50
J. Eur. Opt. Society-Rapid Publ. 22, 6( 2026)
Fig. 3. Geometry of the photon colliding with the cylinder target, the left sub-graph is projection of the interaction process.
possible upward interaction event. Assume L touch is the distance between P i and P touch, and the interaction position P touch can be described by P touch = P i + L touch U. According to the projection of the ray interaction with the cylinder in Figure 3, P’ C is the center of the projection circle, and L’ foot is the distance from P’ i to P’ foot. SinceP’ i =[ y i, z i ] isknownintheyoz plane, the point P’ foot can be represented as [ y i + L’ foot U( y), z i + L’ foot U( z)]. Based on vector
P� C
! 0 P 0 foot is perpendicular to vector� Pi
! 0 P 0 foot, the distance L’ foot and the point P’ foot can be obtained by �!
P 0 C P0 foot ~ e i ¼ 0; ð10Þ where ~ e i ¼½UðyÞ; UðzÞŠ is the unit vector from� Pi
!
0 P 0 foot.
For an interaction event, the distance PC
�!
0 P 0 foot R C( radius of the cross section). The distance L 0 touch is expressed as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 u
�!
L 0 touch ¼ L0 foot � t R 2 c � PC 0
P 0 foot
: ð11Þ
L 0 touch in projecting space is converted to three-dimensional space by
L 0 touch
L touch ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ð12Þ UðyÞ 2 þ UðÞ z 2
and the coordinates of P touch then can also be calculated. If P touch( x) 2 �L C
; L C 2 2 and the geometric distance of this
movement isn’ t less than L touch, the collision eventually occurs.
To calculate the direction after collision, the differential tangent plane around P 0 touch is treated as a Lambertian reflector, so the probability density function of the reflected angle h r can be expressed as
Pðh r Þ ¼ cos h r; ð13Þ
then the propagating direction Ut after collision can be obtained by substituting h r into h of( 7) and the unit
direction vector of PC
�! 0 P 0 touch into U, where the reflected
Fig. 4. Geometry of photon detection of MCT model.( a) Estimation of water signal( green dotted line) and target reflection signal( red dotted line).( b) The shaded area under the target and non-detectable water signal( gray dotted line).
azimuth angle u r = 2pn rnd. For the upward interaction event, U t( z) takes a positive value; for the downward interaction event, U t( z) takes a negative value. The photon’ s weight is reduced to W i = qW i, and q is the surface reflectivity of the target.
2.4 Interaction with the target
Only if the photon’ spositionP i is in the volume of detector’ s field of view( FOV), the signal contribution can be recorded. The condition of the detection is expressed as [ 69 ]
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ P i ðxÞ�P FOV ðxÞŠ 2 þ ½ P i ðyÞ�P FOV ðyÞŠ 2 R 1
><
R 1 ¼ H tan h
FOV þ R tele þ P �
½ iðÞ�H
z Šsin h FOV;
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >:
2
� � sin2 h n 2 w
FOV 2
ð14Þ
where P FOV is the center of the FOV section, and the R tele is the radius of the telescope. For vertical incidence, P FOV( x) andP FOV( y) are equal to x st and y st. For the oblique incidence, P FOV( x) and P FOV( y) are related to P FOV( z). Figure 4a describes the vertical detection situation when the lidar system is at P st = [ 0, 0, 0 ]. If the photon interacts with the target, P i is replaced with P touch to determine whether the interaction position satisfies the detection condition. The signal contribution of jth photon at position P i is estimated by [ 46 ]
Nði; j
Þ ¼ X bh ~ ð p ÞT s W i exp ð�s i Þ; ð15Þ
where bh ~ ð p Þ represents the probability that a photon will be scattered from the current direction to the receiving direction. If the photon interacts with the target, bh ~ ð p Þ is the Lambertian phase function. Otherwise, bhp ~ ð Þ is the phase function of mixed water in equation( 8). T s is the sea surface transmission, W i is the present weight of the photon. The solid angle of the detector DX can be calculated by the detector’ s image position P img =[ x st, y st, �n w H + H ]. s i is the optical depth from position P i to lidar position P st.