Laser communications and radar systems employ conventional telescopes as optical antennas. These telescopes, such as Cassegrain telescopes, are always illuminated by an on-axial emission semiconductor laser and frequently have reflection caused by the secondary mirror center is shown in Fig. 1(a). Intensity patterns of an input Gaussian laser beam and truncated Gaussian beam emitted from the antenna are given in Fig. 1(b) and (c), respectively. Because of the on-axial transmission, energy loss caused by the secondary mirror center reflection will greatly depress the emission efficiency in the optical communication system [1-3].

In recent years, fiber lasers attract very much attention [4, 5]. They are capable of producing stable,high power, allow design freedom, and produce good coherence radiation. They may prove to be an important new light source for applications in medical imaging, sensing, bio-sensing and optical communication systems[6, 7]. In April 2012, an important development in this area: a radial radiation laser using a hollow-core Bragg fiber in combination with organic dye-doped water plugs placed inside the fiber core, was reported in Nature Photonics[8]. This kind of fiber laser is on-axial pumped, resulting in a unique radiating field pattern characterized by cylindrical symmetry and a fixed polarization pointed in the azimuthal direction[9]. This new capability may prove to be an important new light source for applications in space optical communication systems.

In this paper, a novel optical antenna with ellipsoid-paraboloid mirror surfaces is constructed, which is assumed to be illuminated by a radial radiation fiber laser (as shown in Fig. 2(a)), producing stable, on-axially symmetric ring-like radiation (as shown in Fig. 2(b)).

By geometrical optical theory, the design details are proposed and the resultant curves that display the energy loss of the communication system due to defocus between primary and secondary mirrors of emitting antenna are presented. This new type of antenna illuminated by a radial radiation fiber laser, which produces a hollow beam of high precision collimation, may effectively avoid the energy loss caused by the secondary mirror central reflection in the traditional on-axial propagation optical antenna. The investigation results will offer fundamental research for the collimation accuracy and propagation efficiency enhancement of the free space optical communication.

Figure 3 shows the two-dimensional structure model of the ellipsoid mirror surface in the xy plane, which performs as a secondary mirror, and is illuminated by a radial radiation fiber laser. The center of the ellipse is at origin point O(0,0). The transmission axis of the pump light, which is along the fiber core, points in the x-axis direction. The radial radiation that surrounds the fiber on-axially symmetric leads to an azimuthally divergent angle labeled θ in the xy plane. By using the geometrical optical theory, here we directly image the laser light at the xy plane.

The radial emitting region of the fiber laser, which can be regarded as a point course, is located at the right side focus of the ellipse, its coordinate is A(x₀, y₀), where x₀=c, y₀=0, , a and b represent the lengths of long half-axis and short half-axis of the ellipse, respectively. The equations of emitting light and the ellipse reflector as shown below

where is the slope of the emitting light .

Two sets of solutions , derived by solving Eq. (1), represent two intersections of emitting light and the ellipsoid mirror surface.

By deriving the ellipse equation the slope of the tangent at point B(x₁, y₁) can be obtained as tan

The slope angle of reflecting light at point B(x₁, y₁) can be written as .

The two-dimensional structure of the ellipsoid-paraboloid mirror surfaces is shown in Fig. 4. The equations of reflecting light and the paraboloid mirror, which performs as a primary mirror of the emitting antenna, is shown below

where k"|_B(x1,y1) k"| tanα_B is the slope of the reflecting light , point p(x_p,y_p) is the apex of the parabola. Solution D(x₃, y₃) derived by solving the Eq. (2) represents the intersection coordinate of reflected light and the paraboloid mirror. By deriving the parabola equation the slope of the tangent at point D(x₃, y₃) can be obtained as

The divergence angle of the outer edge reflecting light below the x axis (labeled 1 in Fig. 4) is

The divergence angle of the outer edge reflecting light above the x axis (labeled 1′ in Fig. 4) can be written as

The divergence angles of inner edge reflecting light below and above the x axis (labeled 2 and 2′ respectively in Fig. 4) can be obtained by the same method depicted above.

In any case, when β＞0 and, γ＞0 then light converges, toward the x axis, otherwise β＜0 and γ＞0 means that light diverges from the x axis.

Suppose the focus of the parabola overlaps with the left focus of the ellipse, it means that the coordinate of the parabola apex is . Two-dimensional ray tracing simulation results, and the curves of the divergence angles of the edge emitting light are shown in Fig. 5.

In Fig. 5(a), color of lines from red to blue represent energy changes from high to low level. Fig. 5(b) shows that the divergence angles of inner and outer edge laser beam overlapped each other and all approach to 0 degrees, which means a precision collimated laser beam, which hopefully approaches the diffraction limit, will be produced by the confocal optical emitting antenna. This kind of novel antenna could greatly enhance the emission efficiency and collimation accuracy of optical antenna system for long range optical communication.

