## 3 Theory
### Wavelength or Spatial Period of the Interference Pattern for the Star and Planet  
A wavelength may be associated with each cosine-square incidence pattern as \(\Lambda_{i}\), where \(i\) is \(S\) or \(P\), respectively. \[M_{SP}(\rho,\varphi;\varphi_{S},\theta_{S};\varphi_{P},\theta_{P})=M_{S}\bigg{\{} \cos^{2}\bigg{[}\frac{\pi\rho}{\Lambda_{S}}\bigg{]}\bigg{\}}+M_{P}\bigg{\{}\cos ^{2}\bigg{[}\frac{\pi\rho}{\Lambda_{P}}\bigg{]}\bigg{\}} \tag{18}\]  
This wavelength of the fringe pattern may be examined perpendicular to the \(\varphi=\varphi_{P}\) direction. This is the minimum wavelength (spatial period) of the planet fringe. The spatial wavelength is elongated along any other direction. \[\Lambda_{Pmin}(\lambda,\Delta\varphi,\theta_{P})=\frac{\lambda}{2 sin\theta_{P }\ sin(\Delta\varphi)} \tag{19}\]  
The wavelength of the star fringe pattern may also be examined perpendicular to the \(\varphi=\varphi_{S}\) direction. This is the minimum wavelength of the star fringe. \[\Lambda_{Smith}(\lambda,\Delta\varphi,\theta_{S})=\frac{\lambda}{2 sin\theta_{S }\ sin(\Delta\varphi)} \tag{20}\]  
Using the hypothetical values for the star and the planet (illustrated in Figure 5), the minimum fringe wavelength for the star is found to be \(10^{6}\) m, and for the planet 50 m. The shear angle \(\Delta\varphi\) is assumed to be 0.1 rad, and the optical wavelength is 20 \(\upmu\)m. Thus, the minimum star wavelength is twenty-thousand times longer than that of a planet. During each planet period, the fraction of the wavelength corresponding to the star is so short with respect to the planet wavelength that this section results in a nearly straight, nearly horizontal line. The reader will note that neither the star nor the planet wavelength (inverse spatial period) is defined for the zero-shear angle (\(\Delta\varphi=0\)). When the star elevation approaches to zero, the wavelength of the star fringe pattern is no longer defined, as it approaches to infinity. This case corresponds to our earlier theoretical derivation when the star was assumed to be located on the interferometer optical axis. For a general direction \(\varphi\), the fringe wavelength changes with the angular deviation from the planet or the star celestial coordinate. The fringe wavelength is defined as the distance from the brightest point on one fringe through the darkest point on the same fringe again to the brightest point on the next fringe, along any direction,  
\[\Lambda_{S}(\lambda,\Delta\varphi,\varphi;\varphi_{S},\theta_{S})=\frac{\lambda} {2 sin\theta_{S}\ sin(\Delta\varphi)\ \sin(\varphi-\varphi_{S})}, \tag{21}\]  
and  
\[\Lambda_{P}(\lambda,\Delta\varphi,\varphi;\varphi_{P},\theta_{P})=\frac{ \lambda}{2 sin\theta_{P}\ sin(\Delta\varphi)\ \sin(\varphi-\varphi_{P})}. \tag{22}\]  
In cases of both the star and the planet, the wavelength of the fringe pattern is proportional to the spectral wavelength, and inversely proportional to (the sine of) the star or the planet elevation and the sine of the shear angle \(\Delta\varphi\). If the shear angle is decreased, the wavelength of the fringe pattern increases. By increasing the shear angle, the wavelength of the fringe pattern decreases and, therefore, the number of fringes over the field increases. These results represent a generalization of the previously obtained results for the case that the star is centered on the optical axis. Finally, the reader will note that the wavelength of each fringe pattern also increases from some minimum value (Equations (19) and (20)) when \(\varphi-\varphi_{P}=\pi/2\) or \(\varphi-\varphi_{S}=\pi/2\), respectively. This is the direction along which the equal gray-value fringes are formed. The wavelength of the spatial pattern increases when \(\varphi-\varphi_{S}\) decreases, tending to a value of infinity. The relationship between the minimum wavelength of the fringe pattern of the star and the planet is found by dividing Equation (19) by Equation (20).