Directionality of general beams - OSA Publishing

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Xiaoling Ji 1*, Xiaoqing Li1 and Guangming Ji 2 ... A. Yang, E. Zhang, X. Ji, and B. Lü, “Angular spread of partially coherent Hermite-cosh-Gaussian beams.
Directionality of general beams Xiaoling Ji 1*, Xiaoqing Li 1 and Guangming Ji 2 1

Department of Physics, Sichuan Normal University, Chengdu 610068, China College of Information Management, Chengdu University of Technology, Chengdu 610059,China * Corresponding author: [email protected]

2

Abstract: The directionality of general beams propagating in free space and in atmospheric turbulence is studied. Based on the partial-coherence theory, the analytical expressions for the mean-squared width and the angular spread of general beams are derived by using the integral transform technique. It is shown that the mean-squared width and the angular spread depend not only on the weighting factors of all basis modes but also on the weighting factors of the corresponding mode coherence coefficients of the ith and the (i+2)th if the Hermite-Gaussian modes are adopted. It is found that under a certain condition there exist the equivalent general beams which may have the same directionality as a fully coherent Gaussian beam in free space and also in atmospheric turbulence. The result holds true, irrespective of the model of turbulence used. ©2008 Optical Society of America OCIS codes: (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence; (030.4070) Modes.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

A. E. Siegman, Laser (University Science Books Mill Valley, 1986). L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998). E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27-29 (1978). E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293-296 (1978). P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260 (1979). J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203-208 (1980). G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002). A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10-12 (2003). H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere, ” J. Opt. Soc. Am. A 22, 2709-2718 (2005). Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express, 14, 1353-1367 (2006). T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610-612 (2003). X. Ji, X. Chen, and B. Lü, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25, 21-28 (2008). A. Yang, E. Zhang, X. Ji, and B. Lü, “Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence,” Opt. Express, 16, 8366-8380 (2008). K. M. Du, G. Herziger, P. Loosen, and F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081-1093 (1992). A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923-928 (1982). A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).

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17.

1.

A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite–sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2425-2432 (1998).

Introduction

The propagation of laser beams in free space and in atmospheric turbulence is a topic that has been of considerable theoretical and practical interest for a long time [1, 2]. In 1978 Collett and Wolf predicted that Gaussian Schell-model (GSM) beams may have the same directionality as a fully coherent laser beam in free space [3, 4], which was confirmed experimentally later [5, 6]. It means that full spatial coherence is not a necessary condition for highly directional light beams. Recently, much work has been carried out concerning the spreading of laser beams in atmospheric turbulence [7-10]. In 2003 Wolf and his collaborators extended the result in Refs.[3, 4] from free space to atmospheric turbulence [11]. Very recently, we have found that, besides the equivalent GSM beams, there also exist other equivalent partially and fully coherent beams which may have the same directionality as a fully coherent laser beam in free space and also in atmospheric turbulence, such as the equivalent partially and fully coherent Hermite-Gaussian (H-G) beams [12], and the equivalent partially and fully coherent Hermite-cosh-Gaussian (H-ChG) beams and cosh-Gaussian (ChG) [13]. Thus, an interesting question arises: Does the result hold true for general beams (i.e., general partially and fully coherent beams)? The aim of this paper is to study the directionality of general beams in free space and in atmospheric turbulence based on the partial-coherence theory. Some general results are obtained, which are illustrated by examples. 2.

Theoretical model

Within the framework of partial-coherence theory and under the quasi-monochromatic assumptions, the cross-spectral density of general beams is expressed as [14, 15] W ( 0) (r1′, r2′ ) = W ( 0 ) ( x1′ , x′2 )W ( 0) ( y1′ , y2′ ) ,

W ( 0) ( x1′ , x2′ ) =

where

∑ ∑ +∞

+∞

i1=0

j1=0

(1)

λi1, j1ϕi1 ( x1′ )ϕ ∗j1 ( x′2 ) ,

(2)

with λi1,j1 is the corresponding mode coherence coefficients (MCC’s), i1 and j1 are the mode indices. ϕ(x) represents the field distribution, which can be expressed as a series of orthogonal basis modes, for example, the H-G modes at the plane z=0 in the rectangular coordinate system, that is,

[

ϕl ( x) = 21/ 2 /(π 1/ 2 2l w0 h l!)

