Defocus Study of a Novel Optical Antenna Illuminated by a Radial Radiation Fiber Laser
 Author: Jiang Ping, Yang Huajun, Xie Kang, Yu Mingyin, Mao Shengqian
 Organization: Jiang Ping; Yang Huajun; Xie Kang; Yu Mingyin; Mao Shengqian
 Publish: Journal of the Optical Society of Korea Volume 18, Issue5, p485~494, 25 Oct 2014

ABSTRACT
A novel antenna with ellipsoidparaboloid surfaces configuration is designed for matching the incident radial radiation fiber laser distribution for maximum transmission efficiency. The onaxial and offaxial defocus effects on the optical antenna system, resulting in energy loss, are analyzed in detail. Knowledge of the effects of those defocuses on beam divergence, aberration and antenna transmission efficiency is of great importance to the long range communication systems.

KEYWORD
Radial radiation fiber laser , Ellipsoidparaboloid surfaces , Onaxial defocus , Offaxial defocus , Receiving efficiency

I. INTRODUCTION
Laser communications and radar systems employ conventional telescopes as optical antennas. These telescopes, such as Cassegrain telescopes, are always illuminated by an onaxial 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 onaxial transmission, energy loss caused by the secondary mirror center reflection will greatly depress the emission efficiency in the optical communication system [13].
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, biosensing and optical communication systems[6, 7]. In April 2012, an important development in this area: a radial radiation laser using a hollowcore Bragg fiber in combination with organic dyedoped water plugs placed inside the fiber core, was reported in Nature Photonics[8]. This kind of fiber laser is onaxial 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 ellipsoidparaboloid mirror surfaces is constructed, which is assumed to be illuminated by a radial radiation fiber laser (as shown in Fig. 2(a)), producing stable, onaxially symmetric ringlike 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 onaxial propagation optical antenna. The investigation results will offer fundamental research for the collimation accuracy and propagation efficiency enhancement of the free space optical communication.
II. DESIGN OF THE ELLIPSOIDPARABOLOID SURFACES ANTENNA
Figure 3 shows the twodimensional 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 thex axis direction. The radial radiation that surrounds the fiber onaxially symmetric leads to an azimuthally divergent angle labeledθ in thexy plane. By using the geometrical optical theory, here we directly image the laser light at thexy 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 (x _{0},y _{0}), wherex _{0}=c ,y _{0}=0, ,a andb represent the lengths of long halfaxis and short halfaxis of the ellipse, respectively. The equations of emitting light and the ellipse reflector as shown belowwhere 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 (x _{1},y _{1}) can be obtained as tanThe slope angle of reflecting light at point
B (x _{1},y _{1}) can be written as .The twodimensional structure of the ellipsoidparaboloid 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 , pointp (x_{p} ,y_{p} ) is the apex of the parabola. SolutionD (x _{3},y _{3}) 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 pointD (x _{3},y _{3}) can be obtained asThe divergence angle of the outer edge reflecting light below the
x axis (labeled 1 in Fig. 4) isThe divergence angle of the outer edge reflecting light above the
x axis (labeled 1′ in Fig. 4) can be written asThe 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 thex axis, otherwiseβ ＜0 andγ ＞0 means that light diverges from thex axis.III. SIMULATION RESULTS OF EMITTING ANTENNA WITH ELLIPSOIDPARABOLOID SURFACE CONFIGURATION CONFOCAL SIMULATION OF ELLIPSOIDPARABOLOID SURFACES
Suppose the focus of the parabola overlaps with the left focus of the ellipse, it means that the coordinate of the parabola apex is . Twodimensional 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.
3.1. Onaxial and Offaxial Defocus Analysis of Ellipsoid Paraboloid Surfaces
Some mechanical assembly errors could cause the onaxial defocus of the ellipsoidparaboloid mirror surfaces. And some thermal expansion damages may cause offaxial defocus of the two reflectors [10].
Suppose the amount of onaxial defocusing is
s , and the offaxial defocusing amount ist , 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 onaxial 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 thex axis, otherwiseβ ＜0 andγ ＞0 mean that light diverges away from thex axis. It can be inferred from Fig. 6 that the emitting light from the emitting antenna is convergent when the onaxial defocusing amount is negative (as shown in Fig. 6(a) and in the left half of Fig. 6(c)) and is divergence when the onaxial 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 twodimensional ray tracing simulation results of an antenna with two different offaxial defocusing amounts. The emitting light tilts downward when the offaxial defocusing amount is
