Abstract

A rigorous method for modeling received power coupling efficiency (ηF/R) and transmitted power coupling efficiency (ηF/T) in a general-target-illumination ladar system is presented. For our analysis we concentrate on incorporating a single-mode optical fiber into the ladar return signal path. By developing expressions for both ηF/R and ηF/T for a simple, diffuse target, our model allows for varying range, beam size on target, target diameter, and coupling optics. Through numerical analysis ηF/R is shown to increase as the range to target increases and decrease as target diameter increases, and ηF/T is shown to decrease with target range. A baseline signal-to-noise ratio analysis of the system is also provided for varying illumination schemes.

© 1998 Optical Society of America

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References

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  1. M. S. Salisbury, “Sensitivity improvement of a 1 µm ladar system incorporating an optical fiber preamplifier,” Opt. Eng. 32, 2671–2680 (1993).
    [CrossRef]
  2. D. K. Jacob, M. B. Martin, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
    [CrossRef]
  3. H. Hasson, R. Wendt, S. R. Czyzak, “Overview of the field ladar demonstration program developing high-resolution imaging and remote sensing,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 294–308 (1996).
    [CrossRef]
  4. A. W. Synder, “Excitation and scattering of modes on a dielectric of optical fiber,” IEEE Trans. Microwave Theory Tech. MTT-17, 1138–1144 (1969).
    [CrossRef]
  5. J. C. Dainty, Topics in Applied Physics, Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1984).
  6. F. G. Stemler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, New York, 1990).
  7. D. Marcuse, “Loss analysis of singlemode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
    [CrossRef]
  8. J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995).
  9. L. B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, New York, 1990).
  10. F. L. Pedrotti, S. J., L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  11. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  12. J. H. Shapiro, “Target reflectivity theory for coherent laser radar,” Appl. Opt. 21, 3398–3408 (1982).
    [CrossRef] [PubMed]
  13. E. W. Swokowski, Calculus with Analytic Geometry, 4th ed. (PWS-Kent, Boston, 1988).
  14. M. J. Missy, “Analysis of and applications for a liquid crystal optical phased array,” M.S. thesis (University of Dayton, Dayton, Ohio, 1996).
  15. J. A. Overbeck, M. S. Salisbury, M. M. Mark, E. A. Watson, “Required energy for a laser radar system incorporating a fiber amplifier or an avalanche photodiode,” Appl. Opt. 34, 7724–7730 (1995).
    [CrossRef] [PubMed]

1995

D. K. Jacob, M. B. Martin, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

J. A. Overbeck, M. S. Salisbury, M. M. Mark, E. A. Watson, “Required energy for a laser radar system incorporating a fiber amplifier or an avalanche photodiode,” Appl. Opt. 34, 7724–7730 (1995).
[CrossRef] [PubMed]

1993

M. S. Salisbury, “Sensitivity improvement of a 1 µm ladar system incorporating an optical fiber preamplifier,” Opt. Eng. 32, 2671–2680 (1993).
[CrossRef]

1982

1977

D. Marcuse, “Loss analysis of singlemode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

1969

A. W. Synder, “Excitation and scattering of modes on a dielectric of optical fiber,” IEEE Trans. Microwave Theory Tech. MTT-17, 1138–1144 (1969).
[CrossRef]

Buck, J. A.

J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995).

Czyzak, S. R.

H. Hasson, R. Wendt, S. R. Czyzak, “Overview of the field ladar demonstration program developing high-resolution imaging and remote sensing,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 294–308 (1996).
[CrossRef]

Dainty, J. C.

J. C. Dainty, Topics in Applied Physics, Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1984).

Duncan, B. D.

D. K. Jacob, M. B. Martin, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Hasson, H.

H. Hasson, R. Wendt, S. R. Czyzak, “Overview of the field ladar demonstration program developing high-resolution imaging and remote sensing,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 294–308 (1996).
[CrossRef]

J., S.

F. L. Pedrotti, S. J., L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Jacob, D. K.

D. K. Jacob, M. B. Martin, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Jeunhomme, L. B.

L. B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, New York, 1990).

Marcuse, D.

D. Marcuse, “Loss analysis of singlemode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Mark, M. M.

Martin, M. B.

D. K. Jacob, M. B. Martin, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Missy, M. J.

M. J. Missy, “Analysis of and applications for a liquid crystal optical phased array,” M.S. thesis (University of Dayton, Dayton, Ohio, 1996).