]

1/ 2

(

) (

)

exp − x 2 / w02h H l 21/ 2 x / w0 h ,

(l=i1, j1),

(3)

with w0h being the waist width of the basis Gaussian mode, and Hl denoting the lth order Hermite polynomial. W ( 0) ( y1′ , y 2′ ) can be obtained if x, λi1,j1, i1 and j1 are replaced by y, λi2,j2, i2 and j2 in Eq. (2). By using the orthogonality of H-G series, the MCC’s is obtained from Eq. (2), i.e.,

λi1, j1 =

∫∫ ϕ

∗ i1

( x1′ )W ( 0) ( x1′ , x2′ )ϕ j1 ( x2′ )dx1′dx′2 .

(4)

λi2,j2 can be obtained if x, i1 and j1 are replaced by y, i2 and j2 in Eq. (4). Based on the extended Huygens-Fresnel principle, the averaged intensity of the general beam represented by Eq.(1) propagating in atmospheric turbulence at the plane z is given by the expression (Sec.12.2 of Ref. 2)

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I (r, z ) = [k /(2πz )]2

∫∫ d r ′∫∫ d r ′W 2

2

1

{

(0)

2

(r1′, r2′ )

}

× exp − [ik /( 2 z )][(r1′ − r2′ ) − 2r ⋅ (r1′ − r2′ )] 〈exp[ψ ∗ (r, r1′, z ) + ψ (r, r2′ , z )]〉 m , 2

2

(5)

where k is the wave number related to the wave length λ by k=2π/λ, ψ (r, r ′, z ) is the phase function that depends on the properties of the medium, and < >m denotes average over the ensemble of the turbulent medium, which is given by (Sec.12.2.3 of Ref. 2)

[

〈exp ψ

∗ (r, r1′, z ) + ψ (r, r2′ , z )]〉 m = exp{−4π 2 k 2 z

∫ ∫ κΦ (κ )[1 − J (κξ r′ − r′ )]dκdξ } , 1 ∞

n

0 0

0

2

(6)

1

with J0 being the Bessel function of the first kind and order zero, and Φn(κ) being the spatial power spectrum of the refractive-index fluctuations of the turbulent medium. To obtain the analytical results, the new variables of integration are introduced as u = (r2′ + r1′) / 2 , v = r2′ − r1′ , (7) Eq. (5) then becomes I (r, z ) = [k /(2πz )]2

∫∫ d u ∫∫ d vW 2

2

( 0)

(u, v ) exp(− iku ⋅ v / z ) exp(ikr ⋅ v / z ) ∞

× exp{−4π 2 k 2 z ∫ ∫ κΦ n [1 − J 0 (κξv)]dκdξ } . 1

(8)

0 0

The mean-squared beam width in the x direction is defined as [16] wx2 ( z ) = 4 F1 / F2 , F1 =

where

∫∫ x

F2 =

2

(9)

I (r, z ) dxdy ,

∫∫

(10)

I (r, z ) dxdy .

(11)

On substituting from Eq. (8) into Eq. (10), and recalling the integral formulae

∫x

2

exp(−i2πxs)dx = −δ ′′( s) /(2π ) 2 ,

∫ exp(−i2πxs)dx = δ (s) .

(12)

where δ denotes the Dirac delta function, and δ ′′ is its second derivative, we obtain F1 = −( z / k )

2

∑ ∑ ∑ ∑ +∞

+∞

+∞

+∞

i1=0

j1=0

i 2= 0

j 2 =0

λi1, j1 21/ 2 (π 2i1+ j1 w02h i1! j1!) −1/ 2

∫∫ d u ∫∫ d v exp(− 2u / w )exp[− v ]H [2 (u − v / 2) / w ]exp(− 2u / w )

× λi 2, j 2 21 / 2 (π 2i 2+ j 2 w02h i 2! j 2!) −1/ 2

[

× H i1 21 / 2 (u x + v x / 2) / w0 h

[

2

2

2 x

2 0h

1/ 2

j1

x

] [ (

)

2 y

0h

x

] [ (

2 x

/(2 w02h )

]

2 0h

]

)

× exp − v 2y /(2 w02h ) H i 2 21 / 2 u y + v y / 2 / w0 h H j 2 21 / 2 u y − v y / 2 / w0 h exp(− iku ⋅ v / z ) ∞

× exp{−4π 2 k 2 z ∫ ∫ κΦ n (κ )[1 − J 0 (κξv)]dκdξ }δ ′′(v x )δ (v y ) . 1

(13)

