Using a commercial ellipsometer and analytical inversion, we show that both linear and circular birefringence-dichroism pairs can be extracted from a single generalized ellipsometry measurement, providing a complete description of the polarization properties of anisotropic chiral films, which is a distinct advantage over typical circular dichroism measurements. This is demonstrated by measuring the anisotropic optical parameters of post-like and helical composite Ti/Ag thin films fabricated by dynamic shadowing growth. These films are both chiral and highly aligned, and the measured linear and circular birefringence-dichroism pairs scale with the shape anisotropy and chirality. Furthermore, because the total polarization anisotropy is measured through generalized ellipsometry, we are able to determine that the polarization eigenstates can be effectively tuned from purely circular to approximately linear by changing the pitch number, *N*, of plasmonic helices for *N* ≤ 1.

Chirality has recently become an important topic in modern optics, especially through the emergence of chiral plasmonics.^{1} The aim of chiral plasmonics is to understand and engineer the coupling between chiral electromagnetic fields (e.g., left and right circular polarized light) and chiral metal structures. These interactions can have real world applications, such as chiral sensors or wide-angle, polarization-independent absorbers.^{2,3} Many of these applied chiral plasmonic studies employ aligned chiral metal structures, either due to the fabrication method requirements or specific applications. Unfortunately, the use of aligned chiral structures becomes problematic for polarization-dependent optical measurements, such as circular dichroism (CD) spectroscopy, because linear anisotropy resulting from structural alignment can introduce artifacts in circular polarization measurements.^{4} These artifacts complicate experimental analysis and can obscure measurement of the true chiral plasmonic properties. The most popular method to account for these linear artifacts, known as the “front and back” method,^{5} involves making CD measurements through both the front and back of the sample and then taking the average. However, this method may be difficult to implement for some experimental situations (e.g., *in situ*, time-resolved or real time measurements), and worse, it may not give accurate results.^{6} The inaccuracy associated with the front and back methods may not only arise from experimental considerations such as small sample size and non-uniformity, but also because anisotropies may not be additive for strong signals; therefore, averaging will not cancel out the linear effects. Thus, a simple method for reliably extracting and analyzing the circular polarization parameters of aligned or anisotropic chiral plasmonic structures is needed.

Spectroscopic ellipsometry measures the change in polarization state of light, as a function of wavelength, upon transmission or reflection from a sample.^{7,8} Commercial ellipsometry instruments are available, and the technique has long been used for thin film analyses.^{9} Among different ellipsometry measurement strategies, the generalized ellipsometry technique expands upon conventional ellipsometry to include measurement of the normalized off-diagonal Jones matrix elements of an optical system.^{10} Thus, generalized ellipsometry is ideal for the characterization and analysis of homogenous, anisotropic samples. A complete Jones matrix is a comprehensive description of a non-depolarizing optical system.^{11} Notably, when an optical system or sample contains a combination of linear and circular anisotropy, the individual Jones matrix elements are a product of these different retardances and absorbances, whose individual magnitudes can be obtained through matrix decomposition or inversion in certain cases.^{12–14} Recently, Arteaga and Canillas described an analytical method for such an inversion that permits the direct extraction of (*x* – *y*) linear birefringence and dichroism (*LB* and *LD*), 45° birefringence and dichroism (*LB′* and *LD′*), and circular birefringence and dichroism (*CB* and *CD*) from a single Mueller-Jones matrix, experimentally measured using a custom two-modulator generalized ellipsometer (these various retardance parameters are defined in Table I).^{14} This analytical method can easily be extended to encompass the inversion of generalized ellipsometry spectra obtained through ellipsometers with a single compensator, thus providing a readily available method for extracting and analyzing the circular polarization properties of aligned or anisotropic chiral plasmonic structures.

