## Optical Eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions |

Optics Express, Vol. 19, Issue 2, pp. 933-945 (2011)

http://dx.doi.org/10.1364/OE.19.000933

Acrobat PDF (2215 KB)

### Abstract

We report a mathematically rigorous technique which facilitates the optimization of various optical properties of electromagnetic fields in free space and including scattering interactions. The technique exploits the linearity of electromagnetic fields along with the quadratic nature of the intensity to define specific Optical Eigenmodes (OEi) that are pertinent to the interaction considered. Key applications include the optimization of the size of a focused spot, the transmission through sub-wavelength apertures, and of the optical force acting on microparticles. We verify experimentally the OEi approach by minimising the size of a focused optical field using a superposition of Bessel beams.

© 2011 Optical Society of America

## 1. Introduction

2. M. Kac, “Can one hear the shape of a drum?” Am. Math. Mon. **73**, 1–23 (1966). [CrossRef]

3. A. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. **2**, 211 (1993). [CrossRef]

4. P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. **33**, 327–341 (2001). [CrossRef]

5. J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. **78**, 378–381 (1997). [CrossRef]

6. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. **5**, 1550–1566 (1966). [CrossRef] [PubMed]

7. J. Barton, D. Alexander, and S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. **66**, 4594–4603 (1989). [CrossRef]

8. M. Mazilu, “Spin and angular momentum operators and their conservation,” J. Opt. A **11**, 094005 (2009). [CrossRef]

9. F. García-Vidal, E. Moreno, J. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. **95**, 103901 (2005). [CrossRef] [PubMed]

10. A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science **282**, 919 (1998). [CrossRef] [PubMed]

11. M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. **6**, 118 (2010). [CrossRef]

12. L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, “Simulated holographic three-dimensional intensity shaping of evanescent-wave fields,” J. Opt. Soc. Am. B **25**, 849–853 (2008). [CrossRef]

## 2. Method

**E**,

**H**} where

**E**and

**H**denote the electric and magnetic field vectors, respectively. Table 1 provides a list of common examples of such interactions. These interactions may be written in a general quadratic matrix form where we considered a superposition of fields

*A*) labels the light-matter interaction defined in Table 1. The vectors

**a**and

**a**

^{†}are comprised of the superposition coefficients

*a*and their complex conjugates, respectively. The elements are constructed by combining the respective fields {

_{j}**E**

*,*

_{j}**H**

*} and {*

_{j}**E**

*,*

_{k}**H**

*} for*

_{k}*j*,

*k*= 1 . . .

*N*. More precisely, we have: Given the Hermitian form of Eq. (2), we remark that the light-matter interaction

**M**

^{(}

^{A}^{)}defines a spectrum of real eigenvalues

**E**

*,*

_{j}**H**

*} termed here Optical Eigenmode (OEi). Crucially, we may now extremize the light-matter interaction considered; that is, the extremal eigenvalue*

_{j}*m*

^{(A)}while keeping the total field contributions constant

**a**

^{†}

**a**= 1.

*w*of its intensity distribution [13]. Crucially,

*w*can be expressed in terms of

*m*

^{(0)}and

*m*

^{(2)}(see Table 1) or the respective matrix representations Eq. (1) as follows: where

**M**

^{(0)}and

**M**

^{(2)}are termed the

*intensity operator (IO)*and

*spot size operator (SSO)*, respectively. According to Eq. (3), the minimum spot size is obtained by the OEi associated with the smallest eigenvalue

*λ*

^{(2)}of the SSO provided that the IO is simultaneously diagonalized and normalized to 1. Direct evaluation shows that this is precisely achieved by the combined OEi where

**M**

^{(2)}and in the intensity normalised eigenbase

**M**

^{(0)}.

