Abstract

An investigation of the array-tilt aberration for hexagonal, optical phased arrays is presented. The investigation begins with theoretical derivations of the far-zone radiated field, the array factor, and the far-field radiated power for the seven-element hexagonal array with array tilt present. Physical insights gained from this analysis are discussed. An analytical treatment of correlation-based array-tilt estimators is also undertaken. Two novel array-tilt estimation techniques are developed from the analysis. The new techniques are shown to be significantly more efficient computationally than the traditional estimation approach. Simulation and experimental results are presented to validate the new array-tilt estimation methods.

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References

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  1. S. E. Lamberson, “The airborne laser,” Proc. SPIE 4760, 25–33 (2002).
    [CrossRef]
  2. D. T. Kyrazis, “Airborne laser laboratory departure from Kirtland Air Force Base and a brief history of aero-optics,” Opt. Eng. 52, 071403 (2013).
    [CrossRef]
  3. G. Forden, “The airborne laser,” IEEE Spectrum 34, 40–49 (1997).
    [CrossRef]
  4. B. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50  W coherently combined passive laser array,” Opt. Lett. 34, 863–865 (2009).
    [CrossRef]
  5. T. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
    [CrossRef]
  6. R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).
  7. P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
    [CrossRef]
  8. R. J. Pawlak, “Recent developments and near term directions for Navy Laser Weapons System (LaWS) testbed,” Proc. SPIE 8547, 854705 (2012).
    [CrossRef]
  9. P. Bondyopadhyay, “The first application of array antenna,” in IEEE International Conference on Phased Array Systems and Technology (IEEE, 2000), pp. 29–32.
  10. A. Flores, T. Shay, C. Lu, C. Robin, B. Pulford, A. Sanchez, D. Hult, and K. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CFE3.
  11. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14, 12188–12195 (2006).
    [CrossRef]
  12. G. A. Tyler, “Accommodation of speckle in object-based phasing,” J. Opt. Soc. Am. A 29, 722–733 (2012).
    [CrossRef]
  13. M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
    [CrossRef]
  14. C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. (Wiley, 2005).
  15. M. I. Skolnik, Introduction to Radar Systems, 3rd ed. (McGraw-Hill, 2001).
  16. Boulder Nonlinear Systems, Inc., Spatial Light Modulators—XY Series (Retrieved November16, 2013.
  17. Lumenera Corporation, Lu120, Lu125 1.3 Megapixel USB 2.0 Camera (Retrieved November16, 2013 from http://www.lumenera.com/resources/documents/datasheets/industrial/Lu120-Lu125-datasheet.pdf ).

2013 (1)

D. T. Kyrazis, “Airborne laser laboratory departure from Kirtland Air Force Base and a brief history of aero-optics,” Opt. Eng. 52, 071403 (2013).
[CrossRef]

2012 (3)

R. J. Pawlak, “Recent developments and near term directions for Navy Laser Weapons System (LaWS) testbed,” Proc. SPIE 8547, 854705 (2012).
[CrossRef]

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[CrossRef]

G. A. Tyler, “Accommodation of speckle in object-based phasing,” J. Opt. Soc. Am. A 29, 722–733 (2012).
[CrossRef]

2009 (1)

2008 (1)

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

2006 (1)

2005 (1)

T. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

2002 (1)

S. E. Lamberson, “The airborne laser,” Proc. SPIE 4760, 25–33 (2002).
[CrossRef]

1997 (1)

G. Forden, “The airborne laser,” IEEE Spectrum 34, 40–49 (1997).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. (Wiley, 2005).

Bennai, B.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

Berdine, R. W.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

Bondyopadhyay, P.

P. Bondyopadhyay, “The first application of array antenna,” in IEEE International Conference on Phased Array Systems and Technology (IEEE, 2000), pp. 29–32.

Bourdon, P.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

Canat, G.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

Fan, T.

T. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

Flores, A.

A. Flores, T. Shay, C. Lu, C. Robin, B. Pulford, A. Sanchez, D. Hult, and K. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CFE3.

Forden, G.