Effect . | Symbol . | Definition^{a}
. |
---|---|---|

Mean refractive index | Ñ | 2π[(n_{x} + n)_{y}l/λ_{0}] |

Mean absorbance | Ã | ln10(A + _{x}A)/2 _{y} |

(x – y) linear birefringence | LB | 2π[(n_{x} – n)_{y}l/λ_{0}] |

(x – y) linear dichroism | LD | ln10(A – _{x}A)/2 _{y} |

45° linear birefringence | LB′ | 2π[(n_{45} – n)_{135}l/λ_{0}] |

45° linear dichroism | LD′ | ln10(A – _{45}A)/2 _{135} |

Circular birefringence | CB | 2π[(n_{−} – n)_{+}l/λ_{0}] |

Circular dichroism | CD | ln10(A – _{−}A)/2 _{+} |

Effect . | Symbol . | Definition^{a}
. |
---|---|---|

Mean refractive index | Ñ | 2π[(n_{x} + n)_{y}l/λ_{0}] |

Mean absorbance | Ã | ln10(A + _{x}A)/2 _{y} |

(x – y) linear birefringence | LB | 2π[(n_{x} – n)_{y}l/λ_{0}] |

(x – y) linear dichroism | LD | ln10(A – _{x}A)/2 _{y} |

45° linear birefringence | LB′ | 2π[(n_{45} – n)_{135}l/λ_{0}] |

45° linear dichroism | LD′ | ln10(A – _{45}A)/2 _{135} |

Circular birefringence | CB | 2π[(n_{−} – n)_{+}l/λ_{0}] |

Circular dichroism | CD | ln10(A – _{−}A)/2 _{+} |

^{a}

Where *n* is the conventionally defined refractive index; *A* is the standard absorbance in the base *e*; *l* is the path length through the sample; *λ*_{0} is the vacuum wavelength of light; and the subscripts refer to the polarization direction of light as *x*, *y*, 45° to the *x*-axis, 135° to the *x*-axis, right circular (+), or left circular (−).

In particular, by using a spectroscopic ellipsometer in transmission mode, one can obtain the normalized Jones matrix, **J**:

where *t _{xx}*,

*t*,

_{yy}*t*, and

_{xy}*t*are the complex transmission coefficients for orthogonal directions,

_{yx}*x*and

*y*, and

*ψ*and Δ

_{ij}_{ij}are the conventional generalized ellipsometric parameters,

*ψ*= tan

_{ij}^{−1}|

*t*/

_{ij}*t*| and Δ

_{xx}_{ij}= arg|

*t*/

_{ij}*t*| for

_{xx}*i*,

*j*=

*x*,

*y*, which describe the amplitude ratios and phase differences, respectively, for

*x*- and

*y*-polarization upon transmission through a sample. Note that there are different conventions associated with normalization, range of Δ, and signs that will need to be considered when using experimentally obtained data.

^{15}As mentioned above, a Jones matrix can be parameterized by the various polarization dependent birefringence and dichroism values:

^{4,14}

where *χ* = 2(*ñ* − *iÃ*), *T* = (*L*^{2} + *L′*^{2} + *C*^{2})^{1/2}, *L* = *LB − iLD*, *L′* = *LB′ − iLD′*, *C* = *CB – iCD*, and *ñ* and *Ã* are the mean refractive index and mean absorbance, respectively (Table I). Following the method of Arteaga and Canillas, we identify parameters in Eq. (1) with the parameters in Eq. (2):

where *K* is a complex constant. The following relationships are determined by solving the systems of equations created by combining the different matrix elements:

where $\Omega =T\u2009K\u2009/(2\u2009\u2009sin[T/2])$ and $T=2\u2009\u2009cos\u2009\u22121[K(1+tan\u2009\psi yy\u2009ei\Delta yy)/2]$. Equations (4)–(10) provide a direct relationship between the generalized ellipsometry parameters and the birefringent and dichroic properties defined in Table I.

In order to demonstrate the utility of the generalized ellipsometry and analytical inversion, we have applied these methods to six different nanostructured composite Ti/Ag films, labeled as STF1, STF2, STF3, STF4, STF5, and STF6, fabricated using dynamic shadowing growth (DSG), a well-known physical vapor deposition technique.^{15–19} Representative scanning electron microscope (SEM, FEI Inspect F) images of these films are shown in Figures 1(a)–1(f). It can be seen that the nanostructure morphologies of STF1 (Figure 1(a)) resemble conical, vertical posts, which are characteristic of DSG using a relatively fast azimuthal rotation rate and with a fast adatom mobility.^{20} These nanostructures are comparatively achiral and have very little shape anisotropy along the substrate plane. In comparison, the nanostructure morphologies of STF2 resemble helices, while the morphologies of STF3, STF4, STF5, and STF6 resemble slanted, curved posts, or quasi-helices. Thus, STF2 through STF6 are described by chiral nanostructures with increasing and aligned shape anisotropy within the substrate plane. These nanostructures are consistent with DSG growth having a decreasing azimuthal rotation rate. The average morphological parameters, height (*h*), diameter (*d*), radius of curvature (*r*), pitch (*p*), and pitch number (*N* = *h*/*p*) are defined in Figure 2, and the results for the different films are listed in Table II. In general, these plasmonic films have highly aligned nanostructures and varying degrees of chiral and linear anisotropies arising from their morphologies. These mixed anisotropies will result in a nontrivial polarization-dependent optical response and exemplify the difficulties that contraindicate traditional optical characterization of aligned chiral films. Therefore, these films serve as ideal samples to test the generalized ellipsometry method.