*E*is considered in order to determine the IO and SSO as and respectively. These scalar expressions are equivalent to the respective vector versions listed in Table 1 and determined through Eq. (2). The scalar version of the optimimum OEi Eq. (5) explicitly reads

_{i}### 2.1. Smallest focal spot using Laguerre Gaussian beams

14. T. Sales and G. Morris, “Fundamental limits of optical superresolution,” Opt. Lett. **22**, 582–584 (1997). [CrossRef] [PubMed]

15. M. Berry, “Faster than Fourier,” in “*Quantum Coherence and Reality; in celebration of the 60th Birthday of Yakir Aharonov*,”, J. S. Anandan and J. L. Safko, eds. (Singapore, 1994), World Scientific, pp. 55–65. [PubMed]

*k*-vector (gradient of the phase) larger than the spectral bandwidth of the original field. To visualize this effect, in the case of OEi spot size optimized beams, we have calculated the spectral density of the radial wave-vector for the smallest planar spot [16

16. M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett. **33**, 2976–2978 (12). [PubMed]

*∂*arg(

_{r}*u*(

*r*)) where arg(

*u*) defines the phase of the analytical signal

*u*. In this particular case, we observe that super-oscillations occur in the dark region of the beam. Additionally, when the ROI is large compared to the Gaussian beam waist

*w*

_{0}, there are no super-oscillating regions. These only appear when the beam starts to be squeezed.

### 2.2. Smallest focal spot using Bessel beams

17. J. Durnin, “Exact solutions for nondiffracting beams. I. the scalar theory,” J. Opt. Soc. Am. A **4**, 651–654 (1987). [CrossRef]

18. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B-Quantum S. O. **4**, S82–S89 (2002). [CrossRef]

*k*=

_{t}*k*

_{0}sin(

*θ*) and

*k*=

_{z}*k*

_{0}cos(

*θ*) are the transversal and longitudinal wave vectors with

*θ*the characteristic cone angle of the Bessel beam.

**e**

*,*

_{x}**e**

*and*

_{y}**e**

*are the unit vectors in the Cartesian coordinate system. The parameter*

_{z}*ℓ*corresponds to the azimuthal topological charge of the beam while

*α*and

*β*are associated with the polarization state of the beam. The magnetic field

**H**was deduced according to Maxwell’s equations. Figure 4 shows a comparison between the Airy disk, the Bessel beam and the OEi optimized spot considering a numerical aperture of NA= 0.1. As in the case of the LG beams, squeezing the focal spot is accompanied by side bands and a loss in efficiency shown by the Strehl ratio (see Fig. 5).

## 3. Experimental OEi

### 3.1. Experimental implementation of the OEi concept

*E*in the SLM planes. In the following, we indicate the plane of interest by a

_{j}*z*-coordinate along the optical axis where

*z*=

*z*

_{1}and

*z*=

*z*

_{2}refer to the SLM and CCD camera plane, respectively. According to this convention we shape a set of test fields

*E*(

_{j}*z*

_{1}) =

*A*(

_{j}*z*

_{1})

*e*

^{iϕj (z1)}both in amplitude

*A*and phase

_{j}*ϕ*in the SLM plane, and the associated intensities

_{j}*I*(

_{j}*z*

_{2}) ∝ |

*E*(

_{j}*z*

_{2})|

^{2}are detected in the CCD camera plane. The amplitudes

*A*(

_{j}*z*

_{2}) were determined from these intensities by simply taking the square root. We used the three-step phase retrieval algorithm described in Ref. [21] to retrieve the phase modulations

*ϕ*(

_{j}*z*

_{2}). The determination of the phase and amplitude of the beam in the CCD plane allows us to numerically vary the ROI without redisplaying the test fields. Using these fields, the IO and SSO are determined according to Eqs. (5) and (6), respectively.

*a*is used to encode the optimized superimposed field onto the SLM. The CCD camera then detected the intensity

_{i}*I*

_{Exp-S}(

*z*

_{2}) corresponding to this encoded optimized field. The Num-S utilizes the fields

*E*(

_{j}*z*

_{2}), which were individually measured to assemble the OEi operators, in order to

*numerically*determine the intensity distribution as

*I*

_{Exp-S}(

*z*

_{2}) =

*I*

_{Num-S}(

*z*

_{2}). This is indeed observed in our experiments as demonstrated in the following subsection which features a comparison of experimental and numerical intensity distributions.

