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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References

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  1. M. S. Scholl, “Effect of the coating-thickness error on the performance of an optical component,” Infrared Phys. Technol. (to be published).
  2. M. S. Scholl, G. N. Lawrence, “Adaptive optics for inorbit aberration correction: feasibility study,” Appl. Opt. 34, 7295–7301 (1995).
    [CrossRef] [PubMed]
  3. K. Knight, “Methods of detecting extrasolar planets. I. Imaging,” Icarus 30, 422–428 (1977).
    [CrossRef]
  4. C. Roddier, F. Roddier, J. Demarcq, “Compact rotational shearing interferometer for astronomical applications,” Opt. Eng. 28, 66–73 (1989).
    [CrossRef]
  5. M. S. Scholl, “Infrared signal generated by a planet outside the solar system discriminated by rotating rotationally-shearing interferometer,” Infrared Phys. Technol. (to be published).
  6. M. S. Scholl, G. N. Lawrence, “Diffraction modeling of a space relay experiment,” Opt. Eng. 29, 271–278 (1990).
    [CrossRef]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1972), pp. 510–514.

1995 (1)

1990 (1)

M. S. Scholl, G. N. Lawrence, “Diffraction modeling of a space relay experiment,” Opt. Eng. 29, 271–278 (1990).
[CrossRef]

1989 (1)

C. Roddier, F. Roddier, J. Demarcq, “Compact rotational shearing interferometer for astronomical applications,” Opt. Eng. 28, 66–73 (1989).
[CrossRef]

1977 (1)

K. Knight, “Methods of detecting extrasolar planets. I. Imaging,” Icarus 30, 422–428 (1977).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1972), pp. 510–514.

Demarcq, J.

C. Roddier, F. Roddier, J. Demarcq, “Compact rotational shearing interferometer for astronomical applications,” Opt. Eng. 28, 66–73 (1989).
[CrossRef]

Knight, K.

K. Knight, “Methods of detecting extrasolar planets. I. Imaging,” Icarus 30, 422–428 (1977).
[CrossRef]

Lawrence, G. N.

M. S. Scholl, G. N. Lawrence, “Adaptive optics for inorbit aberration correction: feasibility study,” Appl. Opt. 34, 7295–7301 (1995).
[CrossRef] [PubMed]

M. S. Scholl, G. N. Lawrence, “Diffraction modeling of a space relay experiment,” Opt. Eng. 29, 271–278 (1990).
[CrossRef]

Roddier, C.

C. Roddier, F. Roddier, J. Demarcq, “Compact rotational shearing interferometer for astronomical applications,” Opt. Eng. 28, 66–73 (1989).
[CrossRef]

Roddier, F.

C. Roddier, F. Roddier, J. Demarcq, “Compact rotational shearing interferometer for astronomical applications,” Opt. Eng. 28, 66–73 (1989).
[CrossRef]

Scholl, M. S.

M. S. Scholl, G. N. Lawrence, “Adaptive optics for inorbit aberration correction: feasibility study,” Appl. Opt. 34, 7295–7301 (1995).
[CrossRef] [PubMed]

M. S. Scholl, G. N. Lawrence, “Diffraction modeling of a space relay experiment,” Opt. Eng. 29, 271–278 (1990).
[CrossRef]

M. S. Scholl, “Infrared signal generated by a planet outside the solar system discriminated by rotating rotationally-shearing interferometer,” Infrared Phys. Technol. (to be published).

M. S. Scholl, “Effect of the coating-thickness error on the performance of an optical component,” Infrared Phys. Technol. (to be published).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1972), pp. 510–514.

Appl. Opt. (1)

Icarus (1)

K. Knight, “Methods of detecting extrasolar planets. I. Imaging,” Icarus 30, 422–428 (1977).
[CrossRef]

Opt. Eng. (2)

C. Roddier, F. Roddier, J. Demarcq, “Compact rotational shearing interferometer for astronomical applications,” Opt. Eng. 28, 66–73 (1989).
[CrossRef]

M. S. Scholl, G. N. Lawrence, “Diffraction modeling of a space relay experiment,” Opt. Eng. 29, 271–278 (1990).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1972), pp. 510–514.

M. S. Scholl, “Effect of the coating-thickness error on the performance of an optical component,” Infrared Phys. Technol. (to be published).

M. S. Scholl, “Infrared signal generated by a planet outside the solar system discriminated by rotating rotationally-shearing interferometer,” Infrared Phys. Technol. (to be published).

