Abstract

We have studied how turbulence influences one’s ability to determine the spatial-frequency composition of an incoherent source by use of irradiance or speckle interferometry. We have found that results obtained by using these methods are not significantly affected by the turbulence for the case of high spatial frequencies; however, the lower frequencies may be affected considerably.

© 1976 Optical Society of America

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  1. Z. Feizulin and Y. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).
  2. A. Gurvich and V. Tatarski, Radio Sci. 10, 3 (1975).
  3. A. Kon and Z. Feizulin, Radiophys. Quantum Electron. 13, 51 (1970).
  4. Note that Eq. (4) has been obtained from Eq. (11) of Ref. 1, by using the fact that Dx + DS = D1 and the definition Dx(ρ-ρ′, ρ12)=〈[χ(ρ,ρ1)-χ(ρ′,ρ2)]2〉 =2σ2x-2Bx(ρ-ρ′,ρ12), where Bx(ρ-ρ′,ρ12)≡(χ(ρ,ρ1)χ(ρ′,ρ2)〉-〈χ〉2.
  5. R. Fante, J. Opt. Soc. Am. 66, 74 (1976).
  6. For weak turbulence the proof of this statement is readily seen from Eq. (5) of Ref. 5. For strong turbulence the proof follows from a realization that measured data2 indicate the field statistics are nearly log normal. If we compare the Markov approximation solution for 〈u〉 = 〈exp[χ+ iS]〉 with the solution obtained by assuming χ and S are Gaussian variables we find that 〈χS〉 = 0 is required for the two solutions to be identical. For nearly log-normal field statistics it can be shown that 〈χS〉 ≪ 1.
  7. A formal definition for ε is given by ε2=(∫L0s2C2n(s)ds) (∫L0dsC2n(s))-1, where s=L - x is the distance measured from the surface of the earth. In this paper we shall use Hufnagel's model for C2n(s). For a description of this model see R. Hufnagel, OSA Topical Meeting on Propagation through Turbulence, Paper WA1-1, Boulder, Colo., July 1974.
  8. For weak turbulence the proof of this statement is readily seen from Eq. (5) of Ref. 5. For strong turbulence the proof follows from a realization that measured data2 indicate the field statistics are nearly log normal. If we compare the Markov approximation solution for 〈u〉 = 〈exp[χ+ iS]〉 with the solution obtained by assuming χ and S are Gaussian variables we find that 〈χS〉 = 0 is required for the two solutions to be identical. For nearly log-normal field statistics it can be shown that 〈χS〉 ≪ 1.
  9. A formal definition for ε is given by ε2=(∫L0s2C2n(s)ds) (∫L0dsC2n(s))-1, where s=L - x is the distance measured from the surface of the earth. In this paper we shall use Hufnagel's model for C2n(s). For a description of this model see R. Hufnagel, OSA Topical Meeting on Propagation through Turbulence, Paper WA1-1, Boulder, Colo., July 1974.
  10. 8From Eq. (19) is is quite clear that (27) is the appropriate form for all interferometer spacings such that DR ≫ ελκm = 5. 91ελ/l0. Assuming λ ~0.6 × 10-6 m, l0 ~ 10-3 m, and ε ~ 103 m, we see that for DR ≫3 m, ρN is the correct parameter. For DR ≪3 m it is correct to use the form in Eq. (24). It is therefore only for relatively small values of DR that the form used in Eq. (27) is only approximate.
  11. D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).
  12. D. Korff, J. Opt. Soc. Am. 63, 971 (1973).
  13. Let us consider r= ρ0(F/L) | f | and see what value we get for r if λF | f | = ρp. For visible light (λ ~ 0.6 × 10-6 m) and typical atmospheric turbulence (C2n ~ 10-15 m-2/3, ε ~ 103 m) we find that r= 6.5 cos11/5 η. Therefore the requirement that λF | f | ≫ ρ0 means that r ≫ 6.5 cos11/3 η. For η = 80° we then find that r must be greater than 0.14. Next let us consider the restrictions placed on r by the condition λF | f | ≫K-1m. For the aforementioned atmospheric conditions this leads to the requirement that r≫0.025 cos8/5 η.
  14. J. Shapiro (private communication, 1975).

Fante, R.

R. Fante, J. Opt. Soc. Am. 66, 74 (1976).

Feizulin, Z.

Z. Feizulin and Y. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).

A. Kon and Z. Feizulin, Radiophys. Quantum Electron. 13, 51 (1970).

Gezari, D.

