2-D wavelength demultiplexer with potential for >= 1000 channels in ...

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3. Theory Assuming dispersion in x and dispersion in y are independent, the intensity distribution dependence in space and wavelength is just the product of the distribution from the VIPA [9] and the distribution from the grating [6] ( ) ( ⎥ ) ( ) ⎥ 2 ⎞ ⎤ ⎛ 2 − ~ 2 ⎡ ∝ ~ f c x 1 ( ⎟ y ⎜ αω) I out x, y, λ − exp − ⎜ I in ( ω) ⎢ exp 2 ⎛ 2 2 2 f W ⎟ 2 2 ⎦ k ⎞ ⎢ ∆ w 1 − ⎠ ⎝ Rr ⎣ + 4 ⎟ ⎜ Rr sin 0 2 ⎠ ⎝ (1) cos( θ ig ) 2 fλ λ f where, w0 = is the focused beam waist in y, α = is the cos( θ d ) πW 2π cd cos( θ d ) grating dispersion 2 tan parameter, ( ) ( θin ) cos( θ i ) x cos( θ in ) x ∆ = 2t nr cos θin − 2t − t is the VIPA 2 f nr f dispersion parameter, ) ~ I in (ω is the input spectrum, ~ ω = ω − ωo is the offset from the center ω 2π frequency, and k = = is the wave vector. c λ Equation (1) is the master equation that predicts the wavelength demultiplexing characteristics in both space and spectrum dimensions as well as the demultiplexing linewidth. The spatial dispersion of the VIPA disperser is [9] tan( θ ) cos( θ ) x 1 1 x λ (2) ⎤ ⎡ ⎥ ⎢ ⎣ ⎦ p − λ0 = −λ0 in i nr cos( θ in ) f + 2 2 nr 2 2 f where, mλ 0 = 2tnr cos( θ in ) , θ in = θi nr is the beam angle inside the VIPA, and λp(x) is a wavelength whose intensity distribution peaks at position x. The grating spatial dispersion is y − y0 λ p − λ0 = d cos( θ d ) 0 f (3) where, d sin( θ d ) + d sin( θ ) = λ 0 ig 0 , and y0 is the spatial position of peak wavelength λ0. As the grating filter has a -3dB bandwidth of tens of GHz, which is an order of magnitude larger than that of the VIPA filter, the -3dB bandwidth is mainly determined by the VIPA demultiplexer according to master Eq. (1): c 1− Rr FWHM / frequency = , 2π t n cos( θ ) Rr FWHM / wavelength r λ0 1− Rr = 2π t n cos( θ ) Rr #4392 - $15.00 US Received 20 May 2004; revised 14 June 2004; accepted 15 June 2004 (C) 2004 OSA 28 June 2004 / Vol. 12, No. 13 / OPTICS EXPRESS 2898 r 2 i i (4)
4. Experiments Fig 3. The 2-D wavelength demultiplexing. Peak wavelengths are demultiplexed to different (x, y). The red solid lines are approximate linear fittings to the demultiplexing data. The blue ellipses show the size at half maximum of the peak intensity for two selected peak wavelengths. For the 2-D demultiplexing, it is interesting to display the peak wavelengths corresponding to various 2-D positions (x, y). In our testing, we look at 41 peak wavelengths ranging from 1549.20 nm to 1550.80nm with a step of 0.04 nm (5GHz). Figure 3 shows the (x, y) corresponding to all these chosen peak wavelengths. The x and y are readings from the micrometer with 0.01 mm resolution. The red solid lines in the plot are approximate linear fittings for the wavelength dispersion data, and it results from multiple diffraction orders of the VIPA. The beam incident angle θi into the VIPA is 4.2 o ± 0.2 o , which results in an approximate linear dispersion dominated by the first term on the right side in the Eq. (2). The blue ellipses in Fig. 3 show qualitatively the spatial intensity distribution of two specific wavelengths, where the boundary indicates positions with half maxima compared to the peak. The key point is that adjacent different orders of the VIPA are clearly separated in space. Experiments show 80 ± 10 um and 50 ± 10 um in diameter for the ellipse in both x and y direction, which agree well with our theory. #4392 - $15.00 US Received 20 May 2004; revised 14 June 2004; accepted 15 June 2004 (C) 2004 OSA 28 June 2004 / Vol. 12, No. 13 / OPTICS EXPRESS 2899
Page 1 and 2: 2-D wavelength demultiplexer with p
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Page 7 and 8: Figure 4(b) shows the normalized li

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