. | Height, h (nm)
. | Diameter, d (nm)
. | Radius of Curvature, r (nm)
. | Pitch, p (nm)
. | Pitch number, N
. |
---|---|---|---|---|---|

STF1 | 160 ± 60 | 80 ± 60 | … | … | … |

STF2 | 110 ± 30 | 70 ± 30 | 90 ± 20 | 100 ± 5 | 1.1 ± 0.3 |

STF3 | 240 ± 20 | 100 ± 70 | 260 ± 30 | 560 ± 50 | 0.43 ± 0.05 |

STF4 | 200 ± 50 | 80 ± 30 | 540 ± 20 | 900 ± 200 | 0.22 ± 0.07 |

STF5 | 200 ± 20 | 80 ± 30 | 1120 ± 90 | 1900 ± 200 | 0.11 ± 0.02 |

STF6 | 220 ± 20 | 60 ± 30 | 2000 ± 200 | 4200 ± 400 | 0.052 ± 0.007 |

. | Height, h (nm)
. | Diameter, d (nm)
. | Radius of Curvature, r (nm)
. | Pitch, p (nm)
. | Pitch number, N
. |
---|---|---|---|---|---|

STF1 | 160 ± 60 | 80 ± 60 | … | … | … |

STF2 | 110 ± 30 | 70 ± 30 | 90 ± 20 | 100 ± 5 | 1.1 ± 0.3 |

STF3 | 240 ± 20 | 100 ± 70 | 260 ± 30 | 560 ± 50 | 0.43 ± 0.05 |

STF4 | 200 ± 50 | 80 ± 30 | 540 ± 20 | 900 ± 200 | 0.22 ± 0.07 |

STF5 | 200 ± 20 | 80 ± 30 | 1120 ± 90 | 1900 ± 200 | 0.11 ± 0.02 |

STF6 | 220 ± 20 | 60 ± 30 | 2000 ± 200 | 4200 ± 400 | 0.052 ± 0.007 |

The generalized ellipsometry spectra of the Ti/Ag composite films were measured using a commercial ellipsometer (J. A. Woollam, Inc., M-2000) in transmission mode at normal incidence, and then the polarization dependent retardances and absorbances were extracted from these measurements using Eqs. (4)–(10). These results are plotted in Figure 3. The measured linear birefringence-dichroism pairs (*LB-LD* and *LB′-LD′*) are consistent with expectations given the nanostructure morphology and alignment with the experimental coordinate system (Figure 1). For example, the nanostructures of STF5 are primarily aligned with the *x*-axis, and therefore show the most significant *LB* and *LD* (Figures 3(a) and 3(d)), with *LB ∼ *1.1 at wavelength *λ* ∼ 600 nm and *LD ∼ *1.4 at *λ* ∼ 1000 nm. On the other hand, the nanostructures of STF4 and STF6 are very much in congruence with the 45°-axis, and therefore have relatively small values for *LB* and *LD* but have large values for *LB′* and *LD′*, *LB′ > *1 at *λ = *600 nm and *LD > *1.2 at *λ* = 1000 nm (Figures 3(b) and 3(e)). In comparison, the nanostructures of STF1, STF2, and STF3 have a higher degree of symmetry with respect to both the *x*- and 45°-axes and therefore have smaller values for *LB*, *LD*, *LB′*, and *LD′* than the more anisotropic films. As shown in Figures 3(c) and 3(f), the films with helical or quasi-helical morphologies, STF2 through STF6, exhibit significant circular polarization effects, as expected, while the vertical posts of STF1 show very slight *CB* and *CD*. Due to its small radius of curvature and full helicity, STF2 shows a characteristic bisignate *CD* response over the measured wavelengths. STF3 through STF6 shows broadband *CD* peaks that alternate between positive and negative, in agreement with the alternating handedness of their helicities. Interestingly, STF4 is a quarter-turn quasi-helical film and has stronger *CB* and *CD* than STF2 and STF3, which contain full-turn and half-turn helices, respectively. This result is surprising. However, a number of factors can contribute to the large obtained CB and CD magnitudes of STF4. Plasmonic optical activity can be enhanced not only by a highly chiral shape, but also through efficient coupling with the incident light and between adjacent nanostructures and also through efficient mixing of different plasmonic modes.^{21} Additionally, recent studies have demonstrated that noble metals shaped into arcs or incomplete helices can exhibit substantial plasmonic optical activity.^{22,23}

