## A direct analysis of photonic nanostructures

Optics Express, Vol. 14, Issue 8, pp. 3472-3483 (2006)

http://dx.doi.org/10.1364/OE.14.003472

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### Abstract

We present a method for directly analyzing photonic nano-devices and apply it to photonic crystal cavities. Two-dimensional photonic crystals are scanned and reproduced in computer memory for Finite Difference Time Domain simuations. The results closely match experimental observations, with a fidelity far beyond that for idealized structures. This analysis allows close examination of error mechanisms and analytical error models.

© 2006 Optical Society of America

## 1. Introduction

1. S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics **81**, 283 – 293 (2005). [CrossRef]

2. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic Crystal waveguides: role of fabrication disorder and photon Group velocity.” Phys. Rev. Lett. **94**, 033,903 – 4 (2005). [CrossRef]

3. A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B. **72**, 153,102 – (2005). [CrossRef]

4. P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature **424**, 852–55 (2003). [CrossRef] [PubMed]

## 2. Photonic crystal cavities

*γ*

_{∥}and

*γ*

_{⊥}, respectively. These losses are inherent to the PC cavity design for the particular mode, so we will refer to them as intrinsic. These losses remain even if the structure were fabricated perfectly. In contrast, extrinsic losses are comprised of material absorption (with rate

*γ*

_{M}) and scattering from fabrication imperfections (with rate

*γ*

_{S}). The total loss rate is expressed as the sum,

*Q*=

*ω*〈

*E*

_{mode}〉 / 〈

*P*〉, which measures the mode confinement as the fractional energy loss per resonator cycle, where

*ω*is the confined mode frequency and 〈

*P*〉 and 〈

*E*

_{mode}〉 are the averaged radiated power and mode energy. Eq.1 then becomes

*γ*= 2

*ω*/

*Q*and defined

*Q*

_{M}= 4

*πn*/

*λα*

_{M}, denoting the commonly used optical absorption coefficient

*α*

_{M}=

*γ*

_{M}/(

*c*/

*n*), with

*c*/

*n*the speed of light in the material with refractive index

*n*.

*V*

_{mode}= 0.5(

*λ*/

*n*)

^{3}, modified to maximize the quality factor

*Q*for the

*x*-dipole mode (Fig. 1(d)). At the design stage, only intrinsic losses were considered. The optimal structure was obtained by changing the six holes neighboring the defect, yielding

*Q*

_{∥}= 58,000 and

*Q*

_{⊥}= 45,000 for structures with seven hole rings around the defect [5

5. J. Vuckovic and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot.” Appl. Phys. Lett. **82**, 2374 – 6 (2003). [CrossRef]

*Q*

_{∥}can always be increased arbitrarily high by addition of hole rings, the total

*Q*is limited by

*Q*

_{⊥}— in this case, to

*Q*~ 45,000.

*λ*

_{0}= 960 nm and lattice periodicity

*a*= 256nm. The design was written with ten hole rings, a number sufficient to guarantee

*Q*

_{⊥}≪

*Q*

_{∥}and

*Q*~

*Q*

_{⊥}. The fabrication consisted of electron-beam lithography (step size 1.5 nm) on Poly-methyl-methacrylate (PMMA) resist, development of the resist, and pattern transfer by dry etching. Subsequently, a sacrificial Al

_{0.94}Ga

_{0.06}As layer was removed (by wet etching) to create the PC membrane. The PC membrane also contained a central layer of self-assembled InAs quantum dots (QDs) with density 200/

*μ*m

^{2}, which served as a convenient broad-band internal illumination source. The absorption due to the QD layer was negligible in comparison to other loss mechanisms, as we will confirm later.

6. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. **95**(013904) (2005). [CrossRef] [PubMed]

^{2}), the QD emission broadened across a wide spectrum (900-950 nm), revealing the Lorentzian lineshape of the cavity as QD emission sees both higher coupling to PC leaky modes, and is emitted faster, as given by the Purcell effect (see Appendix).

*Q*

_{s1}~ 2512, falls well short of the predicted value of 45,000, and the resonance at frequency

*a*/

*λ*

_{s1}= 0.290 differs significantly from the target wavelength of

*a*/

*λ*

_{0}= 0.2847, where

*a*is the lattice periodicity and

*λ*the wavelength in air.

