## Detection limits of confocal surface plasmon microscopy |

Biomedical Optics Express, Vol. 5, Issue 6, pp. 1744-1756 (2014)

http://dx.doi.org/10.1364/BOE.5.001744

Acrobat PDF (1217 KB)

### Abstract

This paper applies rigorous diffraction theory to evaluate the minimum mass sensitivity of a confocal optical microscope designed to excite and detect surface plasmons operating on a planar metallic substrate. The diffraction model is compared with an intuitive ray picture which gives remarkably similar predictions. The combination of focusing the surface plasmons and accurate phase measurement mean that under favorable but achievable conditions detection of small numbers of molecules is possible, however, we argue that reliable detection of single molecules will benefit from the use of structured surfaces. System configurations needed to optimize performance are discussed.

© 2014 Optical Society of America

## 1. Introduction

**2.** Localized analyte detection in a confocal imaging system

6. B. Zhang, S. Pechprasarn, and M. G. Somekh, “Quantitative plasmonic measurements using embedded phase stepping confocal interferometry,” Opt. Express **21**(9), 11523–11535 (2013). [CrossRef] [PubMed]

6. B. Zhang, S. Pechprasarn, and M. G. Somekh, “Quantitative plasmonic measurements using embedded phase stepping confocal interferometry,” Opt. Express **21**(9), 11523–11535 (2013). [CrossRef] [PubMed]

*p-*polarized so SPs are strongly excited as can be seen from the dip in the back focal plane, in the vertical direction light is

*s-*polarized so no SPs are excited. For radially polarized light, on the other hand, the excitation strength is uniform with azimuthal angle. If we now look at situation that applies to the sample with linear input polarization as shown in Fig. 2(b) we observe the ring where the SPs are excited shown by the shaded annulus. The shading denotes the variation in SP signal strength with azimuthal angle, the strongest excitation corresponding to the direction of

*p-*incident polarization. The radius, R, of the ring is approximately given by

*Δz*tan

*θ*where

_{p}*Δz*is the defocus and

*θ*the angle of excitation of SPs. Figure 2(b) shows the presence of the analyte represented by a blue disc in the figure, this is shown well localized within the annulus of excitation to the optical axis. The effect of the analyte is to change the phase velocity of the SPs in this region so that the phase of the emitted radiation is retarded slightly compared to the case when it is absent. It can be seen immediately that all the rays pass through the sample when it is located on the optical axis, which means that the average phase shift is enhanced. Figure 2(c) shows an unfocused beam passing through a similar region of analyte where we see that a phase shift is only imposed on a small portion of the incident beam, so that the measured mean phase shift is greatly reduced as shown by the dashed line. In order to recover the phase of the SP the reference beam (P1 of Fig. 1(b)) is phase shifted relative to the beam that forms the SP, using a spatial light modulator conjugate to the back focal plane as explained in [6

_{p}6. B. Zhang, S. Pechprasarn, and M. G. Somekh, “Quantitative plasmonic measurements using embedded phase stepping confocal interferometry,” Opt. Express **21**(9), 11523–11535 (2013). [CrossRef] [PubMed]

*V*, to be deposited inside the ring. If the material is uniformly spread throughout the ring this will result in material deposited to a height,

*h*, where

*r(<R)*. In this case the height,

*H*, of the pillar is

*k-*vector for the SP is calculated from

*θ*is the plasmon excitation angle,

_{p}*n*is the refractive index of the couplant between objective and sample. Comparing the value of

_{oil}*k*-vector for the bare gold substrate with a region coated with 1 nm of material of index 1.5 enables one to calculate the change in SP

*k*-vector equal to 1.99x10

^{−5}nm

^{−1}for a free space wavelength of 633nm. If we assume all the rays pass through a disc of diameter 800nm the total additional phase shift will be 0.0159 rads. Throughout this paper we consider an objective lens with numerical aperture 1.65 using coupling fluid of index 1.78, the conclusions for a 1.49 objective and couplant of index 1.52 are very similar.

