Abstract

Hollow ducts compressing and homogenizing the emission of laser diode arrays are cheap, low weight alternatives to lens ducts. An analytic model is presented describing hollow ducts including internal reflection losses, which is useful in optimizing these devices, especially in determining the optimum length for maximum transmission efficiency. As an example, a gold coated duct is presented showing high transmission efficiencies of >90% comparable to lens ducts.

© 2008 Optical Society of America

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References

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  1. R. J. Beach and W. J. Benett, “Lensing duct,” U.S. patent 5,307,430 (26 April 1994).
  2. R. J. Beach, “Theory and optimization of lens ducts,” Appl. Opt. 35, 2005-2015 (1996).
    [CrossRef] [PubMed]
  3. E. C. Honea, R. J. Beach, and S. C. Mitchell, “183-W, M2=2.4 Yb:YAG Q-switched laser,” Opt. Lett. 24, 154-156 (1999).
    [CrossRef]
  4. S. A. Payne, R. J. Beach, S. Mitchell, E. C. Honea, C. Bibeau, J. Lang, D. Maderas, and J. Speth, “Hollow lensing duct,” U.S. patent 6,160,934 (12 December 2000).
  5. “zemax optical design program,” ZEMAX Development Corporation, www.zemax.com.

1999

1996

Beach, R. J.

E. C. Honea, R. J. Beach, and S. C. Mitchell, “183-W, M2=2.4 Yb:YAG Q-switched laser,” Opt. Lett. 24, 154-156 (1999).
[CrossRef]

R. J. Beach, “Theory and optimization of lens ducts,” Appl. Opt. 35, 2005-2015 (1996).
[CrossRef] [PubMed]

R. J. Beach and W. J. Benett, “Lensing duct,” U.S. patent 5,307,430 (26 April 1994).

S. A. Payne, R. J. Beach, S. Mitchell, E. C. Honea, C. Bibeau, J. Lang, D. Maderas, and J. Speth, “Hollow lensing duct,” U.S. patent 6,160,934 (12 December 2000).

Benett, W. J.

R. J. Beach and W. J. Benett, “Lensing duct,” U.S. patent 5,307,430 (26 April 1994).

Bibeau, C.

S. A. Payne, R. J. Beach, S. Mitchell, E. C. Honea, C. Bibeau, J. Lang, D. Maderas, and J. Speth, “Hollow lensing duct,” U.S. patent 6,160,934 (12 December 2000).

Honea, E. C.

E. C. Honea, R. J. Beach, and S. C. Mitchell, “183-W, M2=2.4 Yb:YAG Q-switched laser,” Opt. Lett. 24, 154-156 (1999).
[CrossRef]

S. A. Payne, R. J. Beach, S. Mitchell, E. C. Honea, C. Bibeau, J. Lang, D. Maderas, and J. Speth, “Hollow lensing duct,” U.S. patent 6,160,934 (12 December 2000).

Lang, J.

S. A. Payne, R. J. Beach, S. Mitchell, E. C. Honea, C. Bibeau, J. Lang, D. Maderas, and J. Speth, “Hollow lensing duct,” U.S. patent 6,160,934 (12 December 2000).

Maderas, D.

S. A. Payne, R. J. Beach, S. Mitchell, E. C. Honea, C. Bibeau, J. Lang, D. Maderas, and J. Speth, “Hollow lensing duct,” U.S. patent 6,160,934 (12 December 2000).

Mitchell, S.

S. A. Payne, R. J. Beach, S. Mitchell, E. C. Honea, C. Bibeau, J. Lang, D. Maderas, and J. Speth, “Hollow lensing duct,” U.S. patent 6,160,934 (12 December 2000).

Mitchell, S. C.

Payne, S. A.

S. A. Payne, R. J. Beach, S. Mitchell, E. C. Honea, C. Bibeau, J. Lang, D. Maderas, and J. Speth, “Hollow lensing duct,” U.S. patent 6,160,934 (12 December 2000).

Speth, J.

S. A. Payne, R. J. Beach, S. Mitchell, E. C. Honea, C. Bibeau, J. Lang, D. Maderas, and J. Speth, “Hollow lensing duct,” U.S. patent 6,160,934 (12 December 2000).

Appl. Opt.

Opt. Lett.

Other

S. A. Payne, R. J. Beach, S. Mitchell, E. C. Honea, C. Bibeau, J. Lang, D. Maderas, and J. Speth, “Hollow lensing duct,” U.S. patent 6,160,934 (12 December 2000).

“zemax optical design program,” ZEMAX Development Corporation, www.zemax.com.

R. J. Beach and W. J. Benett, “Lensing duct,” U.S. patent 5,307,430 (26 April 1994).

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Figures (4)

Fig. 1
Fig. 1

(a) Three-dimensional representation of a hollow duct with its equivalent surface approximated by a torus surface. (b) View into that duct along the positive Y axis showing the real shape of the equivalent surface. Notation of the facets is also indicated.

Fig. 2
Fig. 2

(a) Reflectivity of pure and protected gold for s and p polarization, simulated with zemax [5]. (b) Real path of a ray as well as its equivalent straight path in the scope of the method-of-images construction showing two points of reflection R 1 and R 2 .

Fig. 3
Fig. 3

Calculated transmission efficiency of a hollow duct with a perfect sidewall reflectivity and with a protected gold coating for two different orientations of the laser diode array.

Fig. 4
Fig. 4

Calculated transmission efficiency of a hollow duct with an optimum diode alignment compared to the optimum orientation of Fig. 3.

Equations (22)

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r i = a L D a a ,
r 0 = b L D b b r i ,
D ( x d , z d , θ x , θ z ) = 1 π a d b d θ x 0 θ z 0 e ( θ x θ x c / θ x 0 ) 2 ( θ z θ z c / θ z 0 ) 2 .
P t = ( x t y t z t ) = ( ( r 0 + r i cos θ ) cos ϕ ( r 0 + r i cos θ ) sin ϕ r i sin θ )
d 2 Σ = | e x e y e z x t ϕ y t ϕ z t ϕ x t θ y t θ z t θ | d θ d ϕ .
p d = ( x d y d z d )
e r = p t p d | p t p d | .
θ x = arcsin   ( e r · e x ) ,
θ z = arcsin   ( e r · e z ) ,
d 2 Ω t = d 2 Σ · e r | p t p d | 2 .
d 2 T = { D ( x d , z d , θ x , θ z ) d 2 Ω t for   d 2 Ω t > 0 0 otherwise ,
η T = b d / 2 b d / 2 d x d a d / 2 a d / 2 d z d θ = π / 2 θ = π / 2 ϕ = π ϕ = 0 d 2 T .
γ max = arcsin 1 n r ,
Δ ϕ = b r 0 + r i = b b L D , Δ θ = a r i = a a L D ,
| n | = [ | ϕ + π 2 | Δ ϕ + 1 2 ]
| m | = [ | θ | Δ θ + 1 2 ] ,
d 2 T R = { D ( x d , z d , θ x , θ z ) ( 1 α ) | n | + | m | d 2 Ω t for   d 2 Ω t > 0 0 otherwise ,
η T = b d / 2 b d / 2 d x d a d / 2 a d / 2 d z d θ = π / 2 θ = π / 2 ϕ = π ϕ = 0 d 2 T R .
θ 0 = θ FWHM 2 ln 2 .
2 r i a ,
2 ( r 0 + r i ) b ,
L D , min = max ( a 2 ( a a 1 ) , b 2 ( b b 1 ) )

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