t = −17 cm (as shown in Fig. 7(a)) and tilts upward when the offaxial defocusing amount ist =17 cm (as shown in Fig. 7(b)). In this time,β andγ 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 offaxial defocusing
t is between −10 cm and 10 cm, curves of deviation angles vs. offaxial defocusing almost overlapped, from which it can be inferred that the tilted emitting light is still collimated when the offaxial defocusingt 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 onaxial and offaxial defocusing amounts.The curve for the onaxial defocusing presented by the solid line in Fig. 8(a), when the onaxial 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 onaxial defocusing. If the onaxial 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 offaxial defocus case, the emitting light tilts downward when the offaxial defocusing amount
t is negative and tilts upward when thet is positive, the secondary mirror blocking always exists. With increase of the absolute value of offaxial defocusing, emission efficiency is gradually reduced. When the secondary mirror blocking is bypassed, the emission efficiency gradually increases once again. Curve of negative offaxial defocusing amounts vs. emission efficiency (as shown by circle line in Fig. 8(a)) is symmetrically distributed compare with the curve of positive offaxial defocusing amounts (as shown by circle line in Fig. 8(b)). When the onaxial and offaxial defocusing amounts are −28 cm and 17.5 cm, respectively, the emission efficiency reduced to 80%. When the offaxial 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 highprecision 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 onaxial defocusing amounts, i.e.
s ＜0, emitting light is convergent, the inner boundary of the light is blocked by the secondary mirror. Whens =28 cm (as shown in Fig. 9(b)), the emission efficiency approaches to 80％ (corresponding to point B in Fig. 8(a)). Whens =−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 oft ＜0, light tilts downward, but the collimation degree is almost unchanged, serious energy loss is caused by the secondary mirror blocking. Whent =−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)). Whent =−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. Whent =−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 thex axis by 5.3 degrees (as shown in Fig. 7(c)), long range optical communication cannot be realized. The spot diagrams of negative offaxial defocusing amounts are symmetrically distributed as those of the positive offaxial defocusing amounts.By introducing of an berrationfree 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 aberrationfree system. Obviously, convergent or divergent output beam caused by onaxial defocusing may shows spherical aberration at the Gaussian plane. For the offaxial defocusing case, small offaxial defocusing amount cause coma aberration (as shown in Fig. 10(b)), too large offaxial 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.
IV. SIMULATION RESULTS OF 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 andϕ =30 mm，respectively. The obscuration ratio isϕ /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), whens =0 mm, a precision collimated laser beam is completely received by the effective receiving plane. In Fig. 12(c), whens =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 onaxial defocusing (i.e.s ＞0) has greater influence on the receiving efficiency than the negative onaxial defocusing. From Fig. 12(d) to Fig. 12(f), the offaxial 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 distanceL with different defocusing amounts could be obtained, as shown in Fig. 13.It can be inferred from Fig. 13(a), that when the onaxial defocusing is
s =0.1 mm, the maximum transmission distance achievesL =5000 m and the receiving antenna efficiency is reduced to 80％. From Fig. 13(b), when offaxial defocusing ist =0.1 mm, the maximum transmission distance achievesL =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 onaxial defocusing and offaxial 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 offaxial defocusing on the receiving efficiency is greater than that of the onaxial defocusing. Curves of offaxial 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 offaxial defocusing in range of −.05 mm ＜t ＜ 0.05 mm have the same distribution. When the offaxial 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.V. CONCLUSION
A novel antenna with ellipsoidparaboloid 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 offaxial defocusing between the primary and secondary mirrors of emitting antenna on the receiving efficiency is greater than that of the onaxial defocusing. When the absolute value of offaxial 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 offaxial defocusing is less than 0.01 mm, the receiving efficiency is higher than 80％, long range optical communication could be guaranteed.