*k*-vector of the SPs. The standard measure of mass density in most of the SP literature is that one, so called, response unit, is equivalent to a change of 10

^{−6}RIU, which, in turn, is equivalent to a mass coverage on the sample of 1 pgmm

^{−2}[7

7. ICX NOMADICS, “Overview of Surface Plasmon Resonance,” http://www.sensiqtech.com/uploads/file/support/spr/Overview_of_SPR.pdf (accessed 28th Nov 2013).

8. F. Höök, B. Kasemo, T. Nylander, C. Fant, K. Sott, and H. Elwing, “Variations in coupled water, viscoelastic properties, and film thickness of a Mefp-1 protein film during adsorption and cross-linking: A quartz crystal microbalance with dissipation monitoring, ellipsometry, and surface plasmon resonance study,” Anal. Chem. **73**(24), 5796–5804 (2001). [CrossRef] [PubMed]

^{−2}. In the rest of the paper we will use this equivalence of mass coverage, this value is close, but slightly higher than, the values given for proteins so that our estimates while applicable for proteins will give a slightly conservative estimate for mass sensitivity.

## 3. Microscope model

- (i) The first stage is simply calculation of the incident field incident on the sample surface. Assuming the pupil function at the back focal plane is given by:
*P(k*where_{x},k_{y})*k*and_{x}*k*are the allowable spatial frequencies transmitted by the objective. When the sample is defocused the incident plane wave spectrum is given by:_{y}where*k*is the spatial frequency in the axial direction. The caret over the_{z}*E*denotes the angular spectrum of the field. In the system modeled here the spatial light modulator performs a phase stepping measurement so the calculations of the field in image plane (parts iii, iv, and v) use 4 different pupil functions corresponding to each phase step. - (ii) In order to model the effect of the analyte we need to develop a full wave solution to the problem. To this end we use rigorous coupled wave analysis (RCWA) to model the interaction of the sample with the incident radiation. RCWA provides an optimal solution to Maxwell’s equations for a periodic structure, it does, however, have certain limitations that need to be handled carefully if one is to obtain reliable results. The first limitation, of course, is that the solution that is obtained is for a periodic structure represented by a finite number of diffracted orders, the technique is therefore ‘rigorous’ provided only that the number of diffracted orders is sufficient to give an accurate representation of the problem. Another related issue is that since RCWA solves the electromagnetic problem for a periodic structure when we wish to model an isolated feature the size of the repeating unit cells must be sufficiently large that their mutual interaction can be ignored.The mutual dependence between the two limitations above means that it is necessary to exercise care in the application of the RCWA model. Let us make this discussion more definite by referring to the specific structures we need to model. We wish to examine an isolated patch of radius,
*r*, and determine how the incident radiation from the microscope will interact with it. If the patch is to be isolated the effect of reflections from adjacent unit cells must not reflect light back into the illuminated unit cell. In order to evaluate this distance,*d*, we carried out the following test: we consider the microscope response at a remote position from the dielectric patch, at a midpoint as indicated by ‘M’ in Fig. 2(d). Our assertion is that if the results from this region are close to the values obtained by simply applying the Fresnel equations to a uniform substrate we may safely argue that the unit cells are sufficiently large to be considered isolated. This separation value is significant not only from the point of view of the computation but also informs us how closely one can place binding sites on a sensor chip. The calculations showed that a unit cell dimension,*d*, of 10μm gave a maximum phase deviation compared from a truly uniform substrate of 1.3x10^{−5}radians which is less than 1/1200 of the phase difference observed in the region of the patch; we can therefore be confident that unit cell size chosen was sufficient to give results representative of an isolated structure.Unfortunately, the large unit cell also sets another problem since the unit cell size determines the spatial frequency of the diffracted orders, so the larger the size of the unit cell the smaller the value of the spatial frequency associated with each diffracted order. To accurately reproduce the fields from a small structure the highest spatial frequency must be sufficient to replicate the object, this means, of course, that if the chosen unit cell is large many more diffraction orders are needed for a given object size. Increasing the number of diffracted orders has three principal effects in the implementation of the RCWA model (i) Increased storage requirements, which are, of course, particularly severe in 2D implementation of RCWA where the (2*N + 1)*diffracted orders are required, where^{2}*N*is the number of positive and negative diffracted orders (ii) increased computational time and (iii) the possibility of ill-conditioning in the matrices during inversion which was not, in this case, a problem. (i) and (ii) are, of course, strongly dependent on each other especially when we need to use virtual memory as discussed in the next paragraph.In order to implement the RCWA program we used the excellent code written by Kwiecien in Matlab. This appears to be exceptionally stable and fast and has been benchmarked against other available packages such as RODIS [9Fig. 4Log scale of intensity of each diffracted order for N = 13 (white) and N = 50 (black); these were calculated with noil of 1.78 incident at 54 degrees (plasmonic angle) with 633 nm incident wavelength where the sample was a rod array (nrod = 1.5) with grating period of 10 μm, 1 nm thick and 400 nm in radius.] where similar results were obtained, however, the Kwiecien code [109. Photonics Research Group University of Gent, “Gent rigorous optical diffration software (RODIS),” http://www.photonics.intec.ugent.be/research/facilities/design/rodis (accessed 28th Nov 2013).

