1 April 2006 Mechanical means for temperature compensation of planar diffractive optical interconnects: feasibility study
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Abstract
Planar diffractive optical interconnects have many advantages, however, their inherent chromaticity leads to temperature instability due to the wavelength shift of laser diodes' radiation. This shift can be compensated if the optical interconnect is bended in an appropriate direction with curvature proportional to the relative wavelength shift. The bending can be performed by attaching an additional plate to the element with a different thermal expansion coefficient. Theoretical analysis and ray tracing are reported.
Socol: Mechanical means for temperature compensation of planar diffractive optical interconnects: feasibility study

In the semiconductor industry, there is a continually aggravating problem of increasing communication volume. Optical interconnection technology is seen as a prime option for solving this problem.1, 2 In planar optical interconnects (POI) light is injected (by an in-coupling optical element) into a transparent slab (light guide) at a total internal reflection angle ( ψ in Fig. 1) and propagates along it until it meets an out-coupling optical element and goes to the detector. Diffractive lenses as coupling optical elements have many advantages3, 4, 5 and have been suggested also in the context of quasi-optics6 for rapidly expanding terahertz technology.7 However, their intrinsic dependence of optical properties on the wavelength—chromaticity—poses difficult problems.8 Chromaticity leads also to temperature instability. Namely, the wavelength of semiconductor lasers is temperature-dependent. Temperature change causes wavelength shift, therefore the laser beam as deflected by the diffractive in-coupling optical element deviates from its desired path and can miss the out-coupling element. For example, for a typical 850-nm vertical-cavity surface-emitting lasers (VCSEL) source the wavelength shift is dλdT=0.06nmK ,9 i.e., relative wavelength shift is (1λ)(dλdT)7×105K1 . If POI was assembeled at 20°C , the working temperature is 140°C ( Max3905 from Dallas Semiconductors, e.g.), and the POI length is 10cm , the resulting beam deviation is about 1.7mm (see below the calculation). This temperature instability is specific for diffracive POI since other thermal effects are much less: e.g., for fused silica the explicit refraction index temperature dependence is 3×106K1 .10 The implicit temperature dependence (due to the refraction index wavelength dependence dndλ and the laser wavelength shift dλdT ) is even smaller: dndλ=4×105nm1 ,10 leading to dndT=(dndλ)(dλdT)=2.4×107K1 .

Fig. 1

Planar optical interconnect layout. Initial (thick line) and displaced due to the laser-wavelength-shift (thin line) beams are shown.

040502_1_1.jpg

We propose to fix this problem by bending POI in accordance to the temperature change. We suppose that POI and the laser have the same temperature since they are close to each other. A simple bending scheme is absolutely passive and implies attaching to POI a second plate, having different thermal expansion coefficient (in mass production, the attachment may be done by lamination technology11). The curvature of this bending is proportional to the temperature change (as long as the thermal expansion can be considered as linear with temperature). As long as the laser wavelength shift can be also considered proportional to the temperature change, the bending curvature is proportional to the wavelength shift. Thus, with appropriate proportionality between the bending curvature and the wavelength shift, the bending will cause the diffracted beam to propagate along nearly the same path for different wavelength.

Let us consider the problem quantitatively. At some reference temperature t0 , POI is strictly planar and the incident angle of the incoming beam is zero (Fig. 1). Let the propagation angle (within POI) be ψ (see Fig. 1). When temperature changes (t) , POI bends symmetrically in respect to perpendicular plane situated just in the middle between the input and output diffractive elements (Fig. 2).The incident angle α becomes nonzero since the laser beam direction does not change in the lab frame. However, the bending curvature can be tuned in such a way that the propagation angle ψ retains its original value (locally in the incidence/refraction point) due to the incident angle change. We show now how this tuning is achieved.

Fig. 2

Layout of a bended POI: α is the incidence angle of the laser beam, ψ is its propagation angle, t is the POI thickness, and R is the POI bending radius. The shown bending direction corresponds to λ<λ0 , i.e., a shift toward shorter wavelength (i.e., lower temperatures).

040502_1_2.jpg

First, we derive the beam displacement as function of wavelength λ shift. For the reference radiation wave vector k0=2πλ0 , incident angle α0=0 , grating vector K=2πΛ ( Λ is grating period), propagation angle ψ , and propagation vector r we have12

1

rx=K,sinψ=rxnk0=Knk0=λ0nΛ
[see Fig. 3a]. For another wavelength (wave vector k ), the propagation angle ψ is different [Fig. 3b]. We define relative beam displacement as Δll=(tanψtanψ)tan(ψ) , since in one “period” (bounce-back) the beam covers longitudinal distance l=2dtanψ (Fig. 1), where d is the light guide thickness. We suppose that the relative displacement is rather small and consider equal numbers of “periods.” To calculate this relative displacement, we define the wavelength relative change as

2

δ=(kk0)k0(λ0λ)λ0,
where δ is defined as positive for shorter wavelengths (longer wave vectors) in respect to the basic one at reference temperature t0 . Calculation of relative beam displacement in the first order in respect to δ yields: d(tanψ)dλ=[d(tanψ)d(sinψ)][d(sinψ)dλ] . As for the second term, we have from Eq. 1 d(sinψ)dλ=1nΛ , and for the first one d(tanψ)d(sinψ)=d(tanψ)dψ:d(sinψ)dψ , yielding finally

3

ΔllΔλλcos2ψ.
Since usually ψπ4 , Δll2δ . For example, taking the relative wavelength shift (1λ)(dλdT)7105K1 (for 850-nm VCSEL, as mentioned above) and temperature shift 120°C , we get Δll0.017 , i.e., for L=100mm , ΔL=LΔll=1.7mm .

