Abstract

For an ordinary individually addressable microlaser array, a separate control line is used for each microlaser, which requires a large number of control lines for even a small array. An organization that reduces the width of the control stream and simplifies packaging is matrix addressing, in which microlasers are arranged at the crossings of horizontal and vertical control lines. We consider the problem of decomposing arbitrary two-dimensional microlaser patterns into matrix-addressable patterns that are applied time sequentially to realize the target pattern. We present a mathematical model for the decomposition process and present an algorithm for optimal decomposition. We also consider bake factor, in which no more than N microlasers in a neighborhood of M (where N < M) are enabled, which avoids thermal overload by limiting the density of enabled microlasers. We conclude with a case study and show that, for completely arbitrary two-dimensional patterns, the average number of time-sequential patterns is less than the number of rows in a square array.

© 1996 Optical Society of America

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References

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  1. J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. 27, 1332–1346 (1991).
  2. R. A. Morgan, “Vertical cavity surface emitting lasers,” in Miniature and Micro-Optics and Micromechanics, N.C. Gallagher, C. S. Roychoudhuri, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1992 (1993).
  3. H. R. Nahata, M. J. Murdocca, “Decomposition method for matrix addressable microlaser arrays,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 26–28.
  4. E. J. McCluskey, Introduction to the Theory of Switching Circuits (McGraw-Hill, New York, 1965).
  5. T. H. Cormen, R. L. Rivest, C. E. Leiserson, Introduction to Algorithms (MIT, Cambridge, Mass., 1990).
  6. E. Balas, M. Padberg, “Set partitioning,” SIAM (Soc. Ind. Appl. Math.) Rev. 18, 710–760 (1976).
  7. E. Balas, “Some valid inequalities for the set partitioning problem,” in Studies in Integer Programming, I. L. Johnson, B. Korte, G. L. Nemhauser, eds. (North-Holland, Amsterdam, 1977), pp. 13–47.
  8. M. Bellmore, H. D. Ratliff, “Set covering and involutary bases,” Manag. Sci. 18, 194–206 (1971).
  9. G. H. Golub, C. F. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins Press U., Baltimore, Md., 1989).
  10. G. Strang, Linear Algebra and its Applications (Harcourt Brace Jovanovitch, San Diego, Calif., 1980).

1991 (1)

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. 27, 1332–1346 (1991).

1976 (1)

E. Balas, M. Padberg, “Set partitioning,” SIAM (Soc. Ind. Appl. Math.) Rev. 18, 710–760 (1976).

1971 (1)

M. Bellmore, H. D. Ratliff, “Set covering and involutary bases,” Manag. Sci. 18, 194–206 (1971).

Balas, E.

E. Balas, M. Padberg, “Set partitioning,” SIAM (Soc. Ind. Appl. Math.) Rev. 18, 710–760 (1976).

E. Balas, “Some valid inequalities for the set partitioning problem,” in Studies in Integer Programming, I. L. Johnson, B. Korte, G. L. Nemhauser, eds. (North-Holland, Amsterdam, 1977), pp. 13–47.

Bellmore, M.

M. Bellmore, H. D. Ratliff, “Set covering and involutary bases,” Manag. Sci. 18, 194–206 (1971).

Cormen, T. H.

T. H. Cormen, R. L. Rivest, C. E. Leiserson, Introduction to Algorithms (MIT, Cambridge, Mass., 1990).

Florez, L. T.

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. 27, 1332–1346 (1991).

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins Press U., Baltimore, Md., 1989).

Harbison, J. P.

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. 27, 1332–1346 (1991).

Jewell, J. L.

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. 27, 1332–1346 (1991).

Lee, Y. H.

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. 27, 1332–1346 (1991).

Leiserson, C. E.

T. H. Cormen, R. L. Rivest, C. E. Leiserson, Introduction to Algorithms (MIT, Cambridge, Mass., 1990).

McCluskey, E. J.

E. J. McCluskey, Introduction to the Theory of Switching Circuits (McGraw-Hill, New York, 1965).

Morgan, R. A.

R. A. Morgan, “Vertical cavity surface emitting lasers,” in Miniature and Micro-Optics and Micromechanics, N.C. Gallagher, C. S. Roychoudhuri, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1992 (1993).

Murdocca, M. J.

H. R. Nahata, M. J. Murdocca, “Decomposition method for matrix addressable microlaser arrays,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 26–28.

Nahata, H. R.

H. R. Nahata, M. J. Murdocca, “Decomposition method for matrix addressable microlaser arrays,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 26–28.

Padberg, M.

E. Balas, M. Padberg, “Set partitioning,” SIAM (Soc. Ind. Appl. Math.) Rev. 18, 710–760 (1976).

Ratliff, H. D.

M. Bellmore, H. D. Ratliff, “Set covering and involutary bases,” Manag. Sci. 18, 194–206 (1971).

Rivest, R. L.

T. H. Cormen, R. L. Rivest, C. E. Leiserson, Introduction to Algorithms (MIT, Cambridge, Mass., 1990).

Scherer, A.

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. 27, 1332–1346 (1991).

Strang, G.

G. Strang, Linear Algebra and its Applications (Harcourt Brace Jovanovitch, San Diego, Calif., 1980).

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins Press U., Baltimore, Md., 1989).

IEEE J. Quantum Electron. (1)

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, L. T. Florez, “Vertical-cavity surface-emitting lasers: design, growth, fabrication, characterization,” IEEE J. Quantum Electron. 27, 1332–1346 (1991).