] gives far better representation of the evanescent fields for 2D solutions. Nevertheless, the particularly severe conditions required for our computation required some additional modification of the code. We wished to model patches whose diameter approached that of the focus of the SPs, so that, in principle, it is necessary to use10. I. Richter, P. Kwiecien, and J. Ctyroky, “Advanced photonic and plasmonic waveguide nanostructures analyzed with Fourier modal methods,” in

*Transparent Optical Networks (ICTON), 2013 15th International Conference on*(2013), pp. 1–7. [CrossRef]*N≥d/r*which means that*N*should be greater than approximately*25.*Unfortunately, the code as provided only allows for*N = 20*, so we modified the code to allow for*N = 50.*The problem, of course, was that this requires a large amount of memory, so the model was extended to set up a virtual page on the hard drive. This allowed one to compute the diffracted orders up to*N = 50,*unfortunately to do this took 3 days per point and we wish to calculate approximately 200,000 spatial frequencies in the back focal plane to get a good representation of the field. We used this extended model to calculate a single point incident close to the angle necessary to excite SPs and compare the results for the propagating waves (those that will return to the back focal plane) with a model calculated with a more manageable number of diffracted orders. When*N = 13*that is when the total number of diffracted orders was*27*, the propagating diffracted orders for the^{2}= 729*N = 50*are very similar to the*N = 13*case as shown in Fig. 4; this ensures that a reliable result could be achieved even with the reduced number of diffracted orders. The fact that the dielectric object is extremely ‘weak’ giving only a small amount of scattering somewhat relaxes the requirement to use large numbers of diffracted orders. The reduced memory requirements for the*N = 13*case means that the separate cores of the processor could be used for parallel processing without recourse to virtual memory.

- (iii) The diffracted orders can then be propagated to the back focal plane by applying an appropriate phase shift that allows for the displacement and defocus of the object.The incident plane wave spectrum is denoted by the first term in the integral. The scattering term
*S*denotes the diffraction from incident k-vectors_{RCWA}*k*,_{x}*k*to output k-vectors_{y}*k*and_{x}’*k*. This also takes account of the polarization state of the incident light. The effect of defocus and displacement from the origin are expressed in the last two exponential terms. The parameters_{y}’*k*and_{x}’*k*denote spatial frequencies of the scattered light which map to spatial positions in the back focal plane, any magnification terms are omitted for clarity. The terms_{y}’*x*and_{s}*y*represent displacements from the optical axis in x and y directions respectively._{s} - Each point on the back focal plane maps to a plane wave in the image plane whose summation gives the spatial distribution in the image plane. Since we are concerned with a partially coherent imaging system the detected intensity is the integrated intensity over the pinhole. For a small pinhole Eq. (3) is the ideal confocal response.
- (v) The precise field distribution at the image plane allows one to calculate the exact number of photons arriving at the pinhole for well defined input conditions. We calculate this for each phase shift imposed by the spatial light modulator and estimate the noise for each phase step.
- (vi) The noisy input data from each phase step is then used to recover the phase noise.
- (vii) This phase variation is equated to a particular mass change on the sample surface from which the minimum detectable number of molecules can be calculated for different measurement strategies.

