The absorption of seawater mainly includes the absorption of pure seawater, chlorophyll and yellow matter, expressed by the absorption coefficient a_sea(λ); the scattering of seawater mainly includes the scattering of pure seawater, chlorophyll, small particles and large particles, expressed by the scattering coefficient b_sea(λ). Where the scattering of seawater is divided into forward scattering and backscattering, which are expressed by the forward scattering coefficient b_seaf(λ) and the backscattering coefficient b_seab(λ), respectively.

The absorption of pure seawater includes the absorption of pure water and salt in seawater. Assuming that the absorption of salt is negligible in the visible wavelength band [6], the absorption coefficient of pure seawater for light in the visible wavelength band is approximately equal to the absorption coefficient of pure water. In 1997, Pope et al. gave the absorption coefficients of pure water for light in the 380 nm to 727.5 nm band, where the absorption coefficients of pure water are 0.00922 m⁻¹ for blue light at 450 nm and 0.0409 m⁻¹ for green light at 520 nm [3].

The absorption coefficient of chlorophyll a_c(λ) is [11]

a c ( λ ) = a c 0 ( λ ) ( C c C c 0 ) 0.602 (1)

where a c 0 ( λ ) is the specific absorption coefficient of chlorophyll, C_c is the concentration of chlorophyll, C c 0 = 1 mg/m³. Reference [18] gave the values of a c 0 ( λ ) at different light wavelengths, a c 0 ( 450 ) = 0.944   m − 1 and a c 0 ( 520 ) = 0.528   m − 1 .

Yellow matter is a kind of humus produced by the decay of animals and plants, and is a kind of chromophoric dissolvable organic matter (CDOM) with complex structure dissolved in seawater. The absorption coefficient of the yellow substance a_y(λ) is [11]

a y ( λ ) = a f 0 C f exp ( − k f λ ) + a h 0 C h exp ( − k h λ ) (2)

where the specific absorption coefficient of fulvic acid a f 0 = 35.959 m²/mg, the specific absorption coefficient of humic acid a h 0 = 18.828 m²/mg, k_f = 0.0189 nm⁻¹, k_h = 0.01105 nm⁻¹, C_f is the concentration of fulvic, and C_h is the concentration of humic. Haltrin pointed out that C_f and C_h are related to C_c, and can be expressed by C_c [11]. Therefore, the absorption coefficient of seawater a_sea(λ) is

a s e a ( λ ) = a p s ( λ ) + a c ( λ ) + a y ( λ ) (3)

where a_ps(λ) is the absorption coefficient of pure water.

The scattering coefficient of pure seawater b_ps(λ) is [10]

b p s ( λ ) = 8 π 3 2 + δ 1 + δ β ( 90 ˚ ) (4)

where δ = 0.039 is the depolarization rate of pure seawater [9], and the expression of β(90˚) is [8]

β ( 90 ˚ ) = π 2 k B T 2 λ 4 6 + 6 δ 6 − 7 δ [ − g p p g p h d 2 ( n ) + g p g S S h c 2 ( n ) ] (5)

where k_B is the Boltzmann constant, T is the absolute temperature of seawater, λ is the wavelength of light, g_p and g_pp are the first and second order partial derivatives of the Gibbs energy of seawater g(S_A, T_c, p) with respect to the pressure p, g_SS is the second order partial derivative of g(S_A, T_c, p) with respect to the absolute salinity S_A, T_c is the temperature of seawater, and n is the refractive index of seawater. n is related to λ, T_c, S (the salinity of seawater) and p. Reference [8] gave the expression of g(S_A, T_c, p), therefore

g p = ∂ g ( S A , T c , p ) ∂ p = ∑ j = 0 7 ∑ k = 0 6 k g 0 j k w p u k ( T c T u ) j p k − 1 + ∑ j = 0 6 ∑ k = 0 5 [ g 1 j k s ( S A S u ) ln ( S A S u ) 0.5     + ∑ i = 2 7 g i j k s ( S A S u ) 0.5 i ] k p u k ( T c T u ) j p k − 1 (6)

