## Abstract

An infrared rotating rotationallly shearing interferometer can be used for detection of a potential planet orbiting around a nearby star. An expression is derived for the signal generated by a star and its faint companion, and this signal is detected by a rotationally shearing interferometer. It is shown that the planet’s signal can be detected, despite the presence of a much larger star signal, because the planet produces a faint modulation superimposed upon the large star signal when the aperture rotates. In the particular case of a rotating rotationally shearing interferometer, the argument of the cosine term is shown to depend only on the planet’s and the observational parameters. However, the amplitude of the modulation term in the interferometric signal is shown to be proportional to the star’s intensity.

© 1996 Optical Society of America

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### Equations (32)

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(1)
$$E(\mathbf{r})={[I(\mathbf{r})]}^{1/2}\hspace{0.17em}\text{exp}[i\varphi (\mathbf{r})].$$
(2)
$$I(\mathbf{r},\tau )={[I({\mathbf{r}}_{1})I({\mathbf{r}}_{2})]}^{1/2}\{I({\mathbf{r}}_{1})/{[I({\mathbf{r}}_{1})I({\mathbf{r}}_{2})]}^{1/2}+I({\mathbf{r}}_{2})/{[I({\mathbf{r}}_{1})I({\mathbf{r}}_{2})]}^{1/2}+2{\gamma}_{12}(\mathbf{r},\tau )\times \text{cos}[\varphi (\mathbf{r})+\varphi (\mathbf{r}+\mathrm{\Delta}\mathbf{r})+{\beta}_{12}(\mathbf{r},\tau )]\},$$
(3)
$${\mathit{\gamma}}_{12}(\mathbf{r},\tau )={\gamma}_{12}(\mathbf{r},\tau )\text{exp}{(i\beta )}_{12}(\mathbf{r},\tau ).$$
(4)
$${I}_{B}(\mathbf{r})={[I({\mathbf{r}}_{1})I({\mathbf{r}}_{2})]}^{1/2}.$$
(5)
$${I}_{B}(\mathbf{r})={I}_{S}(\mathbf{r})+{I}_{P}(\mathbf{r})={I}_{S}(\mathbf{r}).$$
(6)
$$i({\mathbf{r}}_{1})=I({\mathbf{r}}_{1})/{I}_{B}(\mathbf{r}),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}i=1,2.$$
(7)
$$I(\mathbf{r},\tau )=2{I}_{B}(\mathbf{r})\{1+{\gamma}_{12}(\mathbf{r},\tau )\text{cos}[\varphi (\mathbf{r})+\varphi (\mathbf{r}+\mathrm{\Delta}\mathbf{r})+{\beta}_{12}(\mathbf{r},\tau )]\}.$$
(8)
$$I(\mathbf{r})=2{I}_{B}(\mathbf{r})\{1+{\gamma}_{12}(\mathbf{r})\text{cos}[\varphi (\mathbf{r})+\varphi (\mathbf{r}+\mathrm{\Delta}\mathbf{r})+{\beta}_{12}(\mathbf{r})]\}.$$
(9)
$$\mathrm{\Phi}(\rho +\mathrm{\Delta}\rho ,\phi +\mathrm{\Delta}\phi )=\mathrm{\Phi}(\rho ,\phi )+[\delta \mathrm{\Phi}(\rho ,\phi )/\delta \rho ]\mathrm{\Delta}\rho +\rho [\delta \mathrm{\Phi}(\rho ,\phi )/\delta (\rho \phi )]\mathrm{\Delta}\phi .$$
(10)
$$\mathrm{\Phi}(\rho +\mathrm{\Delta}\rho ,\phi +\mathrm{\Delta}\phi )-\mathrm{\Phi}(\rho ,\phi )=[\delta \mathrm{\Phi}(\rho ,\phi )/\delta \rho ]\mathrm{\Delta}\rho +\hspace{0.17em}\rho [\delta \mathrm{\Phi}(\rho ,\phi )/\delta (\rho \phi )]\mathrm{\Delta}\phi .$$
(11)
$$\mathrm{\Phi}(\rho +\mathrm{\Delta}\rho ,\phi +\mathrm{\Delta}\phi )-\mathrm{\Phi}(\rho ,\phi )=\rho [\delta \mathrm{\Phi}(\rho ,\phi )/\delta (\rho \phi )]\mathrm{\Delta}\phi .$$
(12)