Overbeck, J. A.

Pedrotti, F. L.

F. L. Pedrotti, S. J., L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Pedrotti, L. S.

F. L. Pedrotti, S. J., L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Salisbury, M. S.

Shapiro, J. H.

Stemler, F. G.

F. G. Stemler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, New York, 1990).

Swokowski, E. W.

E. W. Swokowski, Calculus with Analytic Geometry, 4th ed. (PWS-Kent, Boston, 1988).

Synder, A. W.

A. W. Synder, “Excitation and scattering of modes on a dielectric of optical fiber,” IEEE Trans. Microwave Theory Tech. MTT-17, 1138–1144 (1969).
[CrossRef]

Watson, E. A.

Wendt, R.

H. Hasson, R. Wendt, S. R. Czyzak, “Overview of the field ladar demonstration program developing high-resolution imaging and remote sensing,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 294–308 (1996).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

D. Marcuse, “Loss analysis of singlemode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

A. W. Synder, “Excitation and scattering of modes on a dielectric of optical fiber,” IEEE Trans. Microwave Theory Tech. MTT-17, 1138–1144 (1969).
[CrossRef]

Opt. Eng.

M. S. Salisbury, “Sensitivity improvement of a 1 µm ladar system incorporating an optical fiber preamplifier,” Opt. Eng. 32, 2671–2680 (1993).
[CrossRef]

D. K. Jacob, M. B. Martin, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Other

H. Hasson, R. Wendt, S. R. Czyzak, “Overview of the field ladar demonstration program developing high-resolution imaging and remote sensing,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 294–308 (1996).
[CrossRef]

J. C. Dainty, Topics in Applied Physics, Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1984).

F. G. Stemler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, New York, 1990).

J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995).

L. B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, New York, 1990).

F. L. Pedrotti, S. J., L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

E. W. Swokowski, Calculus with Analytic Geometry, 4th ed. (PWS-Kent, Boston, 1988).

M. J. Missy, “Analysis of and applications for a liquid crystal optical phased array,” M.S. thesis (University of Dayton, Dayton, Ohio, 1996).

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Figures (11)

Fig. 1
Fig. 1

Imaging lens system used for the geometric analysis.

Fig. 2
Fig. 2

Geometric received power coupling efficiency (ηF/R) versus target range.

Fig. 3
Fig. 3

Target illumination/return for (a) unresolved-target, multimode return, (b) unresolved-target, single-mode return, (c) resolved-target, multimode return, and (d) resolved-target, single-mode return.

Fig. 4
Fig. 4

General-illumination ladar system demonstrating a multimode return from an unresolved target.

Fig. 5
Fig. 5

Circ function overlap.

Fig. 6
Fig. 6

Geometry used for calculating the focal length for the negative lens l2.

Fig. 7
Fig. 7

Received power coupling efficiency (ηF/R) versus target range for a target diameter of 0.3 m.

Fig. 8
Fig. 8

Flood-illuminated target received power coupling efficiency (ηF/R) versus target diameter at a constant range of 20 km.

Fig. 9
Fig. 9

Transmitted power coupling efficiency (ηF/T) versus target range for a resolved-target, multimode return.

Fig. 10
Fig. 10

Signal-to-noise ratio (SNR) versus target range for a resolved-target, single-mode return.

Fig. 11
Fig. 11

SNR versus target range for a large-resolved-target, multimode return. The beam diameter in the target plane has been set to 15 m.

Equations (55)

