## Optical phased array power penalty analysis

Optics Express, Vol. 15, Issue 8, pp. 5179-5190 (2007)

http://dx.doi.org/10.1364/OE.15.005179

Acrobat PDF (320 KB)

### Abstract

This paper investigates the power penalty from optical phased arrays used for wide-angle beam steering of optical communication signals. The analysis studies the effect of aperture size, data rate, modulation format, and diffraction angle on digital lightwave signals. The results show increasing power penalties for larger angles, aperture sizes, and data rates. At a 10° steering angle, 10-cm aperture, and for both on-off keying (OOK) and differential phase-shift keying (DPSK) the 2.5-Gb/s power penalty is approximately 1.0 dB, while at 10 Gb/s the penalty increases to 7.7 dB for OOK and 7.8 dB for DPSK.

© 2007 Optical Society of America

## 1. Introduction

1. P. F. McManamon et al.,“Optical Phased array technology,” Proc. IEEE **84**, 268–298 (1996). [CrossRef]

2. A. Polishuk and S. Arnon, “Communication performance analysis of microsatellites using an optical phased array antenna,” Opt. Eng. **42**, 2015–2024 (2003). [CrossRef]

## 2. Physical analysis for diffraction gratings

*P*and the observation point to be located at

_{o}*P*. A grating plane with an aperture is placed between these two points. The path between the source point

*P*, the aperture, and observation point

_{o}*P*is shown in Fig. 1. The aperture is located in the

*x*–

*y*plane and has an area equal to

*A*.

*Q*is a point located in the aperture and

*O*is the origin of the

*x*–

*y*plane. The theory of Fraunhofer diffraction will be used to calculate the complex amplitude of the transmitted light intensity at point

*P*. We also denote the first two direction cosines by (

*l*,

_{o}*m*) and (

_{o}*l*,

*m*).

*r*→′

*= (*

_{s}*x*,

_{o}*y*,

_{o}*z*),

_{o}*r*→′

*= (*

_{o}*x*,

*y*,

*z*), and

*s*→= (

*ξ*,

*η*, 0). The Fraunhofer diffraction amplitude is governed be the integral [4]

*A*is defined by the aperture and

*C*is a function of the positions of the source and observation points. The parameters are defined as

*p*=

*l*−

*l*and

_{o}*q*=

*m*−

*m*. For the case of a screen that contains a large number of identical and similarly oriented apertures, the light distribution in the Fraunhofer diffraction is [5]

_{o}*T*(

*ξ*,

*η*). The incident light with input angle

*θ*goes into the diffraction grating and comes out on the other side of the grating diffracted at a different angle. The diffraction is caused by different orders of diffraction which we will discuss later. Here we focus on obtaining the output intensity of the diffraction grating.

_{i}*N*parallel grooves of arbitrary profile. Let

*d*be the distance corresponding to one period in the

*ξ*direction and

*D*be the total length of the grating such that

*D*=

*Nd*. We set

*l*= sin

_{o}*θ*,

_{i}*l*= sin

*θ*,

_{o}*p*=

*l*−

*l*= sin

_{o}*θ*- sin

_{o}*θ*and θ = 0. The amplitude at

_{i}*P*is obtained from Eqn.(3), and the integrand is multiplied by the transmission function

*T*obtained for a single period. In this case, we have

*ξ*=

_{n}*nd*,

*η*= 0, where

_{n}*n*= 0, 1, …,

*N*−1. Substituting the value of

*ξ*and

_{n}*η*into Eqn.(3), we obtain

_{n}*I*of the light is equal to ∣

*U*(

*p*)∣

^{2}.

*I*(

_{o}*p*) = ∣

*U*(

_{o}*p*)∣

^{2}. Using the trigonometric identity cos(2

*x*) = 1 − sin

^{2}(

*x*), the equation for

*I*(

*p*) is simplified to Eqn. (7).

*I*(

*p*) to reach its maximum value,

*kpd*=

*m2π*(

*m*0,±1,±2,…). Therefore,

*p*is

*m*is the diffraction order. Hence

*I*(

*p*) has maximum amplitude of

*CA*and

*BD*is

*CA*-

*DB*=

*d*(sin

*θ*− sin

_{o}*θ*) =

_{i}*dp*, where

*θ*is the incident angle and

_{i}*θ*is the output angle. Therefore, the diffraction angle for each diffraction order can be calculated using Eqn. (9) which is known as grating equation.

_{o}## 3. Phase function of an optical phased array

*T*(

*ξ*) is defined as

*T*(

*ξ*) =

*e*

^{-j2πφ(ξ)}, where φ(

*ξ*) is the phase function. For each period, the phase angle varies between 0 to 2π. The variation of one period is defined as

*k*

_{1}= 2

*πn*/

*λ*, and

*n*is refractive index of the grating. The shape of the phase function

*ζ*(

*ξ*) is shown in Fig. 3. We define

*h*as the height of the phase function,

*d*as the width and

*d*′ as the point where the slope of the function changes. Also we define the parameter

*a*as

*ζ*(

*ξ*) is defined in Eqn.(11).