G. Forden, “The airborne laser,” IEEE Spectrum 34, 40–49 (1997).
[CrossRef]

Hult, D.

A. Flores, T. Shay, C. Lu, C. Robin, B. Pulford, A. Sanchez, D. Hult, and K. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CFE3.

Hyde, M. W.

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[CrossRef]

Jaouen, Y.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

Jolivet, V.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

Kyrazis, D. T.

D. T. Kyrazis, “Airborne laser laboratory departure from Kirtland Air Force Base and a brief history of aero-optics,” Opt. Eng. 52, 071403 (2013).
[CrossRef]

Lamberson, S. E.

S. E. Lamberson, “The airborne laser,” Proc. SPIE 4760, 25–33 (2002).
[CrossRef]

Lombard, L.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

Lu, C.

A. Flores, T. Shay, C. Lu, C. Robin, B. Pulford, A. Sanchez, D. Hult, and K. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CFE3.

Mies, E.

Minden, M.

Motes, R. A.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

Pawlak, R. J.

R. J. Pawlak, “Recent developments and near term directions for Navy Laser Weapons System (LaWS) testbed,” Proc. SPIE 8547, 854705 (2012).
[CrossRef]

Pourtal, E.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

Pulford, B.

A. Flores, T. Shay, C. Lu, C. Robin, B. Pulford, A. Sanchez, D. Hult, and K. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CFE3.

Robin, C.

A. Flores, T. Shay, C. Lu, C. Robin, B. Pulford, A. Sanchez, D. Hult, and K. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CFE3.

Rowland, K.

A. Flores, T. Shay, C. Lu, C. Robin, B. Pulford, A. Sanchez, D. Hult, and K. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CFE3.

Sanchez, A.

B. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50  W coherently combined passive laser array,” Opt. Lett. 34, 863–865 (2009).
[CrossRef]

A. Flores, T. Shay, C. Lu, C. Robin, B. Pulford, A. Sanchez, D. Hult, and K. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CFE3.

Shakir, S. A.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

Shay, T.

A. Flores, T. Shay, C. Lu, C. Robin, B. Pulford, A. Sanchez, D. Hult, and K. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CFE3.

Shay, T. M.

Skolnik, M. I.

M. I. Skolnik, Introduction to Radar Systems, 3rd ed. (McGraw-Hill, 2001).

Spencer, M. F.

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[CrossRef]

Tyler, G. A.

Vasseur, O.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

Wang, B.

IEEE J. Sel. Top. Quantum Electron. (1)

T. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

IEEE Spectrum (1)

G. Forden, “The airborne laser,” IEEE Spectrum 34, 40–49 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

D. T. Kyrazis, “Airborne laser laboratory departure from Kirtland Air Force Base and a brief history of aero-optics,” Opt. Eng. 52, 071403 (2013).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (4)

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[CrossRef]

S. E. Lamberson, “The airborne laser,” Proc. SPIE 4760, 25–33 (2002).
[CrossRef]

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

R. J. Pawlak, “Recent developments and near term directions for Navy Laser Weapons System (LaWS) testbed,” Proc. SPIE 8547, 854705 (2012).
[CrossRef]

Other (7)

P. Bondyopadhyay, “The first application of array antenna,” in IEEE International Conference on Phased Array Systems and Technology (IEEE, 2000), pp. 29–32.

A. Flores, T. Shay, C. Lu, C. Robin, B. Pulford, A. Sanchez, D. Hult, and K. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CFE3.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. (Wiley, 2005).

M. I. Skolnik, Introduction to Radar Systems, 3rd ed. (McGraw-Hill, 2001).

Boulder Nonlinear Systems, Inc., Spatial Light Modulators—XY Series (Retrieved November16, 2013.

Lumenera Corporation, Lu120, Lu125 1.3 Megapixel USB 2.0 Camera (Retrieved November16, 2013 from http://www.lumenera.com/resources/documents/datasheets/industrial/Lu120-Lu125-datasheet.pdf ).