0 0

Recalling the integral formulae ⎧

0

∫ f ( x)δ ( x)dx = f (0) , ∫ exp(− x )H ( x + y)H ( x + z )dx = ⎨⎩2 i! 2

i

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j

i

π

(i ≠ j ) , (i = j )

(14)

Received 30 Jun 2008; revised 20 Aug 2008; accepted 28 Oct 2008; published 31 Oct 2008

10 November 2008 / Vol. 16, No. 23 / OPTICS EXPRESS 18852

∫ exp(− x )H ( x + y) H 2

i

j

( x + z )dx = 2 j π 1 / 2 i! z j − i Lij − i (−2 yz ) ,

( i ≤ j ),

(15)

+∞

∫ f ( x)δ ′′( x)dx = f ′′(0) ,

(16)

−∞

where Lαi denotes the generalized Laguerre polynomial, f is an arbitrary function and f ′′ is its second derivative, and performing the integration with respect to vy, uy, ux and vx in turn in Eq.(13) yields

∑ ∑∑ λ

z +∞ F1 = ( ) 2 λ i 2 ,i 2 k i 2= 0

+∞

+∞

i 1= 0

j 1= 0

i1, j 1

(2

j 1− i1

i1! / j1!) ( − 1/ 2

1 ikw 0 h j 1− i1 − ) f ′′( 0 ) , 2 1 / 2 w0 h 2 3 / 2 z

(17)

A = 1 / w02h + k 2 w02h /(4 z 2 ) ,

where

(

)

(18)

[

1 +∞

]

f (v) = v j1−i1 exp − Av 2 / 2 Lij11−i1 ( Av 2 ) exp{−4π 2 k 2 z ∫ ∫ κΦ n (κ ) 1 − J 0 (κξ v) dκdξ } . 0 0

(19)

On substituting from Eq. (19) into Eq. (17), using the formula λi1,j1=λj1,i1, and considering f ′′(0) = 0 for j1 − i1 ≥ 3 , after some lengthy mathematical manipulations, we obtain



+∞

F1 = (w02h / 4)

i 2 =0

λi 2,i 2

∑ {(1+ 2i1)λ +∞

[

]∑





+ z 2 /(k 2w02h )

+ ⎢(2π 2 / 3) z 3 ⎣

+∞

i 2 =0

+∞

0

λi 2,i2

+ 2[(i1 + 1)(i1 + 2)] λi1,i1+2 1/ 2

i1, i1

i1= 0

∑ {(1+ 2i1)λ +∞

∑ ⎦

κ 3Φ n (κ )dκ ⎤⎥

+∞

i 2 =0

− 2[(i1 + 1)(i1 + 2)] λi1,i1+2 1/ 2

i1,i1

i1= 0

}

λi 2,i 2



+∞

i =0

}

λi1,i1 .

(20)

The right-hand side of Eq.(11) is the total power of the beam at distance z, because of energy conservation, it is equal to the total power of the source. Considering the orthonormality of H-G series, we obtain F2 =



+∞

i1=0

λi1,i1



+∞

i 2= 0

λi 2,i 2 .

(21)

Consequently, in view of Eq. (9), we obtain the mean-squared beam width in the x direction of general beams in atmospheric turbulence, i.e., wx2 ( z ) = w02h

∑ {(1 + 2i)β +∞

i =0

+∞

+ 4z 2 /(k 2 w02h )∑

i =0

i ,i

+ 2[(i + 1)(i + 2)]

{(1 + 2i)β

1/ 2

β i ,i +2

} }



− 2[(i +1)(i + 2)] βi ,i +2 + (8π 2 / 3) z 3 ∫ κ 3Φn (κ )dκ 1/ 2

i ,i

0

(22) where β i ,i = λi ,i



+∞

i =0

λi,i and β i ,i+2 = λi,i +2



+∞

i =0

λi ,i are the weighting factors of all basis

modes and the weighting factors of the MCC’s of the ith and the (i+2)th, respectively. Similarly, we can obtain the mean-squared beam width in the y direction, and omitted here. The first two terms on the right-hand side of Eq. (22) denote the spread of the beam width in free space due to diffraction, which indicate that the mean-squared beam width of general beams in free space is a parabola profile. The third (cubic) term in Eq. (22) presents the spreading of general beams due to atmospheric turbulence, which is independent of the #97771 - $15.00 USD