g p p = ∂ 2 g ( S A , T c , p ) ∂ p 2 = ∑ j = 0 7 ∑ k = 0 6 k ( k − 1 ) g 0 j k w p u k ( T c T u ) j p k − 2 + ∑ j = 0 6 ∑ k = 0 5 [ g 1 j k s ( S A S u ) ln ( S A S u ) 0.5     + ∑ i = 2 7 g i j k s ( S A S u ) 0.5 i ] k ( k − 1 ) p u k ( T c T u ) j p k − 2 (7)

g S S = ∂ 2 g ( S A , T c , p ) ∂ S A 2 = ∑ j = 0 6 ∑ k = 0 5 [ 0.5 S u S A g 1 j k s + ∑ i = 2 7 0.5 i ( 0.5 i − 1 ) S u 0.5 i g i j k s S A ( 0.5 i − 2 ) ] ( T c T u ) j ( p p u ) k (8)

where T_u = 40˚C, p_u = 10⁸ Pa, S_u = 40.188617 g/kg. References [19] [20] gave the values of the coefficients g 0 j k w and g i j k s , respectively. The expressions of h_d(n) and h_c(n) are [8]

h d ( n ) = ( n 2 − 1 ) [ 1 + 2 3 ( n 2 + 2 ) ( n 2 − 1 3 n ) 2 ] (9)

h c ( n ) = 2 n ∂ n ∂ S A (10)

Reference [8] gave the expression of n. The forward scattering coefficient and backscattering coefficient of pure seawater are equal, i.e., b_psf(λ) = b_psb(λ) = 0.5b_ps(λ) [10]. The pressure p of seawater is related to the depth d as [21]

d = g ( S A , T c , p ) − g ( S A , T c , 0 ) g S + 0.5 γ p (11)

where γ = 2.226 × 10⁻⁶ ms⁻²/db, g_S is the first order partial derivative of the seawater Gibbs energy g(S_A, T_c, p) with respect to the absolute salinity S_A.

g S = ∂ g ( S A , T c , p ) ∂ S A = ∑ j = 0 6 ∑ k = 0 5 { g 1 j k s S u ln ( S A S u ) 0.5 + 0.5 g 1 j k s S u + ∑ i = 2 7 0.5 i g i j k s S u ( S A S u ) 0.5 i − 1 }   × ( T c T u ) j ( p p u ) k (12)

The scattering coefficient of chlorophyll b_c(λ) is [11]

b c ( λ ) = 0.3 ( 550 λ ) C c 0.62 (13)

Reference [22] gave the probability of backscattering by chlorophyll b_pc. b_pc multiplied by b_c(λ) is the backscattering coefficient of chlorophyll b_cb(λ). b_c(λ) minus b_cb(λ) is the forward scattering coefficient of chlorophyll b_cf(λ).

The scattering coefficients of small particles b_s(λ) with density 2 g/m³ and large particles b_l(λ) with 1 g/m³ are [11] [23]

b s ( λ ) = b s 0 ( λ ) C s (14)

b l ( λ ) = b l 0 ( λ ) C l (15)

b s 0 ( λ ) and b s 0 ( λ ) are the spectral dependencies for scattering coefficients of small particles and large particles, respectively. And their values were given in reference [11]. C_s and C_l are the concentrations of small and large particles, and they can be expressed by C_c [11]. The probability of backscattering by small particles B_s = 0.039 and large particles B_l = 6.4 × 10⁻⁴ [11]. Therefore, the backscattering coefficients of small particles b_sb(λ) and large particles b_lb(λ) can be obtained by multiplying B_s by b_s(λ) and B_l by b_l(λ), respectively. b_s(λ) minus b_sb(λ) and b_l(λ) minus b_lb(λ) are the forward scattering coefficients of small particles b_sf(λ) and large particles b_lf(λ), respectively.