Some mechanical assembly errors could cause the on-axial defocus of the ellipsoid-paraboloid mirror surfaces. And some thermal expansion damages may cause off-axial defocus of the two reflectors [10].

Suppose the amount of on-axial defocusing is s, and the off-axial defocusing amount is t, the coordinate of the parabola apex can be written as . Twodimensional ray tracing simulation results and the curves of the divergence angles of the edge emitting light with change in on-axial defocusing amounts are shown in Fig. 6.

It has been presented in part II that β＞0 and, γ＞0 mean that light converges toward the x axis, otherwise β＜0 and γ＞0 mean that light diverges away from the x axis. It can be inferred from Fig. 6 that the emitting light from the emitting antenna is convergent when the on-axial defocusing amount is negative (as shown in Fig. 6(a) and in the left half of Fig. 6(c)) and is divergence when the on-axial defocusing amount is positive (as shown in Fig. 6(b) and in the right half of Fig. 6(c)).

Figure 7(a) and Fig. 7(b) shows two-dimensional ray tracing simulation results of an antenna with two different off-axial defocusing amounts. The emitting light tilts downward when the off-axial defocusing amount is t = −17 cm (as shown in Fig. 7(a)) and tilts upward when the off-axial defocusing amount is t=17 cm (as shown in Fig. 7(b)). In this time, β and γ are regarded as the deviation angles between the tilted emitting light and the x axis, as shown in Fig. 7(c).

It can be seen from Fig. 7(c) that when the off-axial defocusing t is between −10 cm and 10 cm, curves of deviation angles vs. off-axial defocusing almost overlapped, from which it can be inferred that the tilted emitting light is still collimated when the off-axial defocusing t is in this range. It also can be inferred from Fig. 6 and Fig. 7 that, light blocking by the secondary mirror occurs in the defocused antenna, which causes energy loss of the emitting antenna system. Figure 8 shows the curves of emission efficiency of emitting antenna vs. different on-axial and off-axial defocusing amounts.

The curve for the on-axial defocusing presented by the solid line in Fig. 8(a), when the on-axial defocusing amount s is negative, the larger absolute value of on -axial defocusing is, the greater divergent emitting light there is. Because of the secondary mirror blocking, the emission efficiency of antenna monotonically decreases with increase of the absolute value of the on-axial defocusing. If the on-axial defocusing ispositive, no blocking of the secondary mirror on the divergent emitting light, emission efficiency remains 100％, as shown by solid line in Fig. 8(b).

For the off-axial defocus case, the emitting light tilts downward when the off-axial defocusing amount t is negative and tilts upward when the t is positive, the secondary mirror blocking always exists. With increase of the absolute value of off-axial defocusing, emission efficiency is gradually reduced. When the secondary mirror blocking is bypassed, the emission efficiency gradually increases once again. Curve of negative off-axial defocusing amounts vs. emission efficiency (as shown by circle line in Fig. 8(a)) is symmetrically distributed compare with the curve of positive off-axial defocusing amounts (as shown by circle line in Fig. 8(b)). When the on-axial and off-axial defocusing amounts are −28 cm and -17.5 cm, respectively, the emission efficiency reduced to 80%. When the off-axial defocusing amount is −28 cm, the emission efficiency reduced to minimum 75%.

Figure 9 shows spot diagrams with different defocusing amounts, obtained at the plane positioned at x =100 cm.

When the confocal condition is satisfied, transmitting light is a high-precision collimated hollow beam, and just in time to avoid the secondary mirror blocking (as shown in Fig. 9(a)). The emission efficiency is 100％ (corresponding to point A in Fig. 8(a)). For the case of negative on-axial defocusing amounts, i.e. s＜0, emitting light is convergent, the inner boundary of the light is blocked by the secondary mirror. When s=-28 cm (as shown in Fig. 9(b)), the emission efficiency approaches to 80％ (corresponding to point B in Fig. 8(a)). When s=−50 cm(as shown in Fig. 9(c), the emission efficiency reduces to 60％ (corresponding to point C in Fig. 8(a)). For the case of t＜0, light tilts downward, but the collimation degree is almost unchanged, serious energy loss is caused by the secondary mirror blocking. When t =−17.5 cm (as shown in Fig. 9(d)), the secondary mirror blocking leads to emission efficiency reduction approaching 80％ (corresponding to point D in Fig. 8(a)). When t =−28 cm (as shown in Fig. 9(e)), emission efficiency reduces to 75％ (corresponding to point E in Fig. 8(a)). Afterwards, the area of the secondary mirror blocking is reduced, and the emission efficiency increases. When t=−50 cm (as shown in Fig. 9(f)), emission efficiency approaches 82％ (corresponding to point F in Fig. 8(a)), but the beam has deviated from the x axis by 5.3 degrees (as shown in Fig. 7(c)), long range optical communication cannot be realized. The spot diagrams of negative off-axial defocusing amounts are symmetrically distributed as those of the positive off-axial defocusing amounts.

