A New Analysis Method for Locating the Focus and for Estimating the Size of the Focus of the Backscatter Light of a LIDAR System ()
1. Introduction
In traditional Lidar systems without optical fiber application and not eye-safe, the low-optical efficiency is due to: 1) The weak optical signal, especially, the Raman return signal; 2) The background noise. In eye-safe Lidar system, the active area of infrared detector (infrared photodiode) is very small, compared with photomultiplier-tube (PMT), only several millimeters [1]. Therefore, to focus the backscatter light completely on the active area of the detector and remove the background noise becomes very important. The size of the field stop (aperture) of telescope at the location of the focus defines the field of view (FOV) of the telescope, dominates the background noise. Smaller field stop greatly reduces the active area of the primary mirror of a telescope [2]; Therefore, the smaller field stop reduces the amount of the Lidar return signal reaching the detector and results in the inefficiency of the Lidar system. Field stop coupling with the size of focus of backscatter light can optimize the optical efficiency of a Lidar system.
Optical fiber has been conveniently and widely used in Lidar measurements. For instance, optical fiber is used in conical-scanning time-correction Lidar system to easily measure wind speed in different directions [3]. Recently, the optical fiber is well used in multi-wavelength Raman-Lidar system for aerosol measurements in the troposphere. The existing fiber to telescope match is discussed in reference [4] in detail. In Lidar measurements, especially, optical fiber-based Raman Lidar measurements, how to improve the Signal to Noise Ratio is a critical issue which has been investigated by many researchers. A novel rotational Raman and Rayleigh Lidar system incorporate a fiber-based optical element to analyze the narrow Raman spectral feature [5]. For noncoaxial Lidar system, how to match the telescope to optic fiber is analyzed in reference [6].
The combination of the Numerical Aperture (NA) (NA defines the maximal acceptance cone of the optical fiber as
, where 
is the refractive index of the fiber core, and 
is the refractive index of the fiber cladding) of optical fiber and the size of the focus of a telescope can dominate the field of view (FOV) of a telescope, therefore, the maximal couple efficiency of the optical fiber and the telescope can be obtained. Reversely, the Numerical Aperture (NA) of optical fiber can also be designed by the size of focus of telescope.
Therefore, accurately locating the focus of the backscatter light of the Lidar system and estimating the size of the focus will contribute to the improvement of the optical efficiency in traditional Lidar system, and in optical fiber coupled telescope Raman or wind Lidar system, as well as in the infrared Lidar system with infrared photodiode.
In general, the researchers estimate the size of the focus of the backscatter light of a Lidar system by the limitation of the diffraction (airy disk) of the primary mirror of the telescope to collimate backscatter light of Lidar to the active area of the detector or the optical fiber. Actually, the size of the focus of the backscatter light is much larger than the airy disk. In this case, much energy of backscatter light will be lost. To locate the focus of a telescope, researchers have considered pointing the telescope to the moon; however, it is not convenient [7]. Furthermore, the location of the focus is not accurate and the observed focus is blurry.
This paper presents a new analysis method and relative experiment to accurately locate the focus of the backscatter light of a Lidar system and to estimate the size of the focus.
2. Theoretic Analysis for Locating the Focus and Estimating the Size of the Focus of the Backscatter Light of a Lidar System
Theoretically, the size of the focus of the backscatter light of a Lidar system is identical to the airy disk, relative to the wavelength of the backscatter light and diameter of the primary mirror of a telescope (the size of airy disk is defined as
, where 
is the diameter of the primary mirror, 
is the wavelength of backscatter light, 
is the focus of telescope). Actually, the blurry of image (or image error) of telescope optical systems always occurs. The divergence of incident light is not zero; therefore, the field of view of a telescope is not zero. The size of the focus of a telescope is much larger than the size of airy disk; therefore, we cannot regard the actual focus as an airy disk. This paper analyzes the location uncertainty of backscatter light’s focus and the uncertainty of the focus size, according to the image optical path of a telescope.
