The deflection amplification accelerating, different from that mentioned above and carried out by a curved surface mirror, is a new optical accelerating technique in which the small deflecting angle is to be amplified, continuously.
The key point of the accelerating is to mainly make the incident light-beam with tiny deflecting angles or tiny displacements sequentially be projected on different locations of a spatially curved surface to form different incident angles during the deflecting period. Based upon the law of reflection, the reflective angle and its incident angle are equal and continuously changing. The change range of the reflective angles is limited by the incident light-beam locations on the curved surface and the curvature of the curved surface which can both be designed according to requirements. Therefore, both scanning of the incident light-beam and amplifying deflection angles can be performed. The optical accelerating factor β is related to structural parameters, such as the relative location of the system configured for generating the original tiny deflections and displacements, and the curvatures of the curved surface.
As a rule, in this situation, the optical accelerating equation can be expressed by
{Ω=ωdγ/dφ
β=dγ/dφ(4)
where, as shown in Fig.3, φ is an original tiny deflection angle, γ is formed by the outgoing beam with the incident beam, Ω is the accelerated angular velocity and ω is the original angular velocity generated by an electro-optical deflector. Here, the optical accelerating factor β is around 2 to 200 usually. The optical information capacity of this kind of imaging technique which is now with advantage in the nanosecond and sub-nanosecond imaging field, can be increased by 1 to 2 orders of magnitude.
Fig.3 Schematic diagram of optical deflecting amplification accelerating
图3 光学偏转加速示意图
As shown in Fig.3, r is a radius of the cylindrical reflector and its curvature is ρ=1/r. Its normal crosses x-axis at point C. An incident angle at an incident point T is θ, thus its deflection angle relative to an incident optical axis is γ=2θ+φ, and θ=φ+α. Now the optical accelerating factor β can be deduced as follows
β=dγ/dφ=2((L+r)/r)(cos φ)/(cos θ)-1(5)
As shown, β is not a constant from Eq.(5)for this reflective mirror. When the original deflection angle φ is tiny,α and θ are small enough, β can be approximately written by
β=dγ/dφ=2ρL+1(6)
where L is a distance between an original deflection point and a vertex of the cylindrical-surface reflective mirror. As it is seen from Eq.(6), the optical accelerating factor is determined by ρ and L, and β is a constant in this supposed condition.
Now we can further study the optical accelerating factor β. As it is shown, β is not a constant from Eq.(5)for the cylindrical reflector. In general, if the reflective curved mirror is some transcendental curved cylindrical-surface, described by the following differential Eq.(7), the optical accelerating factor could be a constant:
(dx)/(dy)=tan(((β-1)/2)tan-1(y/x))(7)
where the initial condition is(L,0). And as tan-1(y/x)～y/x, then Eq.(7)can be deduced to
(dx)/(dy)=y/x((β-1))/2(8)
The analytic solution of Eq.(8)is
x2-y2(β-1)/2=L2(9)
As shown in Fig.4, the blue curve for the analytic solution of Eq.(9)is in very good agreement the black curve for the numeric solution of Eq.(7)near y=0 when β=50.
Fig.4 The analytic solution of Eq.(9)and the numeric solution of Eq.(7)(x-axis normalized by length L)
图4 方程(9)的解析解和方程(7)的数值解(横坐标轴长度L已归一化处理)
According to Eq.(6), the optical accelerating factor β can be increased more if ρ increases. However, when ρ increases, the radius of the curved-surface reflective mirror will be decreased. Thus the curced-surface mirror is difficult to process and apply. Generally speaking, the light-beam deflection angle or the deflection speed can be improved by one to two orders of magnitude. In addition, no mechanical moving device is found in this accelerating device, such as the rotating polyhedron mirror and the retroreflector. Thus it is simple and more suitable for cooperating with the electro-optical deflection and the Goos-Hänchen displacement[12-13] to accelerate the light-beam deflection.
It is found that the deflection amplification accelerating can be used to improve the temporal resolution of the crystal-streak camera[14] 20.0 ps to 2.5 ps[15] at least.