Manag. Sci. (1)

M. Bellmore, H. D. Ratliff, “Set covering and involutary bases,” Manag. Sci. 18, 194–206 (1971).

SIAM (Soc. Ind. Appl. Math.) Rev. (1)

E. Balas, M. Padberg, “Set partitioning,” SIAM (Soc. Ind. Appl. Math.) Rev. 18, 710–760 (1976).

Other (7)

E. Balas, “Some valid inequalities for the set partitioning problem,” in Studies in Integer Programming, I. L. Johnson, B. Korte, G. L. Nemhauser, eds. (North-Holland, Amsterdam, 1977), pp. 13–47.

G. H. Golub, C. F. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins Press U., Baltimore, Md., 1989).

G. Strang, Linear Algebra and its Applications (Harcourt Brace Jovanovitch, San Diego, Calif., 1980).

R. A. Morgan, “Vertical cavity surface emitting lasers,” in Miniature and Micro-Optics and Micromechanics, N.C. Gallagher, C. S. Roychoudhuri, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1992 (1993).

H. R. Nahata, M. J. Murdocca, “Decomposition method for matrix addressable microlaser arrays,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 26–28.

E. J. McCluskey, Introduction to the Theory of Switching Circuits (McGraw-Hill, New York, 1965).

T. H. Cormen, R. L. Rivest, C. E. Leiserson, Introduction to Algorithms (MIT, Cambridge, Mass., 1990).

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Figures (8)

Fig. 1.
Fig. 1.

(a) Individually addressable microlaser array, (b) matrix-addressable microlaser array.

Fig. 2.
Fig. 2.

(a) Examples of symmetric patterns, (b) a binary matrix that is not a symmetric pattern.

Fig. 3.
Fig. 3.

Example of decomposition of a nonsymmetric pattern into three symmetric patterns.

Fig. 4.
Fig. 4.

Existence of decomposition shown as a sum of symmetric patterns that have only a single nonzero point.

Fig. 5.
Fig. 5.

Decomposition need not be unique.

Fig. 6.
Fig. 6.

ANDor Boolean algebra.

Fig. 7.
Fig. 7.

XOR Boolean algebra.

Fig. 8.
Fig. 8.

Decomposition results for randomly generated patterns on an 8 × 8 matrix. 1000 sample points are taken for each number of enabled microlasers (1–63).

Equations (34)

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( i , j ) , ( k , l ) p i , j = 1 ,      p i , l = 1 ,             p k , l = 1 ,      p k , j = 1 ,
= [ 1 1 0 0 ] , 𝒞 = [ 0 1 0 1 ] , Q = [ , 𝒞 ] = T × 𝒞 = [ 1 1 0 0 ] × [ 0 1 0 1 ] = [ 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 ]
[ 0 1 0 1 1 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 ] = [ 1    1    0    0    1    0 ] 𝒞 = [ 0    1    0    1    1    1 ] ,                          [ 0 0 0 0 0 1 0 0 0 ] = [ 0    1    0 ] 𝒞 = [ 0    0    1 ] .               
P 1 = P 2 1 = 2 , 𝒞 1 = 𝒞 2 .
[ 1 , s 0 0 0 ] ,
II × P × Φ = [ 1 , s 0 0 0 ] .
( 2 n 1 ) × ( 2 n 1 ) + 1 .           number of non - zero  R  vectors number of non - zero  C  vectors                the zero      symmetric pattern
( 2 n 2 ) ( 2 n 2 ) + 1 .
[ ( p , q ) 0 < p , q n p q = m ] ( n p ) ( n q ) .
1 = 2 { = 1 𝒞 = 𝒞 1 + 𝒞 2 , 𝒞 1 = 𝒞 2 { 𝒞 = 𝒞 1 = 1 + 2 .
min   w . p i s
i = 1 w P i = .
𝒜 = 𝒜 + .
P i P j P k I P i + P j P k .
[ 1 0 1 0 0 0 1 0 1 ] = [ 1 0 0 0 0 0 0 0 0 ] + [ 0 0 1 0 0 0 0 0 0 ] + [ 0 0 0 0 0 0 1 0 0 ]                         + [ 0 0 0 0 0 0 0 0 1 ] .
min 𝒞 2 l ( 𝒞 P 𝒞 P = h H h = ) .
min X { 0 , 1 } m ( W T X A X e ) ,
w k t ( P ) = x = 1 n k + 1 y = 1 n k + 1 ( i = x x + k j = y y + k p i , j ) t ,
w 2 3 [ 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 ] = 30 ,     w 2 3 [ 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 ] = 153 .
w k , T ( P ) = x = 1 n k + 1 y = 1 n k + 1  min [ 1 , max ( 0 , i = x x + k j = y y + k p i , j T ) ] ,
w 2 , 3 [ 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 ] = 0 ,     w 2 , 3 [ 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 ] = 2 .
[ 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 ] = [ 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 1 ] [ 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 ] .
𝒰 = [ u 1 , u 2 , ·    ·    ·    , u m ] m × m 𝒱 = [ v 1 , v 2 , ·    ·    ·    , v n ] n × n ,
𝒰 T 𝒜 𝒱 = = diag ( σ 1 , σ 2 , ·    ·    ·    , σ p ) m × n ,            P = min { m , n } ,   
σ 1 σ 2       σ p 0 ;
σ 1 σ 2       σ r > σ r + 1 =       σ p = 0 ,
𝒜 = i = 1 r σ i u i v i T .
σ 1 = σ 2 =       σ r = 1 , σ r + 1 =       σ p = 0 .
[ 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 ] .
[ 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 ] + [ 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ] + [ 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 ] .
[ 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 1 ] [ 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 ] .
[ 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 ] .
[ 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 ] + [ 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 ] .
[ 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 ] [ 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 ] [ 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 ] .

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