The forward scattering coefficient b_seaf(λ) and backscattering coefficient b_seab(λ) of seawater are

b s e a f ( λ ) = b p s f ( λ ) + b c f ( λ ) + b s f ( λ ) + b l f ( λ ) (16)

b s e a b ( λ ) = b p s b ( λ ) + b c b ( λ ) + b s b ( λ ) + b l b ( λ ) (17)

a_sea(λ) plus b_sea(λ) is the extinction coefficient of seawater c_sea(λ).

c s e a ( λ ) = a s e a ( λ ) + b s e a ( λ ) = a s e a ( λ ) + b s e a f ( λ ) + b s e a b ( λ ) (18)

Copernicus Marine Service provides the pressure, temperature and salinity values of seawater in most of the global sea area. In this paper, the characteristics of blue-green light during NLOS transmission at different depths in the 24.5˚N, 23.8˚W sea area are analyzed as an example. Assuming the chlorophyll concentration C_c = 0.25 mg/m³, Figure 1 shows the forward scattering coefficients and backscattering coefficients of 450 nm blue light and 520 nm green light by seawater in the depth range of 600 m to 700 m at 24.5˚N, 23.8˚W sea area.

Figure 1 shows the forward scattering coefficients and backscattering coefficients of seawater on blue-green light are in an overall decreasing trend with increasing depth, indicating that the scattering effect of seawater on blue-green light is decreasing with increasing depth. The forward scattering coefficients of seawater for blue-green light at the same depth is greater than the backscattering coefficients, indicating that the scattering of seawater for blue-green light tends to be forward scattering.

An important aspect of the transmission of blue-green light in seawater is the determination of the scattering angle of the scattered photons from seawater. In this paper, the scattering angle of the scattered photons from seawater is calculated using the HG phase function [24] [25].

χ β = ∫ 0 θ s P H G ( θ , g ) sin θ d θ = ∫ 0 θ s 1 − g 2 4 π ( 1 + g 2 − 2 g cos θ ) 3 2 sin θ d θ (19)

where the upper limit of integration θ_s is the scattering angle to be sought, θ is the scattering angle of the scattered photons from seawater, P_HG(θ, g) is the HG phase function, χ_β is a random number uniformly distributed on [0, π], and g is an asymmetry factor. g is related to the backscattering ratio b_p of the seawater [25].

b p = 1 − g 2 g ( 1 + g 1 + g 2 − 1 ) (20)

The backscattering ratio is defined as the ratio of the backscattering coefficient to the scattering coefficient.

b p = b s e a b ( λ ) b s e a ( λ ) (21)

Bringing Equations (20) and (21) into Equation (19) yields

cos θ s = 1 + g 2 − [ 1 − g 2 1 + g − 4 π g χ β ] 2 2 g (22)

The backscattering coefficients and scattering coefficients of seawater at different depths are different, so the backscattering ratio b_p and asymmetry factor g are different at different depths, and thus the scattering angles of scattered photons from seawater are different at different depths.

After determining the absorption coefficient, the scattering coefficients at different depths and the scattering angle of scattered photons from seawater, this paper will analyze the forward coverage range for blue-green light NLOS transmission to ensure that the receiver is within that range, making blue-green light NLOS transmission possible for communication.

There are two forms of blue-green light NLOS transmission in seawater as follows, named as class a form and class b form, as shown in Figure 2.

In this paper, r indicates the separation distance between the laser and the PD, i.e., the communication distance for blue-green light NLOS transmission, θ_t denotes the transmitting deflection angle of the laser, θ_r denotes the receiving deflection angle of the PD, φ_t denotes the beam divergence angle of the laser, and φ_r denotes the receiving field-of-view angle of the PD. In the form of class a, θ_t and θ_r cannot be 90˚ at the same time (we take 0˚ < θ_t < 90˚ and θ_r = 90˚), in class b form, 0˚ < θ_t < 90˚, 0˚ < θ_r < 90˚. The projection of the scattering coverage for a class a form of NLOS transmission is shown in Figure 3 [26].

