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All-Optical Logic Gates

Geoffrey D. Jenkins

EOP 604:

Integrated OpticsFall 2007


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Introduction

A logic gate is a device that performs a certain Boolean logic operation on one ormore logical inputs and produces a single logical output. Boolean algebra is

comprised of operations that give true or false as a result. In this lies the

foundation of digital electronics and computing. It is standard to express true and

false as 1 and 0, respectively. Logic gates are bistable devices, that is, they may

yield one of these two possible stable outputs. In digital electronic systems it is

common to see 1 correspond to 5 Volts and 0 corresponds to 0 Volts. Standard

notation and operations are displayed below:

The relationship between input and output signals are displayed in what are known

as truth tables. The truth tables for the devices above are displayed here:

AND

Input Output

0 0 0

0 1 0

1 0 0

1 1 1

OR

Input Output

0 0 0

0 1 1

1 0 1

1 1 1

NOT

Input Output

0 1

1 0

- -

- -

There are several other logic gates not mentioned above, however, that is beyond

the scope of this paper. These devices, accompanied with feedback, can be

connected together in ways that create more complex devices such as latches and

flip-flops, which are basic electronic memory storage devices.


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These devices and Boolean logic are not limited to electronics. They have been

demonstrated with optics in many different implementations with various

switching mechanisms. The interest the optics community has in these devices has

spawned from the fact that conventional computational speeds are approaching

limits. [1] Mentions that the miniaturization attempts in lithography (in an attempt

to fit as many transistors as possible on a single chip) are becoming problematic.

Optical computing has, or will have, many advantages over electronic computing

[1]. Some examples include:

Immunity to electronic interference Lighter, more compact systems Immunity to short circuits Lower-loss transmission

Significantly more bandwidth Easier/cheaper parallel computing

To put things in perspective, it was said that a computation that would take an

electronic computer over eleven years to complete would take an optical computer

less than an hour [1]. This technology seems far off, but hybrid electro-optical

systems are becoming more popular every day. The focus of this paper is all-

optical logic gates, where light is controlled by light rather than electronics.

Logic devices must have a switching mechanism; otherwise the output would be

monostable. Analogous to transistors in electronics, these devices turn light on or

off to certain thresholds. In the aforementioned hybrid devices, the electro-optic

effect is utilized to vary the refractive indices of optical waveguides to make

voltage-controlled switches. Again, these devices are limited by the switching

speed of electronics, so how is this done all-optically? Nonlinear optics is the key.

Consider a two channel directional coupler fabricated from highly nonlinear

optical materials, as first proposed in [2].

d

n2

n1

a

a


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TheoryIn the linear scheme, these waveguides are phase matched ( = 0) and 100 %

coupling is achieved from channel one to channel two. From coupled-mode theory,

the field amplitude in each waveguide can be derived from the coupled-modeequations [3]:

= 1 2

where u1 and u2 are the transverse mode distributions supported by channels one

and two, respectively, and 1,2 represents the propagation constants of the modes.

Applying a boundary condition stating that at the input end of these waveguides

and the coupled-mode equations above can be solved:


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The optical power in each waveguide is proportional to the square of the field,

therefore

As mentioned before, when the waveguides are phase-matched and = 0, perfect

coupling is achieved and the equations above simplify to


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In the nonlinear scheme, e.g. when a phase mismatch is induced, the coupling

properties of the device are altered. A power transfer ratio may be expressed as a

P2 / P1, which becomes

This term decreases as LC increases like shown in the following figure:

To make practical use of this, a phenomenon called the Kerr effect is used. The

Kerr effect is a change of the refractive index of a medium that is directly

proportional to the square of the optical field residing in that medium. Accounting

for nonlinear index changes, the refractive index of a medium may be expressed as

where is the linear index, is the nonlinear index coefficient [m2/W], E is

the field amplitude, andZ0is the free space impedance, or 377.


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All three logic gates mentioned earlier, the AND, OR, and NOT gates can be

implemented using the Kerr effect and this directional coupler configuration. Let

the inputs of the device be labeled I1 and I2 and the outputs be labeled O1 andO2.

NOT GateWhen a signal beam is launched into I1 and it is not intense enough to cause any

nonlinearities, it will fully couple to the other waveguide at the designed coupling

length,LCas shown in the BPM simulation [4] below.