Fig. 3

Beam deviation as a result of wavelength shift and its correction. The laser beam with wave vector k0 enters the interconnection plate along the z -axis (the incidence angle α=0 ) at reference temperature t0 ; k0 , k , and r , r are the incident and refracted wave vector lengths at reference temperature t0 and at different temperature t correspondingly, and K is the grating vector of the diffractive optical element. (a) Reference temperature t0 ; the propagation angle is ψ ; (b) temperature changes to t , incident wave vector is longer k=k0(l+δ) , propagation angle is ψ<ψ ; (c) temperature t , after bending: the incidence angle is α ; due to the incident beam tilt, the propagation angle is again ψ .

040502_1_3.jpg

In order to correct this deviation we want to get the same propagation angle ψ at a different temperature t . To achieve this we must take nonzero incident angle α [Fig. 3c]. Therefore rx=K+ksinα and sinψ=rxnk . We obtain

4

Knk0=K+ksinαnk.
Substituting Eq. 1 into this equation and making use of δ defined in Eq. 2 yields

5

sinα=(δ(1+δ))nsinψ.
For bending curvature radius R and the interconnect length L (distance between input and output) we obviously have (Fig. 2) α=L2R . Now we have

6

ϵ=2dαL,
where α is given by Eq. 5, ϵ is defined as ϵ=dR , and d is the POI thickness. Positive values of dimensionless curvature ϵ and radius R correspond to curvature center from the laser/detector side (see Fig. 2).

Let us calculate now the change of the beam trajectory in the curved element is respect to the original planar. Consider one beam reflection. In Fig. 4, EÂO=πψ . In the triangle ΔEAO , AÊO=π(πψ)β=ψβ (it should be noted here that the second incidence angle AÊO is smaller than ψ and can be below the total internal reflection threshold). In ΔEBA , tan(ψβ)=ABEB . Considering circumference with origin O we get EB=EC+CB=d+R(1cosβ) and AB=Rsinβ . We have therefore

7

d+R(1cosβ)=Rsinβcot(ψβ)
and finally

8

ϵ+1cosβ=sinβcot(ψβ).
We solve this equation by iterations, supposing that for small ϵ , β will be also small. Expanding Eq. 8 in respect to β and keeping the leading term only, we have

9

β1=ϵtanψ.
Within this approximation, the distance between input and output of 1 bounce-back is 2Rβ1=2R(dR)tanψ=2dtanψ , exactly as in the planar case (Fig. 1). Though the actual beam deviation in space is nonzero due to the bending, it is of second order in respect to δ . However, actually there is linear with δ deviation. In order to estimate it, we make the second iteration. Substituting β=β1+β2 and keeping the leading order of β2 yields

10

β2=3+cos2ψ2sin2ψβ12.
As mentioned above, the unperturbed input-output distance is l=2dtanψ=2Rβ1 , and after wavelength shift and correction bending this distance is 2R(β1+β2) . So the first order approximation to the relative beam displacement is

11

Δll=β2β1=ϵ3+cos2ψ2sin2ψtanψ.
Thus ΔllϵδdL [Eqs. 5, 6].

Fig. 4

Light propagation in bended POI—1 “period” (bounce-back). Notation for derivation of Eqs. 7, 8.

040502_1_4.jpg

Without bending, as mentioned before [Eq. 3], the relative beam displacement is Δll2δ . Therefore the wavelength-shift-caused beam deviation is reduced by factor of dL , i.e., usually above one order of magnitude. Figure 5 presents ray tracing results of the beam deviation as function of wavelength change—with and without compensation by means of POI bending. Exact ray tracing [numerical solution of Eq. 8] results are indistinguishable from the approximation [Eq. 11].

Fig. 5

Ray tracing results for strictly planar [initial, Eq. 3] and bended (after correction) planar optical interconnect. The bending radius fits the wavelength change. The results “after correction” obtained by exact solving of Eq. 8 and by the approximate formula in Eq. 11 are indistinguishable within the given scale. The parameters are: ψ=45° , t=0.1mm , L=10mm .

040502_1_5.jpg

The proposed scheme works only when the source and the detector are situated from one side of POI, but this seems to be the common case. One can realize this scheme for rectangular or circular arrays of sources/detectors. In the latter case, each pair source-detector should be situated diameterally and the bending curvature center should be at the line normal to the array circle and crossing its center. Linear behavior of curvature in respect to temperature should take place also in this case.

Finally, speaking about diffractive optical elements for VCSEL, it should be mentioned that there is an unavoidable spread of nominal VCSEL wavelengths (at a given temperature) from one laser array (chip) to another of usually up to around 10nm or more. The problem of this “bias” may be solved at the assembling stage by off-axis adjusting of the laser array, as in Fig. 3c. The off-axis angle α is given by Eq. 5; its magnitude is about αΔλλ0.01 . As far as α1 , the effects are linear and this adjusting should not affect the above results regarding the temperature compensation.

Acknowledgements

We are grateful to the anonymous referee, who made many valuable comments, leading to considerable improvement of this letter.

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© (2006) Society of Photo-Optical Instrumentation Engineers (SPIE)
Yehoshua Socol, "Mechanical means for temperature compensation of planar diffractive optical interconnects: feasibility study," Optical Engineering 45(4), 040502 (1 April 2006). https://doi.org/10.1117/1.2189869 . Submission:
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