*B*

_{z}∣ component of the confined field at the slab center. From the energy loss relation

*Q*= ω〈

*E*

_{mode}〉/〈

*P*〉, the quality factor is estimated as

*Q*factor as vertical losses are not accounted for. In this calculation, the 3D structure is approximated by an “effective” 2D structure by calculating an effective modal index

*n*

_{eff}[7

7. K. Kawano and T. Kitoh, *Introduction to Optical Waveguide Analysis: Solving Maxwells Equation and the Schrödinger Equation* (Wiley-Interscience Publications, New York, 2001). [PubMed]

*Q*resonances (e.g., see small unpolarized peak just right of the dipole peak). The

*Q*values for these resonances were obtained from 3D simulations to account for the dominant out-of-plane loss.

*E*⃗ denotes the electric field of the resonant mode at frequency

*ω*of the ideal PC structure

*ε*, then the small change Δ

*ω*is given by the first-order perturbation

*ε*is averaged over neighboring cells, so that

*E*⃗ can be approximated as continuously varying[8

8. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E **65**(066611) (2002). [CrossRef]

*λ*/

*λ*

_{0}~ -0.024, or Δ

*λ*~ -23 nm, close to the actual difference of

*λ*

_{meas}-

*λ*

_{design}= 943 - 960 nm= -17 nm.

## 2.1. Simulating Electromagnetic Fields in Fabricated Structures

*Q*= 46,000 and mode volume

*V*

_{mode}= 0.5(

*λ*/

*n*)

^{3}. This design was optimized using a parametric search on the positions of the cavity’s six nearest-neighbor holes, constraining the hole diameters to be within 10% of the unchanged values. Over 300 simulation parameter points were calculated using an automated, iterative optimization procedure; in Fig. 2(b), we show just two slices in parameter space where the four holes in the ΓJ directions were shifted by Δ

*x*, Δ

*y*, and their radii varied. The design that results in fabricated cavities with highest

*Q*is one that has reasonably high theoretical

*Q*and is tolerant to errors in the hole radii and positions. For the dipole cavity mode, we identify the best design as the one shown in Fig. 2(a).

*Q*factors that are easy to fabricate. We will now analyze a set of seven instances of this cavity design. In Fig. 3(b), we show one such fabricated cavity (S2) with its measured PL spectrum (a). We analyzed this and the other structures in the following way.

*a*= 256 nm. In 2D - FDTD simulations, the image was directly converted to a dielectric with 50 program points per

*a*; in 3D simuations, the SEM images were averaged down to 25 pts/

*a*. In both cases, the native resolution limit in the SEM scans requires a calibration for SEM images for differentiating between air and dieletric. This calibration is found from the pixel intensity histogram (Fig. 4(b)) of structure S2 (scaled from 0:black to 1:white), where a threshold value in intensity distinguishes between GaAs and vacuum. This threshold is varied until simulated and observed resonances match (see Fig. 4(a)). For the calibration structure S2, best agreement is found with the threshold at 0.55. Note that the

*Q*values are roughly unchanged and are near the actual value of 2100, except for the lowest threshold value where a lower

*Q*results because the cavity resonance cames close to the bandgap edge. In that case, in-plane confinement suffers and

*Q*drops.

*Q*and

*λ*

_{0}. Simulations of S2 showed that discretizing periods into more than 25 points (or ~ 10 nm) did not considerably change the simulation outcomes.

^{18}~ 260,000) so that the simulated spectrum reliably reproduced the measured one (Fig. 3(c)). To obtain cavity loss rates and filter accurate mode patterns, the simulation was repeated in 3D with fewer time steps (2

^{12}). This yielded

*Q*= 2434 for the dipole mode (Fig. 3(d)).

*Q*values are summarized in Fig. 4(b,c). All experimental resonances matched the directly simulated ones within 2%, far better than the idealized structure simulations (Fig. 4(b)). For

*Q*values, the SEM-based simulations show this improvement even more dramatically (Fig. 4(c)).

## 3. Analyzing loss in photonic crystal cavities

### 3.1. Material Losses

*γ*

_{sim}=

*γ*

_{∥}+

*γ*

_{⊥}+

*γ*

_{S}or

*Q*.

*Q*and

*Q*/

*V*. To answer this question, we return to an earlier analysis that considered the

*Q*of a near-optimal Gaussian cavity field pattern [9

9. D. Englund, I. Fushman, and J. VučkoviĆ, “General Recipe for Designing Photonic Crystal Cavities,” Opt. Express **12**, 5961–75 (2005). [CrossRef]

*k*-space distribution of the cavity mode, which results in lower overlap with the lossy light-cone where TIR does not confine light in the vertical direction. Without extrinsic losses, it was found that

*Q*and

*Q*/

*V*increased roughly exponentially with cavity extent

*σ*

_{x}, as shown in Fig. 5(a). If losses are taken into account according to Eq.2, then

