, the relay lenses play an important role in imaging and aberration correcting. As shown in Fig. 7(a) , relay lenses are similar with eyepiece in structure and the aperture stop is regarded as the entrance pupil of the eyepiece. There is an 18° angle between exit rays and optical axis. The aperture stop diameter is 5mm and the distance between the internal virtual focus and the aperture stop is about 50mm. It is usual to increase the FOV of the relay lenses and the entrance pupil diameter considering of the matching of the system units. Thus, we select a 6mm entrance pupil diameter and 22° FOV when we calculate the original structure of the relay lenses. Figure 7(b) illustrate that a 6 pieces eyepieces structure is finally used as the original structure of the relay lens.

## 4. Optimization and analysis

Put the entrance pupil of the relay lenses coincide with the exit pupil of the PAL to construct the original structure of the panoramic lens. As shown in Fig. 8 , the structure is compact, has a 360° × (45°~85°) FOV, 6mm aperture stop diameter, 9.17mm image height and 134mm total length.

However, the FOV and the aberrations of the original structure can’t meet our design specification. Figure 9(a) illustrates that MTF is unacceptably low. Figure 9(b) shows that the RMS of the spot diagram is 90 um. Thus, we must optimize the panoramic lens to enhance the imaging quality.

The optimization process can be mainly divided into two steps. The first step is a primary aberration correcting. This process aims to correct the aberrations of the optical system and make the system meet the design specification. We firstly set the parameters of the relay lens as the chief optimization variables. If there is no ray spill out, we then set the parameters of the PAL as the variables step by step. Operand REAY is used to constraint the imaging relationship. Operand DMAL and TTHI are used to constraint the geometrical structure. The second step is a more accurate optimization to enhance the imaging quality. During this process, we can simultaneously set the parameters of PAL and relay lenses as variables to optimize the system. Meanwhile, we can correct the chromatic transverse aberration to satisfy the specification by changing the glass.

## 5. Design results

#### 5.1 The ultimate structure

As shown in Fig. 10 , the ultimate two catadioptric panoramic lens is composed of a PAL and 6 pieces relay lenses. This two catadioptric panoramic lens has a 7mm aperture stop diameter, 10mm image height, 360° × (45°~85°)FOV, 148mm total length and 5.9mm rear intercept.

Obviously, this structure is simple and has many advantages. The 5.9mm rear intercept makes the install of the CCD easier. The margin of the surface 1 and the surface 3 is very close and the marine of the surface 4 and the surface 2 is connected. This design benefits the single point diamond turning. Moreover, the aperture stop is 6.7mm away from the PAL and 4.5mm from the relay lenses. It benefits the install of the aperture stop.

#### 5.2 Imaging quality analysis

Figure 11(a) shows that the MTF of the panoramic lens is greater than 0.4 at 50 lp/mm after optimization. Figure 11(b) illustrates a near 5um RMS radius, which is much smaller than the CCD pixel. Figure 11(c) shows the field curvature and distortion of this two catadioptric panoramic lens. Since we compressed the 360° × (0°~45°) non-imaging rays based on the${y}^{\prime}={f}_{p}\omega -h$imaging relationship, the curvature and distortion are nonlinear. The field curvature of 360° × (45°~85°) imaging area has been reduced to 0.2mm. In Zemax software, the reference ray height is${y}^{\prime}={f}_{p}\mathrm{tan}\omega $, which is different from the imaging relationship${y}^{\prime}={f}_{p}\omega -h$. Thus, the distortion of 360° × (45°~85°) imaging area has a shift.

Furthermore, the imaging relationship is figured out to be${y}^{\prime}=10.375\omega -5.4917$ based on the data of the field and image heights at imaging area from the Zemax. Figure 12(a) is the imaging relationship at imaging area, where${f}_{p}$is 10.375mm and $h$is 5.4917mm. However, if we use the relationship ${y}^{\prime}=f\omega $ to image, focal length is only 6.74mm.That means a 1.54 times focal length have been achieved by using the new imaging relationship${y}^{\prime}={f}_{p}\omega -h$.

Figure 12(b) is the distortion of the panoramic lens, which is lower than 2% at the 360° × (45°~85°) imaging field. We can see that the distortion has been well corrected at the imaging area.

#### 5.3 Optical system and its picture

As shown in Fig. 13(a) , a panoramic imaging system composed of a panoramic lens and a CCD has been fabricated. Figure 13(b) shows a raw annular image taken by the panoramic imaging system. We can see that the CCD has a high utilization and the panoramic lens has a sharp imaging quality. Figure 13(c) shows the unwrapped image by use of the two-dimensional linear expansion.

## 6. Conclusions

A new method improving the design of panoramic lens with a long focal length is proposed. We use an ogive surface and aspheric surface to realize a special conjugation between the aperture stop and “annular entrance pupil”. This conjugation helps to correct the chromatic transverse aberrations. Moreover, we use a new imaging relationship${y}^{\prime}={f}_{p}\omega -h$ to realize a 1.54 times focal length and improve the CCD utilization compared with the conventional imaging relationship.

## Acknowledgments

This work is supported by the National Science and Technology Major Project under Grant No.2011ZX02402.

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