PROPAGATION AND COMPENSATION FOR ATMOSPHERIC TURBULENCE
5.3 ADAPTIVE OPTICS
86 PROPAGATION AND COMPENSATION FOR ATMOSPHERIC TURBULENCE
a Gaussian exponential proposed by Tatarski [155]. For extending into the lower wave number (low spatial frequency) past the large-scale limitκ=1/L0, we di- vide by a factor (κ2+κ02), whereκ0=C0/L0withC0=2π, 4π, or 8πdepend- ing on the application [4]. The reason for the variety in selectingC0is that above the large-scale limitL0, the field is no longer statistically homogeneous—the eddies may be distorted by atmospheric striations—so approximation is mainly to make the equations more tractable. The resulting modified Von Karman spec- trum is
n(κ)= 0.033C2n (κ2+κ20)11/6exp
−κ2
κ2m for κm=5.92/ l0, κ0=2π/L0
(5.33)
5.2.3 Atmospheric Temporal Statistics
When wind blows across the path of a laser beam in the atmosphere, the turbulence will move at wind speedV⊥across the beam. Because the eddy pattern takes many seconds to change, the motion of noticeable wind L0/V⊥ is much faster. This is known as the frozen turbulence hypothesis and is similar to cloud patterns caused by air turbulence passing across the sky. Therefore, we can convert spatial turbulence due to eddy currents into temporal turbulence that incorporates the effects of the component of wind at right angles to the path.
u(R, t+τ)=u(R−V⊥τ, t) (5.34)
5.2.4 Long-Distance Turbulence Models
For an optical beam traversing many miles, the strength of turbulence may vary over the path. For example, if a beam moves over the land past cliffs over sea, the turbulence strength will vary because the land and sea have different temperatures and wind striking the cliff will generate strong turbulence. Such situations may be handled by partitioning the path into regions, each with its own constant turbulence.
Further in numerical computations, the effect of turbulence in each region may be equated to a phase screen. Such phase screens were implemented around a sequence of optical disks for a hybrid test facility for the airborne laser (Section 12.2.7.2).
ADAPTIVE OPTICS 87
A
B
C
D
Detector arrays Wave
front
Lenslet array
FIGURE 5.2 Hartmann wave front sensor.
5.3.1.1 Wave Front Sensor The wave front sensor measures the shape of the wave front [144, 160, 163]. Many approaches are possible. Figure 5.2 illustrates how wave front shape is measured with a popular Hartmann wave front sensor. AtAin Figure 5.2, the wave front is tilted up substantially and the small lens will focus the light into the upper row of the associated sensor array. AtBthe wave front is tilted up less, causing light to be focused to the next from top row of the sensor array. The lower half of the wave front focuses to the lower parts of the sensor array. The data from the wave front sensor arrays indicate the local direction in transversexandyof the wave front.
The wave front shape is fed to a computer that calculates the settings for the deformable mirror to give a desired correction to the wave front. The computation allows the discretization of the wave front sensor and the deformable array.
5.3.1.2 Deformable Mirror Device The deformable mirror consists of a flex- ible mirror supported by an array of pistons that can raise the mirror up locally under computer control. Moving the mirror up locally advances the phase of the reflected light at this point because the light travels less far than that at neighboring points.
The light emanating from most high-power lasers has poor spatial coherence because of nonlinear effects that are significant at high powers (Section 8.1.2).
Figure 5.3a shows the distorting effect of a beam of poor spatial coherence. Some parts of the wave front, labeledA, B, C, D, andE, are in the direction of the beam.
But other points on the wave front point out of the beam alongF, G, H, andJ and degrade the quality of the beam with distance. In a beam cleanup, the wave front is flattened into a plane wave by setting the deformable mirror to the conjugate phase of the wave front. For example, we can move to the right the mirror element of the deformable mirror that lines up with the wave front in the direction along the beam at C(Figure 5.3a). This will retard the wave fromC, so it has to travel farther than the one fromB, which brings waves fromBandCinto line. Then waves propagating at an angle to the main beam (F, G, H, J) will be eliminated. In practice, adaptive optics
88 PROPAGATION AND COMPENSATION FOR ATMOSPHERIC TURBULENCE
A B
C
D E
F G
H J
Wave front Deformable mirror
(a) (b)
Deformable mirror
Beam splitter
Wave front sensor Computer
Main beam
Cleaned up main beam
FIGURE 5.3 (a) Poor spatial coherence of a beam and (b) beam cleanup adaptive optics system.
will shape the wave front into a desired smooth function in space while improving beam spatial coherence.
5.3.1.3 Adaptive Optics Cleanup System Figure 5.3b illustrates the prin- ciples of an adaptive optics beam cleanup system to improve spatial coherence and shape the wave front for a high-power laser. The adaptive optics beam cleanup system has an advantage over a spatial filter (Section 1.3.6), which cannot focus a poor quality beam (such as the one distorted by turbulence or from very high-power gas lasers) through a pinhole without losing considerable power. The spatial filter is adequate for cleaning up distortion caused by a Nd:YAG power amplifier (Section 13.2.1). The main beam to be cleaned up is deflected by a deformable mirror that will be set to per- form the cleanup [83, 163]. The deformable mirror sends some of the reflected waves to the wave front sensor via a beam splitter so that the computer can determine how to set the deformable mirror to achieve beam cleanup and transform the wave front to that desired, as specified in the computer. Note that the beam size can be modified to operate with deformable mirrors of different sizes. For example, the 1000×1000 mirror array in the digital light projector (DLP) used in projectors, manufactured by Texas Instruments, is very small because it is a modified memory chip [83, 113]. The adaptive beam cleanup system (Figure 5.3), is incorporated into the airborne laser (ABL) system in Section 12.2.2.
5.3.1.4 Principles of Adaptive Optics Compensation for Turbulence The main beam reflection from a missile cannot be relied on to provide adequate information for estimating turbulence. For this reason, a separate beacon laser is used that is coincident with the path of the main laser beam. The beacon laser light strikes the target and, unlike the main beam, is reflected back to the aircraft for the pur- pose of estimating the turbulence (Section 5.2). The beacon laser is a 1.06m diode pulsed semiconductor pumped solid-state laser (Section 8.2). Beacon lasers are used in adaptive optics for many astronomical telescopes. In this case, the beacon laser acts
COMPUTATION OF LASER LIGHT THROUGH ATMOSPHERIC TURBULENCE 89
Deformable mirror compensates for atmospheric turbulence Beam splitter Cleaned up
main beam Wave front
sensor
Computer
Cassegrain telescope
To target - all beams From target - beacon Beacon
laser
Dichroic mirror
FIGURE 5.4 Principle of adaptive optics for compensating for turbulence.
like the orange ranging beam that measures distance to set focusing before taking a photograph in many digital cameras.
The beacon laser is activated to allow compensating corrections to the wave front with a deformable mirror,DMturbin Figure 12.4, before shooting a pulse of the main high-power beam. Figure 5.4 shows the principle of using a beacon laser to estimate