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MCC’s. From Eq. (22), the angular spread in the x direction of general beams propagating through atmospheric turbulence turns to be

θ x ( z ) ≡ lim wx ( z ) / z z →∞

∑ {(1 + 2i)β

⎧ = ⎨4 /(k 2 w02h ) ⎩

+∞

i =0

}

− 2[(i + 1)(i + 2)] βi ,i +2 + (8π 2 / 3) z 1/ 2

i ,i



+∞

0

1/ 2

κ 3Φ n (κ )dκ ⎫⎬ . ⎭

(23) The first term on the right-hand side in Eq. (23) represents the angular spread of general beams in free space, which is dependent on not only βi,i but also βi,i+2 if the H-G modes are adopted; while the second term describes the effect of turbulence on the angular spread, which does not depend on the MCC’s, i.e., is independent of the beam parameters. For i=0, Eq. (22) reduces to the mean-squared width of a fully coherent Gaussian beam specified by subscripts ‘Gs’, which is given by wx2 ( z )

+∞

Gs

2 = w02Gs + 4 z 2 /(k Gs w02Gs ) + (8π 2 / 3) z 3 ∫ κ 3Φ n (κ )dκ ,

(24)

0

and Eq.(23) reduces to the angular spread of the fully coherent Gaussian beam, i.e., +∞

2 θ x ( z ) Gs = [4 /( k Gs w02Gs ) + (8π 2 / 3) z ∫ κ 3 Φ n (κ )dκ ]1/ 2 .

(25)

0

From a comparison of Eqs. (23) and (25), we conclude that a general beam will generate the same angular spread as a fully coherent Gaussian beam both in free space and in turbulence if the condition

(k

2

w02h

) ∑ {(1 + 2i)β −1

i = +∞ i =0

i ,i

− 2[(i + 1)(i + 2)]

1/ 2

}

2 β i,i +2 = (k Gs w02Gs )

−1

(26)

is satisfied. The result is independent of the spatial power spectrum Φn(κ). Such general beams are called the equivalent general beams [3, 4]. 3. Examples

The examples are given to illustrate the result obtained above, in which fully coherent beams (e.g., Sinh-Gaussian beams) and partially coherent beams (e.g., GSM beams) are considered. If the modified Von Karman spectrum is adopted as a model of the atmospheric turbulence, i.e., [2] −11 / 6 Φ n (κ ) = 0.033C n2 κ 2 + 1 / L20 exp − κ 2 / κ m2 , (27) where κ m = 5.92 / l 0 , l0 and L0 are the turbulence inner scale and outer scale, respectively. Cn2 is the refraction index structure constant, which describes how strong the turbulence is. Letting l0=0.01m and L0=10m yields

(

)

(

)

+∞

F = (8π 2 / 3) ∫ κ 3Φ n (κ )dκ = 18.846Cn2 .

(28)

0

(Ⅰ) Sinh-Gaussian beam The cross-spectral density function of Sinh-Gaussian (ShG) beams at the plane z=0 is expressed as [17] 2 2 W ( 0) ( x1′ , x 2′ ) = exp − x1′ / w02 sinh (Ω 0 x1′ ) exp − x ′2 / w02 sinh (Ω 0 x 2′ ) , (29) where w0 is the waist width of the Gaussian part, Ω0 is the parameter associated with the sinh part. On substituting from Eqs. (3) and (29) into Eq. (4), after very tedious integral

(

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(

)

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10 November 2008 / Vol. 16, No. 23 / OPTICS EXPRESS 18854

calculations, the analytical expression for the MCC’s can be derived as ⎛π ⎞ ⎟ ⎝ 2⎠

1/ 2

λi, j = ⎜

γw0h ⎛ 2 −γ ⎞ ⎜ ⎟ 1/ 2 ⎜ (2 + γ )(i! j!) ⎝ 4 + 2γ ⎟⎠

(i + j ) / 2



exp⎜⎜

δ

⎝ 1+ γ

⎤ ⎡ ⎤ ⎞ ⎡ γδ γδ ⎟⎟ H i ⎢ Hj⎢ , 2 1/ 2 ⎥ 2 1/ 2 ⎥ ⎠ ⎣ (4 − γ ) ⎦ ⎣ (4 − γ ) ⎦

(30)

where i, j=odd numbers, δ = w02 Ω 02 and γ = 2( w0 / w0 h ) 2 . Figure 1 gives the mean squared beam width wx(z) of an equivalent ShG beam and of the corresponding fully coherent Gaussian beam propagating both in free space and in atmospheric turbulence. As can be expected, the equivalent ShG beam exhibits the same directionality as the corresponding fully coherent Gaussian beam in free space and also in atmospheric turbulence. (Ⅱ) GSM beam The cross-spectral density function of GSM beams at the plane z=0 is characterized by W ( 0 ) ( x1′, x2′ ) = exp[− ( x1′ 2 + x2′2 ) / w02 ]exp[− ( x1′ − x2′ ) 2 /(2σ 02 )] ,

