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. . . . . .

Waveguide Composed of Pinhole Array

M. Morinaga and D. Kouznetsov

Institute for Laser Science, UECChofu, Tokyo 182-8585, JAPAN

2012-10-05 IWLS

pinholewaveguide

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. . . . . .

abstract

A waveguide composed of an array of pinholes is proposed.Intensity loss at each pinhole is shown to be proportional toL1.5 where L is the distance between adjascent pinholes. Thusby increasing the number of pinholes 4 times, the loss for unitlength becomes a half. Experimental demonstration of suchwaveguide is also given.

pinholewaveguide

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. . . . . .

guiding of wave (light, atoms,...) with

pinholes/slits

λ =1064nm, 532nm

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. . . . . .

wave

In In+1 In+2I0

L

2d

In+1 =

{1 − β

(λLd2

) 32

}In

By reducing the pinhole/slit spacing 1/4 the loss par unitlength becomes a half!

pinholewaveguide

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. . . . . .

Propagation through the pinhole array

Propagation process can be devided into two parts:

..1 Free propagation between two neighboring slits

TL =

∫ ∞

−∞dk |k⟩eikzL⟨k| = eiϕ0

∫ ∞

−∞dk |k⟩e−iαk2⟨k|

k20 = k2z + k2 → kz ∼ k0 − k2

2k0→ α = L

2k0

..2 Masking of wavefunction when the wave pass through aslit

(TMψ)(x) =

{ψ(x) |x| < d0 otherwise

..3 T ≡ TM TL

..4 Propagation through many slits: Tn

..5 Find eigenvalue and eigenstate of T

pinholewaveguide

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. . . . . .

Wavefunction just after a slit

• Takes nonzero value only inside the opening

• Can be expanded with sin/cos function that are 0 on theslit boundary:

ψn(x) =

c0 cos knx (n : even)c0 sin knx (n : odd)

(−d ≤ x ≤ d)

0 (otherwise)

kn = n+12d π (n = 0, 1, 2, ...)

c0 = 1√d: normalization constant

(ψn are eigenfunctions of “Particle in a Box”)

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. . . . . .

1 − |Tnn|

1e-014

1e-012

1e-010

1e-008

1e-006

0.0001

0.01

1

100

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 10

"diag_.txt" u 1:2"diag_.txt" u 1:3"diag_.txt" u 1:4"diag_.txt" u 1:5"diag_.txt" u 1:6"diag_.txt" u 1:7

f(1,x)f(2,x)f(3,x)f(4,x)f(5,x)f(6,x)

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. . . . . .

off diagonal elements v.s. difference of the

diagonal elements of T

1e-012

1e-010

1e-008

1e-006

0.0001

0.01

1

100

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 10

"odiag_.txt" u 1:2"odiag_.txt" u 1:3"odiag_.txt" u 1:4"odiag_.txt" u 1:5"odiag_.txt" u 1:6"odiag_.txt" u 1:7

g(1,3,x)g(1,5,x)g(3,5,x)

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. . . . . .

asymptotic form of the operator T

Tmn = δmn exp(−i (m+1)2π2

8Lkd2

)+1+(−1)m+n

2 (m + 1)(n + 1)π32

12

(Lkd2

) 32(−1 − i)

+O

((Lkd2

)2)

amplitude loss par single unit of slit (length L):

1 − |Tmm| = (m + 1)2π

32

12

(L

kd2

) 32

+ O

((L

kd2

)2)

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. . . . . .

eigenstates & eigenvalues of T

• difference of the diagonal elements: O(λLd2

)• off-diagonal elements: O

(λLd2

)1.5

When λLd2

→ 0

• ψn become eigenstates

• Tnn become eigenvalues

pinholewaveguide

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. . . . . .

light power after the pinhole no.9

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7 8 9

tran

smis

sion

inte

nsity

(a.

u.)

number of pinholes

λ = 532nm

ϕ = 0.5mm

L = 45mm

0: pinhole no.9 only,1: pinhole no.9 + no.0,2: pinhole no.9 + no.0 + no.1,...

160 in the vertical axis corresponds to about 50% of light power

transmitted through pinhole no.0.

pinholewaveguide

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. . . . . .

light power after each pinhole

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7 8 9 10

norm

aliz

ed tr

ansm

issi

on in

tens

ity

detector position

532nm1064nm

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. . . . . .

beam profile after pinhole no.0 and no.3

λ = 532nm, ϕ = 0.5mm, L = 45mm

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. . . . . .

beam deflection

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

norm

aliz

ed tr

ansm

issi

on in

tens

ity

displacement of the last pinhole [mm]

experimentgeometrical optics

λ = 532nm, ϕ = 0.5mm, L = 45mm

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. . . . . .

beam focusing

35mm80mm322mm

planewave 1 2 3 image

sensor

λ=1064nmφ=1.0mm

0

20

40

60

80

100

120

140

160

180

200

0 100 200 300 400 500 600 700 800 900 1000

inte

nsity

(a.

u.)

pixel

1 pinhole2 pinholes3 pinholes

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. . . . . .

features of this methodsapplication

• Any kind of wave can be handled• electro-magnetic wave of special wavelength, matter wave,

etc.

• possible to bend• wave travels through a free space

• guide atoms by the light guided by the pinhole array

• manipulation of the transverse mode

• FREE beam profiling program for WDM cameras (USBcameras) is available at: http://m.ils.uec.ac.jp/sbpw/

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. . . . . .

pinhole aligning ROBOT(under construction)

number of pinholes: ∼ 100pinhole spacing: & 2mm

pinholewaveguide

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. . . . . .

naive estimate of the losscoefficient

• projection of Tψ onto ψ

S = ⟨ψ|T|ψ⟩ = ⟨ψ|TL|ψ⟩ =

∫dk |⟨ψ|k⟩|2e−iαk2

• 1 − |Sn|2 = 1 − |⟨ψn|T|ψn⟩|2 gives the intensity losscoefficient for the nth mode when λL

d2≪ 1

• property of f(k) ≡ |⟨ψ|k⟩|2:∫ ∞

−∞dk f(k) = 1, f(k) ≥ 0

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. . . . . .

fn(k) ≡ |⟨ψn|k⟩|2fn(k)

k-kc kc0

δk

kc = n+12d π、δk ∼ π

2d。

kc ≡∫ ∞

0dk k{f(k) + f(−k)}

δk ≡(∫ ∞

0dk (k − kc)

2{f(k) + f(−k)}) 1

2

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. . . . . .

S= ⟨ψ|T|ψ⟩=

∫∞−∞ dk f(k)e−iαk2

= e−iαk2c∫∞0 dk {f(k) + f(−k)}e−iα(2kc∆k+∆k2)

∼ e−iαk2c∫∞0 dk g(k){1 − iα(2kc∆k + ∆k2) − 2α2k2c∆k2}

= e−iαk2c{1 + (−iα− 2α2k2c)δk2}

|S|2 ∼ 1 − 4α2k2cδk2 = 1 −

k2cδk2

k20L2

|Sn|2 ∼ 1 −(n + 1)2π4

16k20d4

L2 = 1 −(n + 1)2π2

64

(λL

d2

)2

wrong result!

g(k) = f(k) + f(−k), ∆k ≡ k − kc.