In this paper, A denotes the laser, B denotes the PD, r₁ denotes the distance from the laser to the effective scatterer, and r₂ denotes the distance from the effective scatterer to the PD. In Figure 3(a), AO is the central axis of the laser beam and BK is the central axis of the received field of view of the PD. In Figure 3(b), the class a form forward scattering coverage is elliptical arc E M D F ⌢ , and for backscattering an approximate correction is made using an arc of radius B'D

of the short half-axis of the forward scattering ellipse arc. Take any point M on the elliptical arc E M D F ⌢ , let the coordinates of the point M be (x₀, y₀), the angle between AM and y-axis is α, and let the elliptical equation of the ellipse where the elliptical arc E M D F ⌢ is located be

x 2 a 2 + y 2 b 2 = 1 (23)

where the long semi-axis a and short semi-axis b of the ellipse are

a = E F 2 = B ′ F = tan ( φ t 2 ) r cos ( φ r 2 ) cos ( θ t + φ r 2 ) (24)

b = B ′ D = r tan ( φ t 2 ) cot θ t − tan ( φ r 2 ) (25)

Let the equation of the line AM be

y = k x + c (26)

where the slope k of the line and the intercept c on the y-axis are

k = − cot α (27)

c = − r cos θ t cos ( φ r 2 ) cos ( θ t + φ r 2 ) (28)

From the equation of the ellipse in which the ellipse arc E M D F ⌢ is located and the equation of the line in which the line AM is located, the coordinates of the point A are ( 0 , − r cos θ t cos ( φ r 2 ) cos ( θ t + φ r 2 ) ) . The horizontal coordinate x₀ of the point M is

x 0 = − r cos φ r 2 cos θ t cot α cos 2 φ r 2 ( cot θ t − tan φ r 2 ) 2 + r cos φ r 2 K 2 K 1 (29)

where

K 1 = cos 3 ( θ t + φ r 2 ) + cos ( θ t + φ r 2 ) cot 2 α cos 2 φ r 2 ( cot θ t − tan φ r 2 ) 2 (30)

K 2 = − cos 2 φ r 2 cos 2 θ t ( cot θ t − tan φ r 2 ) 2 cos 2 ( θ t + φ r 2 )     + tan 2 φ t 2 cos 4 ( θ t + φ r 2 ) + tan 2 φ t 2 cos 2 φ r 2 cos 2 ( θ t + φ r 2 )     × cot 2 α ( cot θ t − tan φ r 2 ) 2 (31)

The vertical coordinate y₀ of the point M is

y 0 = − x 0 cot α − r cos θ t cos φ r 2 cos ( θ t + φ r 2 ) (32)

where the maximum value of angle α is half of φ_t. From the equation of the distance between two points, we get

| A M | = x 0 2 + [ y 0 + r cos θ t cos ( φ r 2 ) cos ( θ t + φ r 2 ) ] 2 (33)

where |AM| is the forward scattering coverage of class a form of NLOS transmission, and the radius AP of the backscattering correction circle is equal to the short semi-axis B'D of the forward scattering ellipse, so the backscattering coverage of class a form is approximately B'D. From Equations (33) and (25), it can be seen that |AM| and B'D are independent of the scattering angle as well as the wavelength of light.

The projection of the scattering coverage for the class b form of NLOS transmission is shown in Figure 4 [26].

In Figure 4, φ₁ is the projection of the beam divergence angle of the laser on the horizontal plane, and φ₂ is the angle between BN and BT. As can be seen from Figure 4, the scattering coverage of the class b form increases the triangular coverage area on the basis of the class a form, at which time the backscattering is small and negligible.

For the form of class b, the coordinates of point A is ( 0 , − r cos θ t cos ( φ r 2 ) cos ( θ t + φ r 2 ) ) and the coordinates of point B is (0, r₂cosθ_r). In the class b form, the forward

scattering coverage is the distance from point A to the arc when φ₂ ≥ φ₁, however, when φ₂ ≤ φ₁, the forward scattering coverage becomes the distance from point A to the straight line BN.