It is important to note that like most aligned chiral films, all of the chiral Ti/Ag films show a mixture of anisotropies, meaning that each film is generally defined by elliptical birefringence and dichroism, i.e., the eigenpolarizations are neither linear nor circular. Therefore, it is convenient to define a generalized retardance vector, ** T** = (

*L*,

*L′*, -

*C*), which has absolute magnitude |

**| = (**

*T**LB*

^{2}+

*LD*

^{2}+

*LB′*

^{2}+

*LD′*

^{2}+

*CB*

^{2}+

*CD*

^{2})

^{1/2}in order to quantify and assess the total anisotropy of the individual films. As shown in Figure 4(a), |

**| generally increases for STF1 through STF6, as the nanostructure morphology changes from vertical post-like to helical to quasi-helical, i.e., as the nanostructures exhibit increasing shape anisotropy along the substrate plane. For circular polarization applications, it is also useful to consider the magnitude of circular retardance normalized by the magnitude of total anisotropy,**

*T**C*= |

_{T}*C*| × |

**|**

*T*^{−1}, which quantifies the degree of circular polarization of the transmitted light. The results for the different chiral Ti/Ag films are shown in Figure 4(b). Remarkably, STF2 has

*C*∼ 1 for wavelengths 550 nm ≤

_{T}*λ*≤ 650 nm, indicating that the anisotropy is purely circular in that region. Though STF2 does not exhibit the greatest values of

*CB*or

*CD*of the films examined in this report, it does have among the highest reported values of optical chirality,

^{24}and it will transmit circular polarized light without polarization conversion over 550 nm ≤

*λ*≤ 650 nm, making it ideal for device applications. By comparing the morphological parameters from Table II with Figure 4(b), it is clear that

*C*increases with pitch number,

_{T}*N*, of the quasi-helical Ti/Ag films. In fact,

*C*is approximately equal to the pitch number for

_{T}*N*≤ 1 and

*λ*≥ 650 nm, as shown in Figure 4(c), which plots

*C*at

_{T}*λ*= 650 nm versus

*N*for the quasi-helical films. These results illustrate the possibility of designing chiral plasmonic structures with specific elliptical polarization eigenstates over a large region of wavelengths.

In conclusion, we have demonstrated the advantages of using generalized ellipsometry for polarization-sensitive optical characterization of advanced, highly aligned chiral plasmonic nanostructured thin films. The representative films examined in this report exhibit large circular birefringence and dichroism in addition to large linear anisotropy, all of which can be quantified by a single generalized ellipsometry measurement. This is a distinct advantage over typical CD spectroscopy methods, which require at least two separate measurements and may not provide accurate results. Furthermore, because the total polarization anisotropy is measured through generalized ellipsometry, interesting and useful phenomena of chiral plasmonic nanostructures can be observed that might go unnoticed by solely measuring CD spectroscopy. For example, the results reported here show that the polarization eigenstates can be effectively tuned from purely circular to approximately linear by changing the pitch number, *N,* of the plasmonic helices for *N* ≤ 1. Finally, generalized ellipsometry can provide extensive information from a single measurement, both optical and structural, and the combination of generalized ellipsometry with aligned chiral plasmonic films or coupled nanoparticles might lead to new applications in biosensing or advanced plasmonic rulers.^{25}

The authors thank Mr. Yizhuo He and Professor Jing Wang for their assistance with sample preparation. The authors are grateful for support from the National Science Foundation (Grant No. ECCS-1029609).

## References

Note that some references make the distinction between measurements made in reflection or transmission, calling the former ellipsometry and the latter polarimetry; we do not make this distinction.