$$I(d/2,\phi )=2{I}_{B}(d/2,\phi )[1+{\gamma}_{12}({\rho}_{0},{\phi}_{0})\times \text{cos}((d/2)\{{[\delta \mathrm{\Phi}(\rho ,\phi )/\delta (\rho \phi )]}_{(\rho =d/2)}\}\times \mathrm{\Delta}\phi +{\beta}_{12}({\rho}_{0},{\phi}_{0}))].$$
(13)
$$I(d/2,{\phi}_{0})={I}_{0}=2{I}_{B}[1+{\gamma}_{0}\hspace{0.17em}\text{cos}((d/2)\{{[\delta \mathrm{\Phi}(\rho ,\phi )/\delta (\rho \phi )]}_{(\rho =d/2,\phi ={\phi}_{0})}\}\mathrm{\Delta}\phi )]$$
(14)
$${\mathrm{\Phi}}_{s}(\rho ,\phi )=B\hspace{0.17em}\text{exp}(i{k}_{p}z).$$
(15)
$${\mathrm{\Phi}}_{p}(\rho ,\phi )=b\hspace{0.17em}\text{exp}[i{k}_{p}(lx+nz)].$$
(16)
$$\mathrm{\Phi}(\rho ,\phi )=B\hspace{0.17em}\text{exp}(i{k}_{p}z)+b\hspace{0.17em}\text{exp}[i{k}_{p}(lx+nz)].$$
(17)
$$\begin{array}{l}\delta \mathrm{\Phi}(\rho ,\phi )/\delta \phi =i{k}_{p}l\rho \hspace{0.17em}\text{cos}\hspace{0.17em}\phi \{b\hspace{0.17em}\text{exp}[i{k}_{p}(lx+nz)]\}\\ =i{k}_{p}l\rho \hspace{0.17em}\text{cos}\hspace{0.17em}\phi \mathrm{\Phi}(\rho ,\phi ).\end{array}$$
(18)
$$I(d/2,{\phi}_{0})=2{I}_{B}(d/2,{\phi}_{0})[1+{\gamma}_{0}\hspace{0.17em}\text{cos}((d/2)\{{k}_{p}l\times {[\text{cos}\hspace{0.17em}\phi {\mathrm{\Phi}}_{p}(\rho ,\phi )]}_{(\rho =d/2,\phi ={\phi}_{0})}\}\mathrm{\Delta}\phi )].$$
(19)
$$I(d/2,{\phi}_{0})=2{I}_{B}(d/2,{\phi}_{0})[1+{\gamma}_{0}\text{cos}((d/2)\{{k}_{p}l\times [(\text{cos}\hspace{0.17em}{\phi}_{0}){\mathrm{\Phi}}_{p}(d/2,{\phi}_{0})]\}\mathrm{\Delta}\phi )].$$
(20)
$${I}_{p}(d/2,{\phi}_{0})=2{I}_{B}{\gamma}_{0}(d/2,{\phi}_{0})\text{cos}((d/2)\{{k}_{p}l\times [(\text{cos}\hspace{0.17em}{\phi}_{0}){\mathrm{\Phi}}_{p}(d/2,{\phi}_{0})]\}\mathrm{\Delta}\phi ).$$
(21)
$${(\text{S}/\text{N})}_{\text{st}}={\gamma}_{0}\hspace{0.17em}\text{cos}((d/2)\{{k}_{p}l[(\text{cos}\hspace{0.17em}{\phi}_{0}){\mathrm{\Phi}}_{p}(d/2,{\phi}_{0})]\}\mathrm{\Delta}\phi ).$$
(22)
$$I[(d/2,\phi (t)]=2{I}_{B}[(d/2,\phi (t)]\{1+{\gamma}_{0}\text{cos}[(d/2)({k}_{p}l\times \{[\text{cos}\hspace{0.17em}\phi (t)]{\mathrm{\Phi}}_{p}[d/2,\phi (t)]\})\mathrm{\Delta}\phi ]\}.$$
(23)
$$I[\phi (t)]=2{I}_{B}\{1+{\gamma}_{0}\hspace{0.17em}\text{cos}[(d/2)({k}_{p}l\times \{[\text{cos}\hspace{0.17em}\phi (t)]{\mathrm{\Phi}}_{p}[\phi (t)]\})\mathrm{\Delta}\phi ]\}.$$
(24)
$$\u220a=[(d/2)\mathrm{\Delta}\phi ][{k}_{p}l{\mathrm{\Phi}}_{p}(\phi )].$$
(25)
$$I(\phi )=2{I}_{B}[1+{\gamma}_{0}\hspace{0.17em}\text{cos}(\u220a\text{cos}\hspace{0.17em}\phi )].$$
(26)
$${I}_{\text{min}}(\phi =0)=2{I}_{B}(1+{\gamma}_{0}\hspace{0.17em}\text{cos}\hspace{0.17em}\u220a).$$
(27)
$${I}_{\text{max}}(\phi =\pi /2)=I(\phi =3\pi /2)=2{I}_{B}(1+{\gamma}_{0}).$$
(28)
$${I}_{\text{min}}(\phi =\pi )=2{I}_{B}(1+{\gamma}_{0}\hspace{0.17em}\text{cos}\hspace{0.17em}\u220a).$$
(29)
$$I(t)=2{I}_{B}[1+{\gamma}_{0}\hspace{0.17em}\text{cos}(\u220a\text{cos}\hspace{0.17em}\omega t)].$$
(30)
$$C(\mathrm{\Delta}\phi )=({I}_{\text{max}}-{I}_{\text{min}})/({I}_{\text{max}}+{I}_{\text{min}}).$$
(31)
$$C(\mathrm{\Delta}\phi )=({\gamma}_{0}/2)(1-\hspace{0.17em}\text{cos}\u220a).$$
(32)
$$C(\mathrm{\Delta}\phi )=({\gamma}_{0}/2)\{1-\text{cos}[(d/2){k}_{p}l{\mathrm{\Phi}}_{p}\mathrm{\Delta}\phi ]\}.$$