## 4. Results and tests with the RCWA model

*p-*and

*s-*directions as denoted in Fig. 7 differ substantially falling off much more rapidly as the sample is displaced towards the

*s-*polarization direction. The negative value on the green curve is due to the sidelobe of the SP focus passing through the analyte.

## 5. Estimating minimum detectable signal under shot noise limit

*m*phase step is given by:Where

^{th}*ϕ*is the phase shift associated with the deposited sample. Each of these signals is subject to shot noise proportional to

^{6}times, so that the mean phase error was calculated for different substrates and pinhole opening apertures. This was converted to a specific mass sensitivity from the calculated phase shift given by the RCWA calculations from which the minimum number of detectable molecules was estimated.

11. B. Zhang, S. Pechprasarn, J. Zhang, and M. G. Somekh, “Confocal surface plasmon microscopy with pupil function engineering,” Opt. Express **20**(7), 7388–7397 (2012). [CrossRef] [PubMed]

12. A. V. Kabashin, S. Patskovsky, and A. N. Grigorenko, “Phase and amplitude sensitivities in surface plasmon resonance bio and chemical sensing,” Opt. Express **17**(23), 21191–21204 (2009). [CrossRef] [PubMed]

^{−3}degrees this will correspond to detection of approximately 37 molecules with 100kD mass; these measurements were obtained close to a reflection minimum where the noise is expected to be worse than in our confocal microscope system. We reiterate here that our estimate of the conversion of mass to refractive index units is about 43% more conservative the values presented widely in the literature.

*c.*8%.

^{4}, in which case the mean separation between molecules will be of the order 100 molecular diameters which means even in that case they are isolated, so our extrapolation down to small numbers of molecules is no different from the assumptions made in previous literature [3

3. P. Kvasnička, K. Chadt, M. Vala, M. Bocková, and J. Homola, “Toward single-molecule detection with sensors based on propagating surface plasmons,” Opt. Lett. **37**(2), 163–165 (2012). [CrossRef] [PubMed]

## 6. Techniques to achieve the theoretical sensitivity

14. K. McKenzie, M. B. Gray, P. K. Lam, and D. E. McClelland, “Technical limitations to homodyne detection at audio frequencies,” Appl. Opt. **46**(17), 3389–3395 (2007). [CrossRef] [PubMed]

15. M. A. van Dijk, M. Lippitz, D. Stolwijk, and M. Orrit, “A common-path interferometer for time-resolved and shot-noise-limited detection of single nanoparticles,” Opt. Express **15**(5), 2273–2287 (2007). [CrossRef] [PubMed]

*θ*terms refer to the incident angles for plasmon excitation and reference beam respectively. For a normally incident beam

*θ*is close to zero so cos

_{reference}*θ*is approximately 1. If

_{reference}*θ*is close to

_{reference}*θ*then the microphonic noise terms cancel out. F To address this issue we have developed a system that while conceptually similar to the interferometer described here uses a reference beam that is incident at the angle as the SP, the results from this system will be reported shortly. The four phase steps also impose a requirement on the speed of measurement, so that there is no certainty that each measurement can be obtained under the same condtions. SLMs are available that can achieve 1000 frames per second so that a measurement with 4 phase steps can be achieved in 4ms which overcomes a large proportion of the microphonic noise [14

_{plasmon}14. K. McKenzie, M. B. Gray, P. K. Lam, and D. E. McClelland, “Technical limitations to homodyne detection at audio frequencies,” Appl. Opt. **46**(17), 3389–3395 (2007). [CrossRef] [PubMed]

## 7. Conclusions

16. S. Roh, T. Chung, and B. Lee, “Overview of the characteristics of micro- and nano-structured surface plasmon resonance sensors,” Sensors (Basel) **11**(12), 1565–1588 (2011). [CrossRef] [PubMed]

*k-*vector of surface waves for a given analyte thickness. Developments of these surfaces combined with instrumental developments discussed here mean that detection of single protein molecules should be achievable in the near future.