[FIG. 1.] Model of Cassegrain antenna and energy distributions. (a) The schematic illustration of central reflection caused by the secondary mirror, (b) energy distribution of the input Gaussian beam, (c) energy distribution of the truncated Gaussian beam emits from antenna.

[Fig. 2.] Schematic illustration of novel optical emitting antenna illuminated by a radial radiation fiber laser. (a) Configuration of emitting antenna with ellipsoidparaboloid mirror surfaces and (b) the cross section of the emitting laser beam.

[FIG. 3.] Twodimensional structure of the ellipsoid mirror.

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[FIG. 4.] Twodimensional structure of the ellipsoidparaboloid mirrors.

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[FIG. 5.] Simulation results of the confocal optical emitting antenna, the coordinate of the parabola apex is , p=375 cm, a=75 cm, b=50 cm, and the azimuthal angle of the edge light emitted from the radial radiation fiber laser is θ=30°. (a) Twodimensional ray tracing of the confocal optical emitting antenna. (b) The curves of the divergence angles of inner and outer edge laser beam emitted from confocal optical emitting antenna change with radial radiation azimuthal angle θ, the observation plane is at position of x=100 cm.

[FIG. 6.] Twodimensional ray tracing simulation results and curves of the divergence angles of the edge emitting ray from emitting antenna models vs. different onaxial defocusing amounts. The observation plane is at position of x = 100 cm. (a) Twodimensional ray tracing simulation result of onaxial defocusing amount s = ？25 cm and (b) onaxial defocusing amount s = 25 cm, (c) the curves of the divergence angles of the edge emitting light vs. onaxial defocusing amount s changes from ？50 cm to 50 cm.

[FIG. 7.] Twodimensional ray tracing simulation results and curves of the deviation angles of the edge emitting light of emitting antenna models vs. different offaxial defocusing amounts. (a) Twodimensional ray tracing simulation result of offaxial defocusing amount t = ？17 cm and (b) offaxial defocusing amount t = 17 cm, (c) the curves of the deviation angles of the edge emitting light vs. offaxial defocusing amount t change from ？50 cm to 50 cm.

[Fig. 8.] Curves for defocusing amounts vs. emission efficiency of emitting antenna. (a) Curves of negative defocusing amounts vs. emission efficiency. (b) Curves of positive defocusing amounts vs. emission efficiency.

[FIG. 9.] Spot diagrams with different defocusing amounts. (a) Onaxial defocusing amount is s = 0, (b) s = ？28 cm, (c) s = ？50 cm. (d) Offaxial defocusing amount is t = ？17.5 cm, (e) t = ？28 cm, (f) t = ？50 cm.

[FIG. 10.] Simulation results of emitting antenna aberration. (a) Spot diagram at the Gaussian plane for the confocal case and (b) coma aberration of emitting antenna with offaxial defocusing amount is t = 1 cm, (c) coma and astigmatism aberrations of emitting antenna with offaxial defocusing amount is t = 17 cm.

[FIG. 11.] Schematic diagram of Cassegrain antenna and effective receiving plane. (a) Structure of receiving antenna and (b) effective receiving plane.

[FIG. 12.] Spot diagram on the effective receiving plane at x = 5000 m with different defocusing amounts, (a) s = ？0. 5 mm (b) s = 0 mm, (c) s = 0.2 mm, (d) t = 0.1 mm, (e) t = 0.2 mm, (f) t = 0.3 mm.

[Fig. 13] Relationship between receiving efficiency and transmission distance L. (a) Curves of receiving efficiency vs. transmission distance L with different onaxial defocusing s, (b) curves of receiving efficiency vs. transmission distance L with different offaxial defocusing t.

[FIG. 14.] Curves of offaxial defocusing t vs. receiving efficiency at different transmission distance L in long range optical communication, (a) threedimensional distribution (b) twodimensional distribution.