By introducing of an berration-free ideal converging lens at the exit pupil of the emitting antenna, optical aberration can be exactly visible at its image plane. Figure 10 shows spot diagrams obtained at the Gaussian plane of the ideal converging lens.

For the confocal case, spot diameter at the Gaussian plane of the ideal converging lens is approximately 0.4×10⁻¹² (as shown in Fig. 10(a)), the confocal antenna can be regarded as an aberration-free system. Obviously, convergent or divergent output beam caused by on-axial defocusing may shows spherical aberration at the Gaussian plane. For the off-axial defocusing case, small off-axial defocusing amount cause coma aberration (as shown in Fig. 10(b)), too large off-axial defocusing amounts may terminate coma and astigmatism aberrations at same time (as shown in Fig. 10(c)). Thus the smaller the amount of defocusing is, the higher the optical quality of the antenna system is. According to the Strehl criterion, when the receiving efficiency is greater than 80％, the imaging quality of the optical system can be regarded as perfect.

In order to further study the light transmission efficiency in the whole optical communication system, we must not only consider the impact of defocusing amounts and aberrations, but also consider the impact of the transmission distance on the receiving efficiency of the receiving antenna.

A receiving antenna always uses a typical Cassegrain telescope as shown in Fig. 11(a). The apertures of the primary mirror and secondary mirror are Φ=150 mm and ϕ=30 mm，respectively. The obscuration ratio is ϕ/Φ=0.2. The effective receiving plane can be considered as a concentric ring with outer and inner radii of 150 mm and 30 mm, respectively, as shown in Fig. 11(b).

Figure 12 shows the spot diagrams on the effective receiving plane, which is a distances of 5000 m from the emitting antenna.

In Fig. 12(a), when s =−0.5 mm, the spot diagram at the effective receiving plane is concentrated near the inner ring edge. The inner part of hollow beam energy is blocked by the secondary mirror of the receiving antenna. In Fig. 12(b), when s=0 mm, a precision collimated laser beam is completely received by the effective receiving plane. In Fig. 12(c), when s =0.2 mm, the spot diagram at the receiving plane is near the outer ring edge, the outer edge of the hollow beam failed to enter the effective receiving plane. It can be seen that, positive on-axial defocusing (i.e. s＞0) has greater influence on the receiving efficiency than the negative on-axial defocusing. From Fig. 12(d) to Fig. 12(f), the off-axial defocusing t changes from 0.1 mm to 0.5 mm, the spot diagram at the effective receiving plane moves upward, part of the light is blocked by the secondary mirror of the receiving antenna. By integral calculation of spot energy at the effective receiving plane, and comparison with emitting energy of confocal emitting antenna, the relationship between energy loss of the optical antenna system and transmission distance L with different defocusing amounts could be obtained, as shown in Fig. 13.

It can be inferred from Fig. 13(a), that when the on-axial defocusing is s =0.1 mm, the maximum transmission distance achieves L=5000 m and the receiving antenna efficiency is reduced to 80％. From Fig. 13(b), when off-axial defocusing is t =0.1 mm, the maximum transmission distance achieves L=4680 m and the receiving efficiency is reducesd to 80％. According to the Strehl criterion, to realize long range optical communication (L＞5000 m), the on-axial defocusing and off-axial defocusing must be less than 0.1 mm, which could be caused by thermal expansion damages or mechanical assembly errors of the emitting antenna.

It can be also inferred that curves shown in Fig. 13(a) present linear distribution, and curves shown in Fig. 13(b) present nonlinear distribution, which means the effects of off-axial defocusing on the receiving efficiency is greater than that of the on-axial defocusing. Curves of off-axial defocusing t vs. receiving efficiency at different transmission distances are shown in Fig. 14.

It can be inferred from Fig.14 that for long transmission distance L＞100 km, curves of off-axial defocusing in range of −.05 mm ＜ t ＜ 0.05 mm have the same distribution. When the off-axial defocusing is in the range of −0.01 mm ＜ t ＜ 0.01 mm, the receiving efficiency is higher than 80％, long range optical communication could be guaranteed.

A novel antenna with ellipsoid-paraboloid surfaces configuration is designed for matching the incident radial radiation fiber laser distribution for maximum transmission efficiency. It can be inferred from simulation results that the effects of off-axial defocusing between the primary and secondary mirrors of emitting antenna on the receiving efficiency is greater than that of the on-axial defocusing. When the absolute value of off-axial defocusing is less than 0.05 mm, curves of receiving efficiency for long transmission distance L＞100 km have the same distribution. When the absolute value of off-axial defocusing is less than 0.01 mm, the receiving efficiency is higher than 80％, long range optical communication could be guaranteed.