## Acknowledgment

## References and links

1. | P. P. Markowicz, W. C. Law, A. Baev, P. N. Prasad, S. Patskovsky, and A. Kabashin, “Phase-sensitive time-modulated surface plasmon resonance polarimetry for wide dynamic range biosensing,” Opt. Express |

2. | S.-Y. Wu and H.-P. Ho, “Single-beam self-referenced phase-sensitive surface plasmon resonance sensor with high detection resolution,” Chin. Opt. Lett. |

3. | P. Kvasnička, K. Chadt, M. Vala, M. Bocková, and J. Homola, “Toward single-molecule detection with sensors based on propagating surface plasmons,” Opt. Lett. |

4. | M. Piliarik and J. Homola, “Self-referencing SPR imaging for most demanding high-throughput screening applications,” Sens. Actuators B Chem. |

5. | S. Pechprasarn and M. G. Somekh, “Surface plasmon microscopy: Resolution, sensitivity and crosstalk,” J. Microsc. |

6. | B. Zhang, S. Pechprasarn, and M. G. Somekh, “Quantitative plasmonic measurements using embedded phase stepping confocal interferometry,” Opt. Express |

7. | ICX NOMADICS, “Overview of Surface Plasmon Resonance,” http://www.sensiqtech.com/uploads/file/support/spr/Overview_of_SPR.pdf (accessed 28th Nov 2013). |

8. | F. Höök, B. Kasemo, T. Nylander, C. Fant, K. Sott, and H. Elwing, “Variations in coupled water, viscoelastic properties, and film thickness of a Mefp-1 protein film during adsorption and cross-linking: A quartz crystal microbalance with dissipation monitoring, ellipsometry, and surface plasmon resonance study,” Anal. Chem. |

9. | Photonics Research Group University of Gent, “Gent rigorous optical diffration software (RODIS),” http://www.photonics.intec.ugent.be/research/facilities/design/rodis (accessed 28th Nov 2013). |

10. | I. Richter, P. Kwiecien, and J. Ctyroky, “Advanced photonic and plasmonic waveguide nanostructures analyzed with Fourier modal methods,” in |

11. | B. Zhang, S. Pechprasarn, J. Zhang, and M. G. Somekh, “Confocal surface plasmon microscopy with pupil function engineering,” Opt. Express |

12. | A. V. Kabashin, S. Patskovsky, and A. N. Grigorenko, “Phase and amplitude sensitivities in surface plasmon resonance bio and chemical sensing,” Opt. Express |

13. | R. P. Feynman, R. B. Leighton, and M. Sands, “The origin of refractive index,” in |

14. | K. McKenzie, M. B. Gray, P. K. Lam, and D. E. McClelland, “Technical limitations to homodyne detection at audio frequencies,” Appl. Opt. |

15. | M. A. van Dijk, M. Lippitz, D. Stolwijk, and M. Orrit, “A common-path interferometer for time-resolved and shot-noise-limited detection of single nanoparticles,” Opt. Express |

16. | S. Roh, T. Chung, and B. Lee, “Overview of the characteristics of micro- and nano-structured surface plasmon resonance sensors,” Sensors (Basel) |

**OCIS Codes**

(060.4080) Fiber optics and optical communications : Modulation

(110.0110) Imaging systems : Imaging systems

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(180.0180) Microscopy : Microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: January 13, 2014

Revised Manuscript: March 7, 2014

Manuscript Accepted: March 19, 2014

Published: May 6, 2014

**Citation**

Suejit Pechprasarn and Michael G. Somekh, "Detection limits of confocal surface plasmon microscopy," Biomed. Opt. Express **5**, 1744-1756 (2014)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-5-6-1744