Refractive index distribution along x-axis

Field amplitude vs. device length: Signal =1; Control = 0


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If a second, intense control beam is launched into I1 collinearly with the signal

beam, the signal will see a different refractive index (smaller or larger, depending

on the materials characteristics) than it did when propagating by itself. This causes

a phase mismatch between the two waveguides. If the intensity is at the switching

threshold, the phase mismatch will be great enough to prohibit coupling

completely. This is demonstrated below, where the refractive index of channel 1

was altered due to an intense control beam.

Field amplitude vs. device length with control beam present

Field amplitude vs. device length: Signal =1; Control = 1


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From these CMT simulations, the truth table for a NOT gate is fulfilled between I1

and O2 when considering the control beam as the input. Note that when the control

beam is absent the signal beam crosses channels and becomes the output through

O2. Ports I2 and O1 are neglected.

NOT

Control

Beam

Output

(O2)

0 1

1 0

AND, OR Gates

AND and OR gates may be implemented using the same device but both input and

output ports are used.

An AND gate may be thought of as a multiplier of both inputs, and can be derived

by looking at the truth table. The only scenario in which the AND gate will betrue, or transmitting light, is when both the signal and control beams are

simultaneously incident on the input ports.

OR gates are true if at least one of its inputs is true, therefore it will always

transmit a signal unless there is no signal available to transmit.

Let the control be incident upon I2 and the signal incident upon I1. Although the

signal beam is propagating by itself in channel 1, it will not couple to channel 2

because the control beam in channel 2 will have altered its refractive index,

thereby inducing a phase-mismatch. The control beam will not couple into channel

1 for the same reason; therefore each beam will remain in its original waveguide

for the entire journey through the device.

This is demonstrated in the following figures in two steps; one simulation each for

the signal and control beams.


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Field amplitude through channel 2: Control = Signal = 1

If the signal is launched while the control is not, the signal will couple into the

second channel because there will be a phase match between waveguides due to

the absence of nonlinearities.

Field amplitude through channel 1: Signal = 1; Control = 0


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When the outputs and inputs of this device are tabulated in a truth table, it is

apparent that O1 behaves like an AND gate and O2 behaves like an OR gate.

Inputs Outputs

Signal (I1) Control (I2) O1 O2

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 1

ConclusionAll-optical logic gates of these types are easy to design and cascade, allowing the

construction of more complex structures, as opposed to other reported types [3].

Some designs are fairly complicated and depend on polarization and/or more

complex fabrication processes. Every implementation is sure to have pros and

cons.

One of the main obstacles in the evolution to efficient all-optical computing is the

lack of good nonlinear optical materials [1]. A significant amount of optical power

is needed to induce effective phase mismatches, making practical solutions

difficult. More sensitive materials would lower these power requirements.

Several assumptions were made in these simulations. The phenomenon of self-

focusing was not accounted for, mainly due to the fact that coupled-mode theory is

not accurate under such circumstances [5]. The changes in the refractive index

distribution induced by the control beam were assumed to be constant and uniform,

which may not be the case. Nonlinear BPM software [6] would give more accurateresults for more specific simulation parameters.

It should be possible to run a BPM simulation for one propagation step, perform

Kerr calculations on the resulting field, redefine the refractive index distribution,

and repeat. This would require a large amount of processing power, a project that

would be well suited for an all-optical computer.


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References

[1] Hossin Abdeldayem, Donald O. Frazier, Mark S. Paley, and William K.

Witherow,Recent Advances in Photonic Devices for Optical Computing

[2] S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580

(1982)

[3] Saleh and Teich, Fundamentals of Photonics. (c) 1997

[4] LIGHTS Online Optical Simulation Software

(c) 1997-2007 Andrew Sarangan

[5] D. R. Heatley, E. M. Wright, J. Ehrlich, and G. I. Stegeman,Nonlinear

directional coupler with a diffusive Kerr-type nonlinearity. May 1988 / Vol.

13, No. 5 / OPTICS LETTERS

[6] Takeshi Fujisawa and Masanori Koshiba, Full-Vector Finite-Element Beam

Propagation Method for Three-Dimensional Nonlinear Optical Waveguides.

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 10,

OCTOBER 2002

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