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Figures (10)

Fig. 1
Fig. 1

Number of stars shown as a function of the stellar (visual) magnitude, illustrating the paucity of stars in the vicinity of Earth. There are only 10 stars brighter than stellar magnitude 5 within a distance of 10 parsec from Earth.

Fig. 2
Fig. 2

Typical star at a distance of 10 parsec from Earth. At such a large distance the star–planet distance subtends an angle of 2 μ rad.

Fig. 3
Fig. 3

Number of spectral photons emitted by planets as black-body emitters at several representative temperatures intercepted per unit time per unit collection area at a distance of 10 parsec.

Fig. 4
Fig. 4

Number of spectral photons emitted per unit time by the Sun and by several representative planets in our solar system as a function of wavelength, normalized to the peak solar photon emission. The Sun emits 109 times more photons at its own peak photon emission and 105 times more at the peak photon emission of the planet. The planet Jupiter emits and reflects more radiation than Earth in both the visible and the IR wavelengths.

Fig. 5
Fig. 5

Detecting a faint planet in the vicinity of a bright star. The star and the planet point sources are located in the X, Y coordinate system. The bright star is shown as a disk on the optical axis; however, it can be considered a point source because of the large distance of observation along optical axis Z. The faint, as yet invisible, planet rotates around the star with an unknown rotational velocity. At the specific instant of observation it is located on the X axis (i.e., it defines the X axis), a distance a from the star. The star and the planet plane wave fronts are incident upon two apertures of a rotationally shearing interferometer in the x, y coordinate system. The optical axis, the Z axis, connects the bright star and the interferometer’s center of shear.

Fig. 6
Fig. 6

(a) The representation of the star’s and the planet’s planar spectra on a linear scale does not permit the resolution of either of them. The angular coordinate shown on the abscissa is an arc tangent of the ratio of the star–planet distance divided by the star–Sun separation. (b) The representation of the star’s and the planet’s planar spectra on an expanded scale is required for delineation of the extent of each spectrum. The angular coordinate shown on the abscissa is an arc tangent of the ratio of the star–planet distance divided by the star–Sun separation.

Fig. 7
Fig. 7

The distribution of the plane-wave slopes generated by the star and the planet does not have the azimuthal symmetry because the planet is located at only one radial position.

Fig. 8
Fig. 8

Two-aperture interferometer samples the wave fronts from a star and its planet. The two coordinate systems are X, Y, Z where the star and the planet are located and x, y, z where the interferometer is located. The two coordinate systems have a common z axis along the star–Earth distance. Although the planet indeed rotates around the star, we are defining the X axis through the planet at the time of observation. The planet is located a distance a from the star. The wave front incident from the planet makes an angle α with respect to the z axis. The plane wave front incident from the star has a normal parallel to the z axis at the interferometer’s location.

Fig. 9
Fig. 9

Two apertures may sample the wave front at the same time, but at two different locations (ρ1, φ1) and (ρ2, φ2) expressed in a cylindrical coordinate system.

Fig. 10
Fig. 10

The signal that a rotating rotationally shearing interferometer detects as a function of angle is just a clipped cosine function superimposed upon a large background term.

Equations (32)