The equation of the straight line BN is

x cot ( φ 2 4 ) − y + r sin θ t cos θ r sin θ s = 0 (34)

The equation for the distance from a point to a line gives

| A M | = | A x 0 + B y 0 + C | A 2 + B 2 = | sin ( φ 2 2 ) | | r cos θ t cos ( φ r 2 ) cos ( θ t + φ r 2 ) + r sin θ t cos θ r sin θ s | (35)

Therefore, for φ₂ ≤ φ₁, the forward scattering coverage of the class b form is |AM| in Figure 4. From Equation (35), it can be seen that the forward scattering coverage of the class b form is related to the scattering angle θ_s. The following is an analysis of the received optical power and BER of the PD when blue-green light NLOS transmission in class a form and class b form.

According to Figure 2(a) we can get the single scattering link model when blue-green light NLOS transmission in class a form, as shown in Figure 5.

The volume V_a of the effective scatterer of class a form can be approximated as the volume of the large cone (cone of height h₁) minus the volume of the small cone (cone of height h₂). So, the expression of V_a is [26]

V a = π 3 r 3 tan 2 ( φ r 2 ) [ tan 3 ( θ t + φ t 2 ) − tan 3 ( θ t − φ t 2 ) ] (36)

According to reference [27], we can obtain the received optical power of the PD P_ra in class a form as

P r a = P T b s e a ( λ ) P H G ( θ , g ) cos ξ exp [ − c s e a ( λ ) ( r 1 + r 2 ) ] A r V a Ω t r 1 2 r 2 2 (37)

where P_T is the emitted optical power of the laser, A_r is the area of the receiving antenna, Ω_t is the stereo angle of the emitted beam of the laser, and ξ is the angle between the transmitting light cone and the receiving light cone. We set ξ = 0˚, i.e., the emitting light cone is coplanar with the receiving light cone. The expressions for r₁ and r₂ are easily obtained from the geometric relations of the triangle. The expression of Ω_t is [27]

Ω t = 2 π ( 1 − cos φ t 2 ) (38)

From Equation (37), the received optical power of the PD is related to P_HG(θ,g), b_sea(λ) and c_sea(λ), while P_HG(θ,g), b_sea(λ) and c_sea(λ) are all related to the depth of seawater. The depth of the effective scatterer in seawater h_as is obtained as the depth of the laser in seawater h minus the distance h_s (h_s = r₂ in the class a form) of the effective scatterer from the LOS link between the laser and the PD.

h a s = h − h s = h − r 2 = h − r sin θ t sin ( π − θ t − θ r ) (39)

From Figure 2(b), the single scattering link model of NLOS transmission in class b form is shown in Figure 6.

The volumes of the effective scatterers V A 1 B 1 C 1 D 1 and V E 1 F 1 G 1 H 1 are

V A 1 B 1 C 1 D 1 = π 3 tan 2 ( φ t 2 ) { [ r 1 + r 2 tan φ r 2 ( tan θ s + tan φ r 2 ) cos θ s ] 3 − [ r 1 − r 2 sin φ r 2 sin ( θ s − φ r 2 ) ] 3 } (40)

V E 1 F 1 G 1 H 1 = π 3 tan 2 ( φ r 2 ) { [ r 2 + r 1 tan φ t 2 ( tan θ s + tan φ t 2 ) cos θ s ] 3 − [ r 2 − r 1 sin φ t 2 sin ( θ s − φ t 2 ) ] 3 } (41)

The volume V_b of the effective scatterer in the class b form is the smallest of V A 1 B 1 C 1 D 1 and V E 1 F 1 G 1 H 1 , i.e., V b = min ( V A 1 B 1 C 1 D 1 , V E 1 F 1 G 1 H 1 ) .

By replacing V_a with V_b in Equation (37), the received optical power of the PD P_rb can be obtained when the blue-green light NLOS transmission in class b form. The depth of the effective scatterer in seawater in class b form h_bs is h minus h_s.

h b s = h − h s = h − r sin θ r sin θ t sin ( π − θ t − θ r ) (42)

Assuming that a PIN PD is used at the receiver, the noise current i n 2 of the PIN PD mainly consists of thermal noise i t 2 , scattering noise i s h 2 and dark current noise i d a r k 2 [28].

i n 2 = i t 2 + i s h 2 + i d a r k 2 = 4 k B T PIN B R + 2 e I s B + 2 e I D B (43)

where B is the bandwidth of the PD, assuming that the bandwidth of the PD is two times the information transmission rate, that is, B = 2R_b, R is the load resistance, according to engineering experience R = 50 Ω, T_PIN is the absolute temperature of the PIN PD work, e is the electronic charge, I_s is the signal light reaches the PD generated by the light current and I_D is the dark current.

