Abstract

Based on the vector Rayleigh–Sommerfeld far-field diffraction integral formula, a vector model for describing the far field of a laser diode is proposed beyond the paraxial approximation. Through the analysis of the error yielded by the scalar approximation to the properties of laser diode beams, it is found that the error is only correlative with propagation distance and the coordinate perpendicular to the junction plane, and there is a certain relationship between the error and a space angle.

© 2006 Optical Society of America

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References

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  1. X. Zheng, Zhejun Feng, and Yuying An, "Far-field expression of a high-power laser diode," Appl. Opt. 43, 5168-5172 (2004).
    [CrossRef]
  2. X. Zheng and A. Naqwi, "Far-field distribution of double-heterostructure diode laser beams," Appl. Opt. 33, 4491-4494 (1993).
    [CrossRef]
  3. W. W. Anderson, "Mode confinement and gain in junction lasers," IEEE J. Quantum Electron. QE-1, 228-236 (1965).
    [CrossRef]
  4. T. H. Zachos and J. E. Ripper, "Resonant modes of GaAs junction lasers," IEEE J. Quantum Electron. QE-5, 29-37 (1969).
    [CrossRef]
  5. T. L. Paoli, "Waveguiding in a stripe-geometry junction laser," IEEE J. Quantum Electron. QE-13, 662-668 (1997).
  6. X. Zheng and Yuying An, "Error analysis for the far-field of a small plane source," Opt. Commun. 222, 137-141 (2003).
    [CrossRef]
  7. S. Nemoto, "Experimental evaluation of a new expression for the far field of a diode laser beam," Appl. Opt. 33, 6387-6392 (1994).
    [CrossRef] [PubMed]

2004 (1)

2003 (1)

X. Zheng and Yuying An, "Error analysis for the far-field of a small plane source," Opt. Commun. 222, 137-141 (2003).
[CrossRef]

1997 (1)

T. L. Paoli, "Waveguiding in a stripe-geometry junction laser," IEEE J. Quantum Electron. QE-13, 662-668 (1997).

1994 (1)

1993 (1)

X. Zheng and A. Naqwi, "Far-field distribution of double-heterostructure diode laser beams," Appl. Opt. 33, 4491-4494 (1993).
[CrossRef]

1969 (1)

T. H. Zachos and J. E. Ripper, "Resonant modes of GaAs junction lasers," IEEE J. Quantum Electron. QE-5, 29-37 (1969).
[CrossRef]

1965 (1)

W. W. Anderson, "Mode confinement and gain in junction lasers," IEEE J. Quantum Electron. QE-1, 228-236 (1965).
[CrossRef]

An, Yuying

X. Zheng, Zhejun Feng, and Yuying An, "Far-field expression of a high-power laser diode," Appl. Opt. 43, 5168-5172 (2004).
[CrossRef]

X. Zheng and Yuying An, "Error analysis for the far-field of a small plane source," Opt. Commun. 222, 137-141 (2003).
[CrossRef]

Anderson, W. W.

W. W. Anderson, "Mode confinement and gain in junction lasers," IEEE J. Quantum Electron. QE-1, 228-236 (1965).
[CrossRef]

Feng, Zhejun

Naqwi, A.

X. Zheng and A. Naqwi, "Far-field distribution of double-heterostructure diode laser beams," Appl. Opt. 33, 4491-4494 (1993).
[CrossRef]

Nemoto, S.

Paoli, T. L.

T. L. Paoli, "Waveguiding in a stripe-geometry junction laser," IEEE J. Quantum Electron. QE-13, 662-668 (1997).

Ripper, J. E.

T. H. Zachos and J. E. Ripper, "Resonant modes of GaAs junction lasers," IEEE J. Quantum Electron. QE-5, 29-37 (1969).
[CrossRef]

Zachos, T. H.

T. H. Zachos and J. E. Ripper, "Resonant modes of GaAs junction lasers," IEEE J. Quantum Electron. QE-5, 29-37 (1969).
[CrossRef]

Zheng, X.