(31)

where σ0 the spatial correlation length of GSM beams at the plane z=0. Substituting Eqs. (3) and (31) into Eq. (4), after very tedious integral calculations, we obtain the MCC’s

λi , j

(

)

i

⎧⎪ [1 + 1 /(2α )]2 − γ 2 / 4 1 / 2 ⎫⎪ (i! j!)1/ 2 = λ0 ⎨ ⎬ i+ j i j (i+ j ) / 2 ⎪⎩ 1 + 1 /(2α ) + γ / 2 ⎪⎭ 2 r ( w0 h / 2) p



min( i , j )

×

k =0

⎡ (−1) − k +( i + j ) / 2 2 k ⎢ ⎣ k![(i − k ) / 2]![( j − k ) / 2]!

(2r

2

)

p −1

( i −k ) / 2

(w

2 0h

)

p / 2 −1

( j−k ) / 2 ⎤

⎥, ⎦

(32)

where i, j and k have the same parity, or λ i, j =0 when i and j have different parity; and

λ0 =

⎡ πγ / p ⎢ 1 + 1 /( 2α ) + γ ⎣

{

⎤ ⎥ / 2⎦

1/ 2

, p=

4α 2 [1 + 1 /( 2α ) + γ / 2] − 1 , 4α 2 w02 [1 + 1 /( 2α ) + γ / 2] 2

(33)

}

r = αw0 h [1 + 1 /(2α )] − γ 2 / 4 , α = (σ 0 / w0 ) 2 . (34) The mean-squared width wx(z) of an equivalent GSM beam and of the corresponding fully coherent Gaussian beam versus propagation distance z is shown in Fig. 2. As can be seen, the equivalent GSM beam generate the same directionality as the corresponding fully coherent Gaussian beam not only in free space but also in turbulence as well.

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0. 05

0. 08

0. 04 b

0. 06

wx(z) [m]

wx(z) [m]

0. 10

a

0. 04

b

0. 02

0. 02 0. 00

a 0. 03

0. 01

C n2 = 0 C n2 = 10 −13 m − 2 / 3

0

500

Fig. 1 wx (z ) versus z.

z[m]

1000

1500

a: the corresponding

C n2 = 0 C n2 = 2 × 10 −13 m − 2 / 3

0. 00

0

200

400

z[m]

600

Fig. 2 wx (z ) versus z. a: the corresponding fully

fully coherent Gaussian beam with w0Gs = 9.7 mm , λGs = 1.54μm ; b: an equivalent ShG beam with

coherent Gaussian beam with w0Gs = 7.5mm , λGs = 1.54μm ; b: an equivalent GSM beam with

w0 = 4mm , γ = 0.5 , δ = 6 , λ = 0.6328μm .

w0 = 4mm , γ = 1.28 , α = 3.06 , λ = 0.6328μm .

4. Concluding remarks

The directionality of general beams propagating in free space and in atmospheric turbulence has been studied. Based on the partial-coherence theory, the analytical expressions for the mean-squared width and the angular spread of general beams propagating in atmospheric turbulence have been derived by using the integral transform technique. The propagation of general beams in free space can be treated as special cases. It has been shown that the mean-squared width and the angular spread are dependent not only on the weighting factors of all basis modes but also on the weighting factors of the MCC’s of the ith and the (i+2)th if the H-G modes are adopted, i.e., βi,i and βi,i+2. It has been found that under condition (26) there exist equivalent general beams which may have the same directionality as a fully coherent Gaussian beam both in free space and in turbulence, which is valid, irrespective of the turbulence model characterized by Φn(κ). As compared with the previous publications, the results obtained in this paper are more general, which hold true both for general fully coherent beams and for general partially coherent beams. Acknowledgments

The authors are very thankful to the anonymous reviewers for valuable comments. This work was supported by the National Natural Science Foundation of China under grant 60778048.

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