*Q*asymptotically approaches

*Q*

_{M}. Interestingly, this implies that

*Q*/

*V*has an absolute maximum. Fig. 5(b) shows plots of

*Q*and

*Q*/

*V*vs. cavity extent

*σ*

_{x}for pure GaAs at 900 nm (below bandgap) at room temperature, where

*α*= 10/cm [10

10. J. H.C. Casey, D. D. Sell, and K. W. Wecht, “,” Journal of Applied Physics **46**, 250 (1975). [CrossRef]

*Q*/

*V*reaches a maximum at a cavity length of ~ 1.6

*a*, where

*a*is the PC’s lattice periodicity. Thus, in designing a PC structure to exploit cavity quantum electrodynamic effects, the optimal structure should be only a few lattice periods in size (the optimal length has a roughly logarithmic dependence on the extrinsic loss and is therefore rather insensitive to the exact value of the loss).

### 3.2. Extrinsic Scattering Losses

*S*

_{z}, just above the membrane surface. We compare these results against a simulation where the defect was removed in the image (i.e., the hole is rounded out), as shown in the second column. The differences between true and ‘corrected’ structures are displayed in the third column. From the field pattern, we see that the defect shifts the cavity mode slightly (top row). The defect also causes considerable scattering, as seen from the field’s Fourier transform in the second column. Clearly visible are regions of high scatter where waves are re-directed upward and lost out of the cavity (bottom row, right). From the field patterns, we calculate the difference in the loss

*S*

_{z}, as shown in Fig. 6(bottom right). By integrating this loss, we find that removing the defect causes a 4.3% decrease in losses, corresponding to an increase of

*Q*of ~ 4.5%. This calculation is supported by the simulation of the ‘polished’ structure, which predicts an increase in

*Q*of 5.7%. This increase in

*Q*is accompanied by a frequency increase of Δ(

*a*/

*λ*) ~ 4.4 · 10

^{-4}.

*Q*compared to ideal simulated structures. The largest drop in

*Q*arises from global deviations from the design: the structures considered here have systematic radius deviations on the order of 10% and hole center deviations on the order of 5%. This deviation is enough to change cavity

*Q*by up to five times, according to the parametric search (Fig. 2(b)). These deviations cause increased vertical losses due to imperfect TIR [9

9. D. Englund, I. Fushman, and J. VučkoviĆ, “General Recipe for Designing Photonic Crystal Cavities,” Opt. Express **12**, 5961–75 (2005). [CrossRef]

### 3.3. Intrinsic Scattering Losses

*k*- components of

*B*

_{z}and

*E*

_{z}, evaluated in a plane just above the PC slab. One way to reduce these losses is to reduce scattering of high

*k*components back into the light cone. This can be seen by first considering the infinite PC crystal. The Fourier-transform of the wave-equation is [11]

*k*⃗, where we expanded the dielectric as

*k*⃗ can be divided into many subsets, for each wave vector

*K*⃗, and containing terms of form

*A*⃗ (

*K*⃗) and

*A*⃗(

*K*⃗ -

*G*⃗) with all reciprocal lattice vectors

*G*⃗. Now note that if

*ε*(

*r*⃗) consists of only low Fourier components, then high Fourier- components of the field will be decoupled and eliminated. For example, if

*ε*(

*r*⃗) = cos(

*xπ*/

*a*), then each subset of equations for a particular

*K*⃗ consists of only three equations, for

*K*= 0,

*K*= ±

*π*/

*a*.

*k*- components into the light cone, decreasing vertical losses. We tested this with FDTD. The smoothing was achieved by applying a filtering function

*F*

_{S}(

*l*), where

*l*is the FWHM of a Gaussian convolving envelope. In Fig. 7, we compare the field patterns of a single defect cavity with and without smoothing, where smoothing is applied with

*l*= 0.2

*a*. The field is shown as ∣

*FT*

_{2}(

*H*

_{z})∣ + ∣

*FT*

_{2}(

*E*

_{z})∣ (in logarithmic scaling). It is clear that for the smoothed structure, high

*k*components are sharply diminished.

*Q*value. Thus, to show that smoothing truly results in better confinement, the smoothed structure must be re-optimized by changing hole radii and positions. Though we do not pursue this here, we note that this should lead to a decrease of intrinsic losses in PC cavities.