I s = e η h v P s (44)

where η is the quantum efficiency, h is Planck’s constant, v is the frequency of light and P_s is the received optical power of the PD. In the form of class a, P_s = P_ra, and in the form of class b, P_s = P_rb. The output SNR of the PD is

S N R = e 2 η 2 P s 2 h 2 v 2 i n 2 (45)

The blue-green light is modulated by the external modulator as on-off keying (OOK) signal, and the BER at the receiver of the system is

P e = 1 2 erfc ( S N R 4 ) (46)

After obtaining the forward scattering coverage of blue-green light under different forms of NLOS transmission, we will set the simulation parameters of transmitting deflection angle, beam divergence angle, receiving deflection angle and receiving field of view angle to analyze the characteristics of blue-green light during NLOS transmission in the forms of class a and class b in seawater of different depths.

The simulation results of the received optical power and BER of the PD for 450 nm blue light and 520 nm green light NLOS transmission in class a form are shown in Figure 8. In the simulation, φ_t = 30˚, φ_r = 110˚ and α = 10˚. Before the simulation, it is verified that the position of the PD is located within the forward scattering range of the laser according to Equation (33).

Figure 7 shows the forward scattering coverage of the laser increases with the increase of the emission deflection angle. The location of the PD under the given simulation conditions is always within the forward scattering coverage of the laser, i.e., the PD is able to receive the optical signal.

Figure 8(a) and Figure 8(b) show the received optical power of the PD is in the overall decreasing trend with the increase of the emission deflection angle, and the decreasing trend is similar to the exponential form, so reducing the emission deflection angle of the laser can effectively increase the received optical power of the system. The received optical power of blue-green light increases with the increase of seawater depth for the same emission deflection angle. This is because the deeper the seawater, the smaller the scattering coefficient of seawater, the less scattering effect on blue-green light. With the same emission deflection angle, the received optical power of 520 nm green light is greater than that of 450 nm blue light. Figure 8(c) shows the BER at the receiver for different information transmission rates when the transmitting deflection angle of the laser is 5˚. The BER at the receiver side decreases with the increase of seawater depth for the same information transmission rate. With the BER of less than 10⁻⁶, blue-green light can achieve an information transmission rate of more than 100 Mbps with NLOS transmission in class a form at the depth of 690 m.

The simulation results of the received optical power and BER of the PD for blue-green light NLOS transmission in the form of class b are shown in Figure 10, taking θ_t = 37˚, φ_t = 30˚and φ_r = 110˚. The location of the PD is verified to be within the forward scattering range of the laser according to Equation (35) before the simulation.

Figure 9 shows the forward scattering coverage of the laser is in an overall increasing trend with the increase of the receiving deflection angle. The location of the PD under the given simulation conditions is always within the forward scattering coverage of the laser.

Figure 10(a) and Figure 10(b) show the received optical power of the PD with the increase of the received deflection angle is in the overall decreasing trend, the same decreasing trend is similar to the exponential form, so reduce

the received deflection angle of the PD can effectively increase the received optical power of the system. The same as class a form, when blue-green light NLOS transmission in class b form, the received optical power of blue-green light increases with the depth of seawater under the same receiving deflection angle, and the received optical power of 520 nm green light is greater than that of 450 nm blue light. Figure 10(c) shows the BER at the receiver end for different information transmission rates with a receive deflection angle of 15˚ for blue light and 50˚ for green light. Also with BER less than 10⁻⁶, blue-green light can achieve an information transmission rate of more than 100 Mbps with NLOS transmission in class b form at the depth of 690 m.