X. Zheng, Zhejun Feng, and Yuying An, "Far-field expression of a high-power laser diode," Appl. Opt. 43, 5168-5172 (2004).
[CrossRef]

X. Zheng and Yuying An, "Error analysis for the far-field of a small plane source," Opt. Commun. 222, 137-141 (2003).
[CrossRef]

X. Zheng and A. Naqwi, "Far-field distribution of double-heterostructure diode laser beams," Appl. Opt. 33, 4491-4494 (1993).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (3)

W. W. Anderson, "Mode confinement and gain in junction lasers," IEEE J. Quantum Electron. QE-1, 228-236 (1965).
[CrossRef]

T. H. Zachos and J. E. Ripper, "Resonant modes of GaAs junction lasers," IEEE J. Quantum Electron. QE-5, 29-37 (1969).
[CrossRef]

T. L. Paoli, "Waveguiding in a stripe-geometry junction laser," IEEE J. Quantum Electron. QE-13, 662-668 (1997).

Opt. Commun. (1)

X. Zheng and Yuying An, "Error analysis for the far-field of a small plane source," Opt. Commun. 222, 137-141 (2003).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Facet of a laser chip and the related coordinate system.

Fig. 2
Fig. 2

Comparison of the intensity profile among the results calculated by Eqs. (20)–(22).

Fig. 3
Fig. 3

Variation of the scalar approximation error with respect to x and z.

Equations (26)

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2 E ( r ) + k 2 E ( r ) = 0 ,
E x ( r ) = 1 2 π s E x ( r ) G ( r , r ) z  d s ,
E y ( r ) = - 1 2 π s E y ( r ) G ( r , r ) z d s ,
E z ( r ) = 1 2 π s [ E x ( r ) G ( r , r ) x + E y ( r ) G ( r , r ) y ] d s ,
G ( r , r ) = exp ( i k | r r | ) / | r r | ,
k | r r | 1 , 1 | r r | 1 | r | ,
| r r | = | r | ( | r | ) r = | r | 1 | r | r r .
E x ( x , y , z ) = i z exp ( i k r ) λ r 2 + + E x ( x , y ) × exp [ i k r ( x x + y y ) ] d x d y ,
E y ( x , y , z ) = i z exp ( i k r ) λ r 2 + + E y ( x , y ) × exp [ i k r ( x x + y y ) ] d x d y ,
E z ( x , y , z ) = i exp ( i k r ) λ r 2 + + [ E x ( x , y ) ( x x ) + E y ( x , y ) ( y y ) ] × exp [ i k r ( x x + y y ) ] d x d y ,
E x ( x , y ) = E 0 exp ( p | x | q y 2 ) ,
E y ( x , y ) = 0 ,
E x ( x , y , z ) = i z E 0 λ r 2 π q exp ( k 2 y 2 4 q r 2 ) 2 p r 2 p 2 r 2 + k 2 x 2 × exp ( i k r ) ,
E y ( x , y , z ) = 0 ,
E z ( x , y , z ) = i E 0 λ π q exp ( k 2 y 2 4 q r 2 ) [ 2 p x p 2 r 2 + k 2 x 2 + 4 p k x r i ( p 2 r 2 + k 2 x 2 ) 2 ] exp ( i k r ) .
I x ( x , y , z ) = z 2 u 0     2 π λ 2 q exp ( k 2 y 2 2 q r 2 ) ( 2 p p 2 r 2 + k 2 x 2 ) 2 ,
I y ( x , y , z ) = 0 ,
I z ( x , y , z ) = u 0     2 π λ 2 q exp ( k 2 y 2 2 q r 2 ) { [ 2 p x p 2 r 2 + k 2 x 2 ] 2 + [ 4 p k x r ( p 2 r 2 + k 2 x 2 ) 2 ] 2 } ,
I ( x , y , z ) = I x ( x , y , z ) + I z ( x , y , z ) .
I n ( x , 0 , z ) = I ( x , 0 , z ) / I ( 0 , 0 , z ) ,
I x n ( x , 0 , z ) = I x ( x , 0 , z ) / I ( 0 , 0 , z ) ,
I z n ( x , 0 , z ) = I z ( x , 0 , z ) / I ( 0 , 0 , z ) .
ε = I I x I .
ε = M M + 4 p 2 z 2 ( p 2 r 2 + k 2 x 2 ) 2 ,
ε = x 2 ( a 2 r 2 + x 2 ) 2 + 4 x 2 r 2 / k 2 x 2 ( a 2 r 2 + x 2 ) 2 + 4 x 2 r 2 / k 2 + z 2 ( a 2 r 2 + x 2 ) 2 .
ε = x 2 x 2 + z 2 .

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