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### References

- P. P. Markowicz, W. C. Law, A. Baev, P. N. Prasad, S. Patskovsky, and A. Kabashin, “Phase-sensitive time-modulated surface plasmon resonance polarimetry for wide dynamic range biosensing,” Opt. Express15(4), 1745–1754 (2007). [CrossRef] [PubMed]
- S.-Y. Wu and H.-P. Ho, “Single-beam self-referenced phase-sensitive surface plasmon resonance sensor with high detection resolution,” Chin. Opt. Lett.6(3), 176–178 (2008). [CrossRef]
- P. Kvasnička, K. Chadt, M. Vala, M. Bocková, and J. Homola, “Toward single-molecule detection with sensors based on propagating surface plasmons,” Opt. Lett.37(2), 163–165 (2012). [CrossRef] [PubMed]
- M. Piliarik and J. Homola, “Self-referencing SPR imaging for most demanding high-throughput screening applications,” Sens. Actuators B Chem.134(2), 353–355 (2008). [CrossRef]
- S. Pechprasarn and M. G. Somekh, “Surface plasmon microscopy: Resolution, sensitivity and crosstalk,” J. Microsc.246(3), 287–297 (2012). [CrossRef] [PubMed]
- B. Zhang, S. Pechprasarn, and M. G. Somekh, “Quantitative plasmonic measurements using embedded phase stepping confocal interferometry,” Opt. Express21(9), 11523–11535 (2013). [CrossRef] [PubMed]
- ICX NOMADICS, “Overview of Surface Plasmon Resonance,” http://www.sensiqtech.com/uploads/file/support/spr/Overview_of_SPR.pdf (accessed 28th Nov 2013).
- F. Höök, B. Kasemo, T. Nylander, C. Fant, K. Sott, and H. Elwing, “Variations in coupled water, viscoelastic properties, and film thickness of a Mefp-1 protein film during adsorption and cross-linking: A quartz crystal microbalance with dissipation monitoring, ellipsometry, and surface plasmon resonance study,” Anal. Chem.73(24), 5796–5804 (2001). [CrossRef] [PubMed]
- Photonics Research Group University of Gent, “Gent rigorous optical diffration software (RODIS),” http://www.photonics.intec.ugent.be/research/facilities/design/rodis (accessed 28th Nov 2013).
- I. Richter, P. Kwiecien, and J. Ctyroky, “Advanced photonic and plasmonic waveguide nanostructures analyzed with Fourier modal methods,” in Transparent Optical Networks (ICTON), 2013 15th International Conference on (2013), pp. 1–7. [CrossRef]
- B. Zhang, S. Pechprasarn, J. Zhang, and M. G. Somekh, “Confocal surface plasmon microscopy with pupil function engineering,” Opt. Express20(7), 7388–7397 (2012). [CrossRef] [PubMed]
- A. V. Kabashin, S. Patskovsky, and A. N. Grigorenko, “Phase and amplitude sensitivities in surface plasmon resonance bio and chemical sensing,” Opt. Express17(23), 21191–21204 (2009). [CrossRef] [PubMed]
- R. P. Feynman, R. B. Leighton, and M. Sands, “The origin of refractive index,” in The Feynman Lectures on Physics: Mainly Mechanics, Radiation, and Heat (Basic Books, 2011), Chap. 21.
- K. McKenzie, M. B. Gray, P. K. Lam, and D. E. McClelland, “Technical limitations to homodyne detection at audio frequencies,” Appl. Opt.46(17), 3389–3395 (2007). [CrossRef] [PubMed]
- M. A. van Dijk, M. Lippitz, D. Stolwijk, and M. Orrit, “A common-path interferometer for time-resolved and shot-noise-limited detection of single nanoparticles,” Opt. Express15(5), 2273–2287 (2007). [CrossRef] [PubMed]
- S. Roh, T. Chung, and B. Lee, “Overview of the characteristics of micro- and nano-structured surface plasmon resonance sensors,” Sensors (Basel)11(12), 1565–1588 (2011). [CrossRef] [PubMed]

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