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θos=Dt/L,
θis=MDt/L.
At=reff2πθisfIL22=πMDtfIL2L2=πreff2,
P˜R=|U˜f(ρ¯f)|2πMDtfIL2L2,
P˜sigdρ¯fU˜f(ρ¯f)U01*(ρ¯f)2.
U01(ρ¯f)2πω2 exp-|ρ¯f|2ω2,
ωrc0.65+1.619V3/2+2.879V6,
V=(2πrc/λ)NA.
P˜sig=|U˜f(ρ¯f)|22πω21-exp-rc2ω22.
ηF/R=8(Lω)2(MDtfIL)21-exp-rc2ω22.
f/#=12NA=MfILDR,
P˜sigdρ¯f U˜f(ρ¯f)U01*(ρ¯f)2dρ¯f|U˜f(ρ¯f)|2dρ¯f|U01*(ρ¯f)|2.
P˜sigdρ¯f|U˜f(ρ¯f)|2=|U˜f(ρ¯f)|20r eff02πr drdθ=PR.
DDLS=2.44λL/DR,
h(ρ¯t-ρ¯)=exp(ikL)iλLexpik2L|ρ¯t-ρ¯|2,
Ut(ρ¯t)=Atransdρ ¯Utrans(ρ¯)expik2 f2|ρ¯|2h(ρ¯t-ρ¯),
U˜R(ρ¯R)=Atdρ¯t T˜(ρ¯t)Ut(ρ¯t)h(ρ¯R-ρ¯t),
U˜f(ρ¯f)=ARdρ¯R U˜R(ρ¯R)exp-ik2 f3|ρ¯R|2h(ρ¯f-ρ¯R),
U˜f(ρ¯f)=exp(i2kL)exp(ikf3)(iλf3)(λL)2expik2 f3|ρ¯f|2×dρ¯R WR(ρ¯R)exp-ikf3ρ¯f·ρ¯R×Atdρ¯t T˜(ρ¯t)×expik2L(|ρ¯t|2+|ρ¯R|2-2ρ¯t·ρ¯R)×Atransdρ ¯Utrans(ρ¯)expik2 f2|ρ¯|2×expik2L(|ρ¯|2+|ρ¯t|2-2ρ·¯ρ¯t).
U˜f(ρ¯f)=exp[ik(2L+f3)](iλf3)(λL)2Atdρ¯t T˜(ρ¯t)expi2πλL|ρ¯t|2×Atransdρ¯Utrans(ρ¯)exp-i2πλLρ·¯ρ¯t×dρ¯R WR(ρ¯R)expiπλL|ρ¯R|2×exp-i2πρ¯R·ρ¯tλL+ρ¯fλf3,
Utrans(ρ¯)=Utrans(ρ¯)expik2 f2|ρ¯|2.
E(P˜sig)1(λf3)2(λL)4Atdρ¯t1dρ¯t2E[T˜(ρ¯t1)T˜*(ρ¯t2)]×expi2πλL|ρ¯t1|2-|ρ¯t2|2×Utransρ¯tλL2dρ¯f U01*(ρ¯f)×dρ¯RWR(ρ¯R)expiπλL|ρ¯R|2×exp-i2πρ¯R·ρ¯tλL+ρ¯fλf32,
E[T˜(ρ¯t1)]=0,
E[T˜(ρ¯t1)T˜(ρ¯t2)]=0,
E[T˜(ρ¯t1)T˜*(ρ¯t2)]=λ2T0(ρ¯t1)δ(ρ¯t1-ρ¯t2),
E(P˜sig)1f32(λL)4Atdρ¯tT0(ρ¯t)×Utransρ¯tλL2dρ¯RU01*ρ¯Rλf3WR(ρ¯R)×expiπλL|ρ¯R|2expi2πλLρ¯R·ρ¯t×dρ¯RU01ρ¯Rλf3WR*(ρ¯R)exp-iπλL|ρ¯R|2×exp-i2πλLρ¯R·ρ¯t,
ρ¯0ρ¯R+ρ¯R2,Δρ¯ρ¯R-ρ¯R
dρ¯Rdρ¯R=dρ¯0dΔρ¯,
E(P˜sig)T0f32(λL)4dΔρ¯dρ¯tWt(ρ¯t)×Utransρ¯tλL2 exp-i2πλLρ¯t·Δρ¯×dρ¯0U01*-ρ¯0λf3+Δρ¯2λf3×U01-ρ¯0λf3-Δρ¯2λf3WRρ¯0-12Δρ¯×WR*ρ¯0+12Δρ¯exp-i2πλLρ¯0·Δρ¯.
Utrans(ρ¯)2Ptransπω02 exp-|ρ¯|2ω02.
Utrans(ρ¯)2Ptransπω02 exp-|ρ¯|2ω02expik2 f2|ρ¯|2.
Utransρ¯tλL2=8πPtransf22ω024f22+(kω02)2×exp-8π2f22ω02(λL)2[4f22+(kω02)2]|ρ¯t|2.