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of a seven-element hexagonal array composed of identical circular elements. The diameter of each element is d; the spacing between adjacent elements is D. For the reader’s convenience, the (x,y) element center coordinates are annotated on the figure.

Fig. 2.
Fig. 2.

Γ(0.4,0.15|W˜x,W˜y) for the (a) seven-element hexagonal and (b) 2×2 rectangular arrays.

Fig. 3.
Fig. 3.

True (Wx=0.4 and Wy=0.15) and matched (W˜x=0.38 and W˜y=0.36) I2×2—(a) and (e), respectively; I3×1—(b) and (f), respectively; Ic—(c) and (g), respectively; and Irad—(d) and (h), respectively. The element spacing D=1.1d.

Fig. 4.
Fig. 4.

Parallel method for estimating x and y array tilts.

Fig. 5.
Fig. 5.

Serial method for estimating x and y array tilts.

Fig. 6.
Fig. 6.

Mean and standard deviation of the error versus mean counts K¯.

Fig. 7.
Fig. 7.

Example simulation trial for K¯=0.25—(a) Imeas, (b) Irad traditional, (c) Irad method 1, and (d) Irad method 2. The true value of array tilt is annotated on (a); the estimated values are annotated on (b), (c), and (d).

Fig. 8.
Fig. 8.

Schematic of the experimental setup.

Fig. 9.
Fig. 9.

Example experimental trial—(a) Imeas, (b) Irad traditional, (c) Irad method 1, and (d) Irad method 2. The true value of array tilt is annotated on (a); the estimated values are annotated on (b), (c), and (d).

Tables (2)

Tables Icon

Table 1. Simulation Array-Tilt Estimation Results

Tables Icon

Table 2. Experimental Array-Tilt Estimation Results

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

Erad(r)jkejkrrerad[(d2)2J1(kd2sinθ)kd2sinθ]AF(θ,ϕ),
AF=i=17ejφiδ(rir)ejkr^·rd3r=i=17ejφiejkr^·ri,
φi=2φxD3xi+2φyDyi,
AF=m=22n=11amn(ej2πWymejkr^·y^D2m)(ej2πWxnejkr^·x^D32n),
η1=(01010)Tξ1=(101)Tη2=(10101)Tξ2=(010)T.
AF=(m=22η1mej2πWymejkr^·y^D2m)(n=11ξ1nej2πWxnejkr^·x^D32n)+(m=22η2mej2πWymejkr^·y^D2m)(n=11ξ2nej2πWxnejkr^·x^D32n)=AFy2×2AFx2×2+AFy3×1AFx3×1.
Wx=Wx+p+q2Wy=Wy+pq2,
AF19=AFy5×1AFx5×1+AFy4×2AFx4×2+AFy3×2AFx3×2AF37=AFy7×1AFx7×1+AFy6×2AFx6×2+AFy5×2AFx5×2+AFy4×2AFx4×2,
Irad(r)Erad(r)·Erad*(r)k2r2[(d2)2J1(kd2sinθ)kd2sinθ]2(erad·erad*)AF(θ,ϕ)AF*(θ,ϕ)Iel(θ)|AF(θ,ϕ)|2,
Irad(r)Iel(θ)|AFx2×2|2|AFy2×2|2+Iel(θ)|AFy3×1|2+Iel(θ)2Re(AFx2×2AFy2×2AFy3×1*).
I19rad(r)Iel(θ)|AFx4×2|2|AFy4×2|2+Iel(θ)|AFx3×2|2|AFy3×2|2+Iel(θ)|AFy5×1|2+Iel(θ)2Re(AFx4×2AFy4×2AFy5×1*)+Iel(θ)2Re(AFx3×2AFy3×2AFy5×1*)+Iel(θ)2Re(AFx4×2AFy4×2AFx3×2*AFy3×2*),
arg maxW˜x,W˜y(0.5,0.5]Γ(Wx,Wy|W˜x,W˜y),
Γ=Irad(ρ,L|Wx,Wy)Irad(ρ,L|W˜x,W˜y)d2ρ.
e=|WxW˜x|+|WyW˜y|,

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