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

Gurvich, A.

A. Gurvich and V. Tatarski, Radio Sci. 10, 3 (1975).

Kon, A.

A. Kon and Z. Feizulin, Radiophys. Quantum Electron. 13, 51 (1970).

Korff, D.

D. Korff, J. Opt. Soc. Am. 63, 971 (1973).

Kravtsov, Y.

Z. Feizulin and Y. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).

Labeyrie, A.

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

Shapiro, J.

J. Shapiro (private communication, 1975).

Stachnik, R.

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

Tatarski, V.

A. Gurvich and V. Tatarski, Radio Sci. 10, 3 (1975).

Other (14)

Z. Feizulin and Y. Kravtsov, Radiophys. Quantum Electron. 10, 33 (1967).

A. Gurvich and V. Tatarski, Radio Sci. 10, 3 (1975).

A. Kon and Z. Feizulin, Radiophys. Quantum Electron. 13, 51 (1970).

Note that Eq. (4) has been obtained from Eq. (11) of Ref. 1, by using the fact that Dx + DS = D1 and the definition Dx(ρ-ρ′, ρ12)=〈[χ(ρ,ρ1)-χ(ρ′,ρ2)]2〉 =2σ2x-2Bx(ρ-ρ′,ρ12), where Bx(ρ-ρ′,ρ12)≡(χ(ρ,ρ1)χ(ρ′,ρ2)〉-〈χ〉2.

R. Fante, J. Opt. Soc. Am. 66, 74 (1976).

For weak turbulence the proof of this statement is readily seen from Eq. (5) of Ref. 5. For strong turbulence the proof follows from a realization that measured data2 indicate the field statistics are nearly log normal. If we compare the Markov approximation solution for 〈u〉 = 〈exp[χ+ iS]〉 with the solution obtained by assuming χ and S are Gaussian variables we find that 〈χS〉 = 0 is required for the two solutions to be identical. For nearly log-normal field statistics it can be shown that 〈χS〉 ≪ 1.

A formal definition for ε is given by ε2=(∫L0s2C2n(s)ds) (∫L0dsC2n(s))-1, where s=L - x is the distance measured from the surface of the earth. In this paper we shall use Hufnagel's model for C2n(s). For a description of this model see R. Hufnagel, OSA Topical Meeting on Propagation through Turbulence, Paper WA1-1, Boulder, Colo., July 1974.

For weak turbulence the proof of this statement is readily seen from Eq. (5) of Ref. 5. For strong turbulence the proof follows from a realization that measured data2 indicate the field statistics are nearly log normal. If we compare the Markov approximation solution for 〈u〉 = 〈exp[χ+ iS]〉 with the solution obtained by assuming χ and S are Gaussian variables we find that 〈χS〉 = 0 is required for the two solutions to be identical. For nearly log-normal field statistics it can be shown that 〈χS〉 ≪ 1.

A formal definition for ε is given by ε2=(∫L0s2C2n(s)ds) (∫L0dsC2n(s))-1, where s=L - x is the distance measured from the surface of the earth. In this paper we shall use Hufnagel's model for C2n(s). For a description of this model see R. Hufnagel, OSA Topical Meeting on Propagation through Turbulence, Paper WA1-1, Boulder, Colo., July 1974.

8From Eq. (19) is is quite clear that (27) is the appropriate form for all interferometer spacings such that DR ≫ ελκm = 5. 91ελ/l0. Assuming λ ~0.6 × 10-6 m, l0 ~ 10-3 m, and ε ~ 103 m, we see that for DR ≫3 m, ρN is the correct parameter. For DR ≪3 m it is correct to use the form in Eq. (24). It is therefore only for relatively small values of DR that the form used in Eq. (27) is only approximate.

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

D. Korff, J. Opt. Soc. Am. 63, 971 (1973).

Let us consider r= ρ0(F/L) | f | and see what value we get for r if λF | f | = ρp. For visible light (λ ~ 0.6 × 10-6 m) and typical atmospheric turbulence (C2n ~ 10-15 m-2/3, ε ~ 103 m) we find that r= 6.5 cos11/5 η. Therefore the requirement that λF | f | ≫ ρ0 means that r ≫ 6.5 cos11/3 η. For η = 80° we then find that r must be greater than 0.14. Next let us consider the restrictions placed on r by the condition λF | f | ≫K-1m. For the aforementioned atmospheric conditions this leads to the requirement that r≫0.025 cos8/5 η.

J. Shapiro (private communication, 1975).

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