### 3.2. Results and discussion

*N*= 11 non overlapping amplitude ring masks with a constant phase modulation as fields of interest

*E*(

_{i}*z*

_{1}). After propagation through the Fourier lens 5 (see Fig. 6) the resulting fields

*E*(

_{i}*z*

_{2}) form a set of Bessel beams. Figure 7(a) shows the largest ring modulation encoded onto the SLM with the resulting Bessel beam shown in Fig. (b). As this particular Bessel beam comes along with the highest NA compared to the Bessel beams created with smaller ring modulations, the beam shown in Fig. (b) exhibits the smallest central spot of all beams realized in our experiments. The spot size of the Bessel beam featuring the smallest core is denoted as

*w*

_{B}and used as reference for the measurements presented below. For comparison Figure 7(c) depicts a circular aperture which is encoded onto the SLM in order to observe the Airy disk (see Fig. (d)). The spot size of the Airy disk is approximately 1.5 times larger than the core of the reference Bessel beam as expected [22

22. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics **2**, 501–505 (2008). [CrossRef]

*I*

_{Num-S}(

*z*

_{2}) (top row) and the Exp-S intensity distributions

*I*

_{Exp-S}(

*z*

_{2}) (bottom row) clearly reveals good agreement and thus verifies the linearity of our optical system as elucidated above. For completeness, the central row shows the Exp-S superposition in RGB format as encoded onto the SLM. The color code features a blue channel representing the amplitude modulation from 0 (black) to 1 (blue) and a green channel corresponding to phase modulations from 0 (black) to 2

*π*(green). Next, we conclude from the measured relative spot size

*w*/

*w*

_{B}that the spot size decreases if the ROI size is reduced. The reduced spot size is achieved at the expense of the spot intensity which is redistributed to a ring outside of the ROI similar to the theoretical results presented in section 2.2 and Fig. 4. Referring to the Exp-S data, for

*R*= 7 pixel the spot size is reduced to 72 % of the size of the reference Bessel beam’s core and even further to 50 % for

*R*= 4 pixel. The latter result is somewhat vague, though, due to the low spot intensity which may be truncated by the sensitivity threshold of the CCD detector and thus may appear smaller. However, our experimental results overall clearly verify the OEi concept applied to spot size minimization. Moreover, the results strongly suggest that the OEi optimization may indeed squeeze spots to the subdiffractive regime since the optimal superposition of Bessel beams not only beats the Airy disk but also the reference Bessel beam diameter.

## 4. Applicability of the OEi method to scattering interactions

**E**| for the different cases considered. To implement the OEi method, we determine the matrix operator with the help of Eq. (2). Here, we use angular spectral decomposition [19] of the incident light field corresponding to a numerical aperture of NA=0.8.

*m*

^{(0)}to determine the largest transmission through a sub-wavelength aperture (diameter=200nm) in a thin layer of silver (thickness=200nm). The incident light field considered is linearly polarised and the transmission is determined across the output surface of the aperture.

**E**| of most efficient transmission OEi is shown in Fig. 9a illustrating scattering of the aperture. The OEi on its own in Fig. 9b. The transmission enhancement factor, with respect to the tightest Bessel beam achievable for a numerical aperture of NA=0.8 (Fig. 9c), is 2.1 and 1.55 with respect to the Airy diffraction limited disk with the same numerical aperture.

*m*

^{(F·uz)}as defined in Table 1. The momentum OEi with the largest positive eigenvalue corresponds to the field profile (Fig. 9 e–f) giving the largest optical force on the microparticle. Figure 9d shows the field amplitude |

**E**| of this OEi on its own and scattering of the microparticle (Fig. 9e). The optical force enhancement factor, with respect to the plane wave illumination (Fig. 9f), is of 49.3 and of 1.33 with respect to the Airy diffraction limited disk with the same numerical aperture.

## 5. Discussion and Conclusion

## Annex: Dimensionality study

*N*, taken into account, whilst retaining a fixed numerical aperture. The convergence behaviour is compared to standard phase front correction methods which can also be used to achieve focalised spots in the case of highly aberrated light fields. More precisely, these methods are based on the variable partitioning of the SLM to create

*N*beamlets whose phases are individually changed such that all beamlets constructively interfere in the focal target point [23

23. I. Vellekoop and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. **281**, 3071–3080 (2008). [CrossRef]

24. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics **4**, 388–394 (2010). [CrossRef]

24. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics **4**, 388–394 (2010). [CrossRef]

*η*shown in black in Fig. 10a for both standard methods [23

23. I. Vellekoop and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. **281**, 3071–3080 (2008). [CrossRef]

24. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics **4**, 388–394 (2010). [CrossRef]

*N*.

