16Principles of Adaptive Optics�Aa 2 �I ( P) = �2 ��R �2 1 221 2 2 u蟻 ]�0 0�ei[ k桅捪呄os(胃捪�)�蟻d蟻d胃(1�8)one can see that the exponent contains three basic terms, the wavefront contribution k桅, the tilt contribution 蠀蟻 cos(胃捪�, and the focus contribution u蟻2�If we determine that the wavefront 桅 contains tilt of a magnitude Kx in the x direction, we can perform a coordinate transformation 桅 = 桅�+ Kx蟻 sin 胃�With the tilt term separated into its x搚 components, the focal length represented by f, and the aperture radius, a, the exponent becomes k桅 �+ kK x蟻 sin 胃 �ka ka 1 �a�r 蟻 sin 蠄 sin 胃 �r 蟻 cos 蠄 cos 胃 �kz ��蟻2 f f 2 �f�2(1�9)Removing the tilt from the higher-order wavefront term 桅�and transforming according to x �= x �(R/a)K x , y �= y, z �= z , the exponent becomes ka 1 �a�k桅 �蟻�cos(胃 �蠄 � �kz ��蟻2 f 2 �f�2(1�0)which is the same form as Equation 1��The distribution of light in the image plane will be the same as the original untilted wavefront�The centroid of the image will shift, however, by an amount equal to f Kx/a�A similar transformation can be made for a focal shift�If the wavefront contains the term K z蟻2, then the distribution of light will not change in the focal plane, but the focal distance f will shift an amount proportional to K z�This very important result is fundamental to the measurement of the focus or the tilt in a beam and will be discussed in detail in Chapter 5�1.3.4 interference Interference occurs when two or more coherent light beams are superimposed�White light interference can occur because (incoherent) white light can be thought of as the sum of coherent components that interfere�Basic principles of optical interference can be used for practical applications such as measuring wavefronts in adaptive optics [424]�The intensity is the time-averaged squared magnitude of the electric field�We can begin by expressing the electric field vector of a plane wave as 1 E = [A(r ) e�i蠅t + A �(r ) ei蠅t ] 2 where the vector components of the amplitude are Ax = ax eikr 捨�x(1�1)(1�2)