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E ( r ) = [ I ( r ) ] 1 / 2 exp [ i ϕ ( r ) ] .
I ( r , τ ) = [ I ( r 1 ) I ( r 2 ) ] 1 / 2 { I ( r 1 ) / [ I ( r 1 ) I ( r 2 ) ] 1 / 2 + I ( r 2 ) / [ I ( r 1 ) I ( r 2 ) ] 1 / 2 + 2 γ 12 ( r , τ ) × cos [ ϕ ( r ) + ϕ ( r + Δ r ) + β 12 ( r , τ ) ] } ,
γ 12 ( r , τ ) = γ 12 ( r , τ ) exp ( i β ) 12 ( r , τ ) .
I B ( r ) = [ I ( r 1 ) I ( r 2 ) ] 1 / 2 .
I B ( r ) = I S ( r ) + I P ( r ) = I S ( r ) .
i ( r 1 ) = I ( r 1 ) / I B ( r ) ,             i = 1 , 2.
I ( r , τ ) = 2 I B ( r ) { 1 + γ 12 ( r , τ ) cos [ ϕ ( r ) + ϕ ( r + Δ r ) + β 12 ( r , τ ) ] } .
I ( r ) = 2 I B ( r ) { 1 + γ 12 ( r ) cos [ ϕ ( r ) + ϕ ( r + Δ r ) + β 12 ( r ) ] } .
Φ ( ρ + Δ ρ , φ + Δ φ ) = Φ ( ρ , φ ) + [ δ Φ ( ρ , φ ) / δ ρ ] Δ ρ + ρ [ δ Φ ( ρ , φ ) / δ ( ρ φ ) ] Δ φ .
Φ ( ρ + Δ ρ , φ + Δ φ ) - Φ ( ρ , φ ) = [ δ Φ ( ρ , φ ) / δ ρ ] Δ ρ + ρ [ δ Φ ( ρ , φ ) / δ ( ρ φ ) ] Δ φ .
Φ ( ρ + Δ ρ , φ + Δ φ ) - Φ ( ρ , φ ) = ρ [ δ Φ ( ρ , φ ) / δ ( ρ φ ) ] Δ φ .
I ( d / 2 , φ ) = 2 I B ( d / 2 , φ ) [ 1 + γ 12 ( ρ 0 , φ 0 ) × cos ( ( d / 2 ) { [ δ Φ ( ρ , φ ) / δ ( ρ φ ) ] ( ρ = d / 2 ) } × Δ φ + β 12 ( ρ 0 , φ 0 ) ) ] .
I ( d / 2 , φ 0 ) = I 0 = 2 I B [ 1 + γ 0 cos ( ( d / 2 ) { [ δ Φ ( ρ , φ ) / δ ( ρ φ ) ] ( ρ = d / 2 , φ = φ 0 ) } Δ φ ) ]
Φ s ( ρ , φ ) = B exp ( i k p z ) .
Φ p ( ρ , φ ) = b exp [ i k p ( l x + n z ) ] .
Φ ( ρ , φ ) = B exp ( i k p z ) + b exp [ i k p ( l x + n z ) ] .
δ Φ ( ρ , φ ) / δ φ = i k p l ρ cos φ { b exp [ i k p ( l x + n z ) ] } = i k p l ρ cos φ Φ ( ρ , φ ) .
I ( d / 2 , φ 0 ) = 2 I B ( d / 2 , φ 0 ) [ 1 + γ 0 cos ( ( d / 2 ) { k p l × [ cos φ Φ p ( ρ , φ ) ] ( ρ = d / 2 , φ = φ 0 ) } Δ φ ) ] .
I ( d / 2 , φ 0 ) = 2 I B ( d / 2 , φ 0 ) [ 1 + γ 0 cos ( ( d / 2 ) { k p l × [ ( cos φ 0 ) Φ p ( d / 2 , φ 0 ) ] } Δ φ ) ] .
I p ( d / 2 , φ 0 ) = 2 I B γ 0 ( d / 2 , φ 0 ) cos ( ( d / 2 ) { k p l × [ ( cos φ 0 ) Φ p ( d / 2 , φ 0 ) ] } Δ φ ) .
( S / N ) st = γ 0 cos ( ( d / 2 ) { k p l [ ( cos φ 0 ) Φ p ( d / 2 , φ 0 ) ] } Δ φ ) .
I [ ( d / 2 , φ ( t ) ] = 2 I B [ ( d / 2 , φ ( t ) ] { 1 + γ 0 cos [ ( d / 2 ) ( k p l × { [ cos φ ( t ) ] Φ p [ d / 2 , φ ( t ) ] } ) Δ φ ] } .
I [ φ ( t ) ] = 2 I B { 1 + γ 0 cos [ ( d / 2 ) ( k p l × { [ cos φ ( t ) ] Φ p [ φ ( t ) ] } ) Δ φ ] } .
= [ ( d / 2 ) Δ φ ] [ k p l Φ p ( φ ) ] .
I ( φ ) = 2 I B [ 1 + γ 0 cos ( cos φ ) ] .
I min ( φ = 0 ) = 2 I B ( 1 + γ 0 cos ) .
I max ( φ = π / 2 ) = I ( φ = 3 π / 2 ) = 2 I B ( 1 + γ 0 ) .
I min ( φ = π ) = 2 I B ( 1 + γ 0 cos ) .
I ( t ) = 2 I B [ 1 + γ 0 cos ( cos ω t ) ] .
C ( Δ φ ) = ( I max - I min ) / ( I max + I min ) .
C ( Δ φ ) = ( γ 0 / 2 ) ( 1 - cos ) .
C ( Δ φ ) = ( γ 0 / 2 ) { 1 - cos [ ( d / 2 ) k p l Φ p Δ φ ] } .

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