12. G. C. D.*et al*., “Wet Chemical Digital Etching of GaAs at Room Temperature,” J. Electrochemical Soc. **143**, 3652–56 (1996). [CrossRef]

13. J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. **78**, 1180–2 (2001). [CrossRef]

## 4. Conclusion

### A. Measuring the cavity resonance from Purcell-modified spontaneous emission

*μ*⃗ (at position

*r*⃗ and emitting at wavelength

*λ*) by the Purcell factor

*η*

_{cav}to the objective lens, due to the cavity mode’s coupling to photonic crystal leaky modes[6

6. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. **95**(013904) (2005). [CrossRef] [PubMed]

*η*

_{cav}

*dθ*∫

_{A}

*d*

^{2}

*r*⃗

*ρ*(

*r*⃗,

*λ*,

*μ*⃗)

*F*

_{cav}(

*λ*,

*r*⃗,

*μ*⃗), integrated over the observation region A and dipole angle

*θ*with respect to the cavity, where

*ρ*(

*r*⃗,

*λ*,

*μ*⃗) describes the density of QD emission over position, wavelength, and dipole orientation. In this expression, we assumed a continuous laser pump with intensity far above the saturation intensity of discrete lines, as mentioned in the text. At high pump power,

*ρ*broadens to

*ρ*(

*r*⃗,

*λ*,

*μ*⃗) =

*ρ*

_{1}(

*r*⃗)

*ρ*

_{2}(

*λ*)

*ρ*

_{3}(

*μ*) ∝ =

*ρ*

_{2}(

*λ*) as the other components are roughly constant over position and polarization. Thus, the PL is proportional to

*η*

_{cav}

*ρ*(

*λ*)

*F*

_{cav}(

*λ*), and provided

*ρ*(

*λ*) is smooth on the order of the cavity linewidth, the PL follows a Lorentzian lineshape that gives the cavity

*Q*directly.

## References and links

1. | S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics |

2. | S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic Crystal waveguides: role of fabrication disorder and photon Group velocity.” Phys. Rev. Lett. |

3. | A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B. |

4. | P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature |

5. | J. Vuckovic and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot.” Appl. Phys. Lett. |

6. | D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. |

7. | K. Kawano and T. Kitoh, |

8. | S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E |

9. | D. Englund, I. Fushman, and J. VučkoviĆ, “General Recipe for Designing Photonic Crystal Cavities,” Opt. Express |

10. | J. H.C. Casey, D. D. Sell, and K. W. Wecht, “,” Journal of Applied Physics |

11. | A. Yariv and P. Yeh, |

12. | G. C. D. |

13. | J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(140.3410) Lasers and laser optics : Laser resonators

(140.5960) Lasers and laser optics : Semiconductor lasers

(230.5750) Optical devices : Resonators

**ToC Category:**

Optical Devices

**History**

Original Manuscript: February 17, 2006

Revised Manuscript: April 4, 2006

Manuscript Accepted: April 11, 2006

Published: April 17, 2006

**Citation**

Dirk Englund and Jelena Vučković, "A direct analysis of photonic nanostructures," Opt. Express **14**, 3472-3483 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3472

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### References

- S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B: Lasers Opt. 81, 283-293 (2005). [CrossRef]
- S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic optical scattering loss in photonic Crystal waveguides: role of fabrication disorder and photon Group velocity." Phys. Rev. Lett. 94, 033,903-003,904 (2005). [CrossRef]
- A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B. 72, 153,102- (2005). [CrossRef]
- P. Vukusic and J. R. Sambles, "Photonic structures in biology," Nature 424, 852-855 (2003). [CrossRef] [PubMed]
- J. Vuckovic and Y. Yamamoto, "Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot." Appl. Phys. Lett. 82, 2374-2376 (2003). [CrossRef]
- D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. Vučković, "Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal," Phys. Rev. Lett. 95013904 (2005). [CrossRef] [PubMed]
- K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwells Equation and the Schrödinger Equation (Wiley-Interscience Publications, New York, 2001). [PubMed]
- S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwells equations with shifting material boundaries," Phys. Rev. E 65066611 (2002). [CrossRef]
- D. Englund, I. Fushman, and J. Vučković, "General recipe for designing photonic crystal cavities," Opt. Express 12, 5961-5975 (2005). [CrossRef]
- J. H.C. Casey, D. D. Sell, and K. W. Wecht, "," J. Appl. Phys. 46, 250 (1975). [CrossRef]
- A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley Interscience, 2003).
- G. C. DeSalvo, C. A. Bozada, J. L. Ebel, D. C. Look, J. P. Barrette, C. L. A. Cerny, R. W. Dettmer, J. K. Gillespie, C. K. Havasy, T. J. Jenkins, K. Nakano, C. I. Pettiford, T. K. Quach, J. S. Sewell, and G. D. Via, "Wet chemical digital etching of GaAs at room temperature," J. Electrochem. Soc. 143, 3652-3656 (1996). [CrossRef]
- J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, "Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter," Appl. Phys. Lett. 78, 1180-1182 (2001). [CrossRef]

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