*T*(

*ξ*) for the OPA.

*ξ*and simplification by we obtain the far-field approximation for

*U*.

^{(OPA)}_{o}## 4. Blazed grating

*m*= 1) and minimize the output from other modes. This means that the diffraction efficiency for the light intensity of mode 1 (

*m*= 1) is 100%, and the other modes become zero. To do so, the phase shape function

*ζ*(

*ξ*) is modified to the case of

*a*= 1, for which

*β*is 90

*°*. This type of diffraction grating is called a blazed grating. For a blazed grating, the shape of the phase function takes the form of a right triangle and the second term of Eqn.(12) will vanish. In our development, we will define the limits of integration as [-

*d*/2 ,

*d*/2]. The equation for the intensity of light now becomes

*I*, we need to choose the correct value of

_{o}*h*.

*I*has maximum value in amplitude of 1, when

_{o}*h*must has the value of

*m*= 1, to have maximum output power and other orders to have zero output, then the value for the height is

*λ*refers to the center wavelength. The frequency response of the OPA for a plane wave can be calculated by modifying Eqn.(17) by letting

_{o}*N*is the number of apertures (or elements) for a given sized OPA. The value of

*N*varies by the center wavelength

*λ*, the diffracted angle

_{o}*θ*

_{1}, and the effective diameter of the OPA (

*Nd*). For fixed values of the center wavelength

*λ*and the OPA diameter, the value of

_{o}*N*increases as the diffracted angle increases. Table 1 shows the number of apertures

*N*for the OPA per centimeter and the single aperture length

*d*for a center wavelength of

*λ*

_{0}= 1565 nm.

*d*is calculated by Eqn. (9) with nomal incident angle (

*θ*= 0).

_{i}*k*), and a small variance of input angles in the wave front. The length of one aperture is comparable with the optical wavelength which is much smaller than the width of the Gaussian beam. A single aperture has a wide spectral response. The Gaussian beam can be approximated by discretizing the Gaussian beam as shown in Fig. 4. The discretized step size is the length of an aperture

*d*. For each aperture, the input can be approximated by a piece-wise plane wave with varying intensity which approximates a Gaussian shape. The spectral response for a Gaussian beam is given in Eqn. (21) by modified Eqn. (3)

*U*is the normalized amplitude of the Gaussian beam with variance

_{G}*w*

^{2}

*, and*

_{d}*n*is indexed thorough all the apertures.

_{j}*θ*, for a Gaussian beam are shown in Figs. 5 and 6 as functions of wavelength

_{o}*λ*. The center wavelength

*λ*for all the cases is 1565 nm. The diameter of the OPA is chosen to be 1 cm and 10 cm. From Fig 5, the main lobe width of the spectral response for the smaller aperture diameter (

_{o}*D*= 1

_{opa}*cm*) is much wider than that for the aperture diameter

*D*= 10 cm. For example, the 3-dB bandwidth of spectral response for

_{opa}*D*= 1 cm is 20 nm for a diffraction angle of 1

_{opa}*°*. The 3-dB bandwidth is 2 nm for

*D*= 10 cm with the same diffracted angle which is 10 times less than the case when

_{opa}*D*= 1 cm. Thus, the 3-dB bandwidth decreases as the diffraction angle increases. The slope of the side-lobes decreases faster with larger diffraction angle. From the information given by Table 1, the 3-dB bandwidth decreases as the number of total aperture increases.

_{opa}## 5. Data transmission simulation

*Gb*/

*s*and 10

*Gb*/

*s*, and compute the power penalty of the OPA for different parameters.

^{23}–1-bit pseudorandom bit sequence. The probability for the occurrence of ones and zeros is equivalent (50%). For each bit duration, the center of the Gaussian pulse is located in the center of the bit period. The width of the pulse is one-third the bit duration. The amplitude of the Gaussian pulse is varied to change the average power for the one bit. We also examine the BER for two types of modulation formats: on-off keying (OOK) and differential phase-shift keying (DPSK) [7]. For both modulations we use return-to-zero signaling. For OOK, the zero bit has amplitude of zero, and the one bit has an amplitude greater than zero. For DPSK, the phase difference between one and zero is

*π*.

*G*is used as the optical receiver. Additive white Gaussian (AWG) noise is added to the received signal. The total noise is dominated by amplified spontaneous emission noise (ASE) [8

8. L. G. Kazovsky and O. K. Tonguz “Sensitivity of direct-detection lightwave receivers using optical preamplifiers,” IEEE Photon. Technol. Lett. **3**, 53–55 (1991). [CrossRef]

9. P. A. Humblet and M. Azizoglu,“On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. **9**, 1576–1584 (1991). [CrossRef]

*n*is the spontaneous emission factor (required to ≥ 1);

_{sp}*h*is Planck’s constant;

*f*is the central optical frequency which equals

*c*/

*λ*;

*B*

_{o}is the amplifier bandwidth; and

*G*is the amplifier gain. After the photodetector, the received signal is

*r*(

*t*) = ∣

*G*[

*s*(

*t*)∗

*opa*(

*t*)]+

*n*(

*t*)∣

^{2},where

*s*(

*t*) and

*n*(

*t*) is transmitted signal and the noise, respectively.