For easy discussion, the optical path of Newtown telescope is shown in Figure 1. The refraction index of the object space and the image space is n; the object space and image space of Newtown telescope overlap completely; the propagation direction of incident light in object space reverses in image space. Figure 1 shows that the image distance variation is 
and the image height variation is 
from the variation of incident angle
. 
and 
will be close to zero when 
is close to zero. Variation 
(
) is introduced for easy discussion. From Figure 1, Equations (1) and (2) can be easily deducted as follows:
(1)
(2)
Equation (3) can be obtained from the combination of Equations (1) and (2):
(3)
is the image distance.
The refraction index in object space is identical to that in image space, Equation (4) exists.
(4)
Combine differential calculation of Equations (4) and (3), Equation (5) or (6) is obtained at 
(5)
(6)
Take 
into account, Equation (7) can be obtained:
(7)
where 
is the object distance,
(8)
is amplification ratio of the image.
Equation (9) is obtained from the combination of (7) and (8).
(9)
when the object distance is very large, the amplification ratio of the image very small and we can assume it is zero. In this case, 
occurs. It can be seen in Equation (9) that 
is identical to a certain value when 
is very small. (Incident beam is parallel.) It means that the parallel beam will be focused by telescope.
Equation (10) can be obtained by (2):
(10)
(11) is obtained with consideration of (9):
(11)
Combine Equations (1) and (11), we get Equation (12):
(12)
From the geometrical relationships of Figure 1, we have:
(13)
According to Equations (9–13), we have Equation (14):
(14)
According to the above equations, we build the curve showing the relationship between
, the divergence, and the object distance, as well as the relationship between
, the divergence, and the objective distance. Figure 2 shows the relationship between 
and the divergence. The uncertainty of image distance 
is proportional to the divergence of the incident beam; as a result, 
becomes larger when the divergence of the incident beam becomes larger. And 
is also relative to object distance. 
becomes smaller at larger object distance. When object distance is 1000m (assume it is possible that the axis of the incident beam is parallel to the axis of telescope)，uncertainty of image distance 
is 1 millimeter at divergence of 1 mrad; if object distance is larger than 1000m, 
is less than 1 millimeter. 
is close to zero at object distance at
. Figure 3 shows the relationship between the uncertainty of the size of image disk (
) and divergence of the
Figure 2. The relationship between 
and divergence of incident beam
Figure 3. The relationship between 
and divergence
incident beam. 
is relative to object distance and is proportional to divergence of incident beam. 
becomes larger at larger divergence of incident beam and becomes smaller at larger object distance.
3. Focus Size of Backscatter Light of a Lidar System
According to Equation (14), we can assume that the axis of the incident beam is parallel to the axis of the telescope in the case of
, 
is the divergence of the incident beam. In the case of large object distance (
is close to infinite), we put 
into (14) to obtain
. This indicates that the location of focus does not vary with the variation of divergence of incident beam. In this case, we obtain 
by Equation (12), the uncertainty of the size of focus is double of focal length times divergence of incident beam. The size of focus is 0.3 millimeter at divergence of 0.1 mrad (mili-radian) at
. According to diffraction limitation (airy disk), the diffraction angle is about 10-5 rad (radian) if the diameter of laser beam is 10 millimeter, and the radius of the airy disk is about 0.015 milimeter. The actual size of focus is much larger than that of the airy disk; it is about 10~200 times of the airy disk.
Figure 4 presents the uncertainty of the location and the size of the focus. Figure 5 corresponds to Figure 4, it is image disk at different image distance analyzed by ZEMAX software which is powerful, accurate and affordable software for all aspects of optical system design. The middle image disk 3 TSA (min) (Transverse spherical aberration) corresponds to the focus of the laser beam [5]. Compared Figure 3 to Figures 4 and 5, Figure 3 is analysis result by the above optical path, it shows the size of image disk relative to object distance; the image disk is smaller if the object distance larger; the smallest image disk is the size of focus, corresponding to object distance at infinity. From Figures 4 and 5, we can see that the image disk 3 is smaller than image disk 1, 2, 4, and 5; the image disk 3 correspond to the smallest Transverse spherical aberration, the focus spot. Figure 3 coincides with Figures 4 and 5 analyzed by ZEMAX software.