## Acknowledgments

## References and links

1. | C. Cohen-Tannoudji, |

2. | M. Kac, “Can one hear the shape of a drum?” Am. Math. Mon. |

3. | A. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. |

4. | P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. |

5. | J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. |

6. | H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. |

7. | J. Barton, D. Alexander, and S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. |

8. | M. Mazilu, “Spin and angular momentum operators and their conservation,” J. Opt. A |

9. | F. García-Vidal, E. Moreno, J. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. |

10. | A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science |

11. | M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. |

12. | L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, “Simulated holographic three-dimensional intensity shaping of evanescent-wave fields,” J. Opt. Soc. Am. B |

13. | R. Paschotta, |

14. | T. Sales and G. Morris, “Fundamental limits of optical superresolution,” Opt. Lett. |

15. | M. Berry, “Faster than Fourier,” in “ |

16. | M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett. |

17. | J. Durnin, “Exact solutions for nondiffracting beams. I. the scalar theory,” J. Opt. Soc. Am. A |

18. | K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B-Quantum S. O. |

19. | B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic Systems,” Proc. R. Soc. London, Ser. A. |

20. | R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express |

21. | D. Malacara, |

22. | H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics |

23. | I. Vellekoop and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. |

24. | T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics |

**OCIS Codes**

(090.1970) Holography : Diffractive optics

(120.7000) Instrumentation, measurement, and metrology : Transmission

(140.7010) Lasers and laser optics : Laser trapping

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: October 12, 2010

Revised Manuscript: December 24, 2010

Manuscript Accepted: January 3, 2011

Published: January 7, 2011

**Citation**

M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, "Optical Eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions," Opt. Express **19**, 933-945 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-933

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### References

- C. Cohen-Tannoudji, Quantum Mechanics (Wiley, New York, 1977).
- M. Kac, “Can one hear the shape of a drum?” Am. Math. Mon. 73, 1–23 (1966). [CrossRef]
- A. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2, 211 (1993). [CrossRef]
- P. Bienstman, and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001). [CrossRef]
- J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997). [CrossRef]
- H. Kogelnik, and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1566 (1966). [CrossRef] [PubMed]
- J. Barton, D. Alexander, and S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4603 (1989). [CrossRef]
- M. Mazilu, “Spin and angular momentum operators and their conservation,” J. Opt. A: Pure Appl. Opt. 11, 094005 (2009). [CrossRef]
- F. García-Vidal, E. Moreno, J. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005). [CrossRef] [PubMed]
- A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998). [CrossRef] [PubMed]
- M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118 (2010). [CrossRef]
- L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, “Simulated holographic three-dimensional intensity shaping of evanescent-wave fields,” J. Opt. Soc. Am. B 25, 849–853 (2008). [CrossRef]
- R. Paschotta, Encyclopedia of Laser Physics and Technology (Wiley-VCH, 2008).
- T. Sales, and G. Morris, “Fundamental limits of optical superresolution,” Opt. Lett. 22, 582–584 (1997). [CrossRef] [PubMed]
- M. Berry, “Faster than Fourier,” in Quantum Coherence and Reality; in celebration of the 60th Birthday of Yakir Aharonov, J. S. Anandan and J. L. Safko, eds., (World Scientific, Singapore, 1994), pp. 55–65. [PubMed]
- M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett. 33, 2976–2978 (2008). [PubMed]
- J. Durnin, “Exact solutions for nondiffracting beams. I. The Scalar Theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]
- K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002). [CrossRef]
- B. Richards, and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic Systems,” Proc. R. Soc. Lond. A 253, 357–379 (1959).
- R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913 (2007).
- D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992), 2nd ed.
- H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008). [CrossRef]
- I. Vellekoop, and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008). [CrossRef]
- T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010). [CrossRef]

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