Wt(ρ¯t)=circρ¯tDt1,|ρ¯t|Dt/20,|ρ¯t|>Dt/2.
WR(ρ¯)=circρ¯DR1,|ρ¯|DR/20,|ρ¯|>DR/2.
E(P˜sig)4NT0PtransDR2Dt2a2L2×dΔρ ¯exp(-a2|Δρ¯|2)×dρ¯tcirc(ρ¯t)exp(-Nπ2Dt2|ρ¯t|2)×exp-i2πDRDtλLρ¯t·Δρ¯×dρ¯0circρ¯0-12Δρ¯×circρ¯0+12Δρ¯exp(-4a2|ρ¯0|2)×exp-i2πDRλLρ¯0·Δρ¯,
N8f22ω02(λL)2[4f22+(kω02)2],
a212πωDRλf32=2πωNAopticsλ2.
E(P˜sig)4NT0PtransDR2Dt2a2L2×dΔρ¯exp(-a2|Δρ¯|2)×dρ¯t circ(ρ¯t)×exp(-Nπ2Dt2|ρ¯t|2)cos2πDRDtλLρ¯t·Δρ¯×dρ¯0circρ¯0-12Δρ¯circρ¯0+12Δρ¯×exp(-4a2|ρ¯0|2)cos2πDRλLρ¯0·Δρ¯.
Δρ¯r(cos θ)xˆ+r(sin θ)yˆ,ρ¯txtxˆ+ytyˆ,
ρ¯0xxˆ+yyˆ,
ρ¯t·Δρ¯=rxt cos θ+ryt sin θ,
ρ¯0·Δρ¯=rx cos θ+ry sin θ.
E(P˜sig)128πNT0PtransDR2Dt2a2L2×01dr r exp(-a2r2)×01/2dxt exp(-Nπ2Dt2xt2)×0(1/4-xt2)1/2dyt exp(-Nπ2Dt2yt2)×cos2πDRDtrytλL0(1/2)(1-r2)1/2dx×exp(-4a2x2)0(1/4-x2)1/2-r/2dy ×exp(-4a2y2)cos2πDR2ryλL.
U˜R(ρ¯R)=exp(i2kL)(λL)2Atdρ¯tT˜(ρ¯t)×expik2L(|ρ¯t|2+|ρ¯R|2-2ρ¯t·ρ¯R)×Atransdρ¯Utrans(ρ¯)×expik2 f2|ρ¯|2×expik2L(|ρ¯t|2-2ρ·¯ρ¯t),
I˜R=1(λL)4Atdρ¯tT˜(ρ¯t)expikL|ρ¯t|2×exp-ikLρ¯R·ρ¯tUtransρ¯tλL×Atdρ¯tT˜*(ρ¯t)exp-ikL|ρ¯t|2×expikLρ¯R·ρ¯tUtrans*ρ¯tλL,
E(I˜R)=NT0πPtransL2dρ¯t circρ¯tDtexp(-Nπ2|ρ¯t|2),
E(I˜R)=T0PtransL21-exp-NDt2π24.
E(P˜R)=T0PtransπDR24L21-exp-NDt2π24.
ηF/R=E(P˜sig)E(P˜R)512Na2Dt21-exp-NDt2π24×01dr r exp(-a2r2)×01/2dxt exp(-Nπ2Dt2xt2)×0(1/4-xt2)1/2dyt exp(-Nπ2Dt2yt2)×cos2πDRDtrytλL×0(1/2)(1-r2)1/2dx exp(-4a2x2)×0(1/4-x2)1/2-r/2dy exp(-4a2y2)cos2πDR2ryλL.
f2=R(zeff)=zeff1+πωeff2λzeff2,
ω(zeff)=ωeff1+λzeffπωeff221/2,
ω(L+zeff)=ωeff1+λ(L+zeff)πωeff221/2.
zR=πωeff2/λ=3.42m.
ηF/T=E(P˜sig)Ptrans128NT0a2Dt2DR2πL2×01dr r exp(-a2r2)01/2dxt exp(-Nπ2Dt2xt2)×0(1/4-xt2)1/2dyt exp(-Nπ2Dt2yt2)×cos2πDRDtrytλL0(1/2)(1-r2)1/2dx exp(-4a2x2)×0(1/4-x2)1/2-r/2dy exp(-4a2y2)cos2πDR2ryλL.
SNR=(RJ0ηF/T)2T12eID+RJ0ηF/TT1+4kbTRL,

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