*opa*(

*t*) is the impulse response of the OPA whose Fourier transform is

*U*

^{(b)}(

*λ*). For the OOK simulations, maximum likelihood detection is used to detect the received signals as a one or zero bit. Because of the square law detection in the receiver, the noise distribution for zero and one are changed. The optimal threshold is computed based on the new noise distribution for zero and one for different transmitted powers. The signals are demodulated based on their modulation type. The demodulated signals are compared with the transmitted pattern to calculate the bit-error rate.

## 6. BER simulation results

*°*, 5

*°*, 10

*°*. The total number of elements is calculated by the diffracted angles and the OPA size. The optical amplifier gain

*G*is 27 dB, the amplifier bandwidth

*B*

_{0}is 3.45 GHz for the 2.5-Gb/s system and 13.6 GHz for the 10-Gb/s system. The OPA aperture diameter

*D*is chosen to be 1, 2, and 10 cm. The bit-error rate results are shown in Figs. 8 and 9.

_{opa}*°*first order diffracted angle is the same as the BER without the OPA. The power penalty for the OPA with a 5

*°*diffraction angle is less than 0.5 dB, and the power penalty for the OPA with 10

*°*is 1.5 dB for both modulation formats.

*°*and 5

*°*diffracted angles and aperture diameter

*D*≤ 2 cm is very small and can be neglected for both types of modulation. The power penalty at 5

*°*with a 10-cm diameter is significantly increased to approximately 3 dB for OOK and DPSK. The power penalty for larger diffraction angles such as 10

*°*exceeds that of smaller diffraction angles (≤ 5

*°*) with the same aperture diameter. In the worst case, the power penalty for 10

*°*with

*D*= 10 cm is 8 dB for both modulation formats. The simulation results also show that the power penalties for OOK and DPSK closely match each other for the simulation parameters used here. As expected, the simulation results show that the power penalty increases with data rate.

## Acknowledgments

## References and links

1. | P. F. McManamon et al.,“Optical Phased array technology,” Proc. IEEE |

2. | A. Polishuk and S. Arnon, “Communication performance analysis of microsatellites using an optical phased array antenna,” Opt. Eng. |

3. | A. A. Oliner and G. H. Knittel, |

4. | M. Born and E. Wolf, |

5. | M. Born and E. Wolf, |

6. | T.K. Gaylord and M.G. Moharam, “Analysis of optical diffraction by gratings,” Proc. IEEE |

7. | B. Sklar, |

8. | L. G. Kazovsky and O. K. Tonguz “Sensitivity of direct-detection lightwave receivers using optical preamplifiers,” IEEE Photon. Technol. Lett. |

9. | P. A. Humblet and M. Azizoglu,“On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1960) Diffraction and gratings : Diffraction theory

(060.4510) Fiber optics and optical communications : Optical communications

(280.5110) Remote sensing and sensors : Phased-array radar

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: December 7, 2006

Revised Manuscript: February 12, 2007

Manuscript Accepted: March 7, 2007

Published: April 13, 2007

**Citation**

Jing M. Tsui, Charles Thompson, and Jeffrey M. Roth, "Optical phased array power penalty analysis," Opt. Express **15**, 5179-5190 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5179

Sort: Year | Journal | Reset

### References

- P. F. McManamon et al.," Optical Phased array technology," Proc. IEEE 84, 268-298 (1996). [CrossRef]
- A. Polishuk and S. Arnon, "Communication performance analysis of microsatellites using an optical phased array antenna," Opt. Eng. 42, 2015-2024 (2003). [CrossRef]
- A. A. Oliner and G. H. Knittel, Phased Array Antennas (Artech House, Massachusetts, 1972) pp. 243-253.
- M. Born and E. Wolf, Principle of Optics (University Press, Cambridge, 7th ed., 2002) p. 428, Eqn. (36).
- M. Born and E. Wolf, Principle of Optics (University Press, Cambridge, 7th ed., 2002) p. 448.
- T.K. Gaylord and M.G. Moharam, "Analysis of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985). [CrossRef]
- B. Sklar, Digital Communications Fundamentals and Applications (Prentice Hall PTR, New Jersey, 2nd ed., 2004) pp. 194-204.
- L. G. Kazovsky and O. K. Tonguz "Sensitivity of direct-detection lightwave receivers using optical preampli- fiers," IEEE Photon. Technol. Lett. 3, 53-55 (1991). [CrossRef]
- P. A. Humblet andM. Azizoglu,"On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1584 (1991). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.