Marius Henrich

	The sensitivity of a high-resolution Raman imaging system is crucial to the quality of the acquired information. The spectral and spatial resolutions are among the primary factors that influence the obtainable results. The limits of resolution are defined theoretically by the laws of physics, but are experimentally determined by the instrument parameters. In this article, the theoretical background and the possibilities in practical applications are discussed.
	 

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Key Elements of Confocal Raman Microscopy for High-Resolution Imaging

The sensitivity of a high-resolution Raman imaging system is crucial to the quality of the acquired information. The spectral and spatial resolutions are among the primary factors that influence the obtainable results. The limits of resolution are defined theoretically by the laws of physics, but are experimentally determined by the instrument parameters. In this article, the theoretical background and the possibilities in practical applications are discussed.

Confocal Raman microscopes are the instruments of choice for many Raman measurements in a wide variety of applications ranging from geosciences (1–3), biology (4–6), nanocarbon materials (7–9) to pharmaceutical compounds (10,11), just to name a few. This article sheds light on the possibilities and, in part, the origins in terms of spectral and spatial resolution for confocal Raman systems in general. 

Any confocal Raman system will have a spectral resolution which is mainly determined by the following parameters:  					

the focal length of the spectrometer (the longer the focal length, the higher the spectral resolution)

the grating (the higher the groove density, the higher the spectral resolution)

the pixel size on the charge-coupled device (CCD) camera (the smaller the pixels, the higher the spectral resolution)

the entrance slit or pinhole (the smaller the slit or pinhole the higher the spectral resolution)

the line shape preservation (equals imaging quality) of the spectrometer.

In some cases, one of the parameters can put limitations on the spectral resolution. If, for example, the projection of the pinhole onto the CCD is already large compared to the pixel size on the CCD camera, then a further reduction of pixel size will not increase the spectral resolution. 

Please note that the microscope components such as the objective used for collection of the signal should not influence the spectral resolution if the entrance slit or pinhole is the limiting element. This is preferential with confocal Raman microscopes. 

The determination of the spectral resolution is often a point of debate. First, one should clearly differentiate the spectral resolution from the sensitivity of the system to detect shifts of individual peaks. Relative peak shifts can be detected with a much higher accuracy using fitting algorithms as has been demonstrated with a sensitivity down to 0.02 rel. 1/cm standard deviation of the peak shift of a Si peak (12). The maximum achievable fit accuracy depends heavily on the number of detected photons and the width of the peak that is fitted. This shift analysis is especially relevant for examining stress within a sample, but may not be taken as a measurement for the spectral resolution. 

The spectral resolution, which determines how the system can measure (that is, full width at half maximum [FWHM] of a narrow peak or how well overlapping peaks can be differentiated), needs to be addressed separately from the peak shift sensitivity. There are various ways to state the spectral resolution, and some of the most common ones are outlined below. 

 is the difference in wavenumbers when moving from one pixel on the CCD camera to the next and is independent of factors such as slit width or peak width of the detected peak. This can only be seen as the true resolution limit if the pixel size and not the size of the entrance slit or pinhole is the limiting factor. For example, if the image of the slit or pinhole on the CCD camera is 100 µm in diameter and the pixel size on the CCD camera is 26 µm, then the resolution would be significantly worse than the distance (in wavenumbers) between two pixels. Since wavenumbers are measured in reciprocal space, it also needs to be noted that the pixel resolution will differ depending on the spectral position where it is determined. The resolution close to the Rayleigh line can, in this way, differ by almost a factor of two from the pixel resolution near 3500 rel. 1/cm in the case of 532-nm excitation. 

For this criterion two times the pixel resolution is taken. The logic behind this is that to discriminate two neighboring peaks one needs to have one pixel on one peak, one in the minimum between the peaks, and a third one on the next peak. This criterion is analogous to the Nyquist theorem in signal processing. The same limitations as outlined for the pixel resolution criterion apply in this case.  

Full Width at Half Maximum of Atomic Emission Lines

Atomic emission lines are typically much narrower than any Raman line. Their narrow width makes them a good probe to check the resolution. Figure 1 shows an atomic emission line of mercury near 579.07 nm. The 

-axis is given in units of rel. 1/cm assuming a 532.00-nm excitation laser. The spectrum was recorded using a mercury and argon calibration lamp coupled via a 10-µm core diameter multimode fiber to a UHTS300 spectrograph (WITec GmbH) equipped with both 1800- and 2400 grooves/mm gratings (BLZ at 500 nm) and a Newton electron multiplying charge coupled device (EMCCD) camera with a pixel size of 16 µm. The integration time was 0.1 and 0.24 s, respectively, for the spectra. The FWHM derived through this approach is a good measure of the resolution, but care must be taken to ensure that enough points are available within the curve to ensure a good fit to the curve. 

Figure 1: Spectra of the mercury atomic emission line at 579.066 nm plotted as a function of wavenumber assuming a 532.00-nm excitation. The red line is the fitted curve (pseudo voight function) and the blue diamonds are the measured data points for the 1800-grooves/mm grating. The green triangles and the purple curve show the results obtained using the 2400-grooves/mm grating.			

Measurement of Peak Resolution on Known Reference Samples

There are a few samples that are established standards to demonstrate spectral resolution. The most prominent is probably CCl

. Figure 2 shows two spectra of this substance recorded with different spectral resolutions. It can clearly be seen, that the peaks are nicely separated in the red spectrum whereas the separation is not as clear for the purple spectrum. 

Figure 2: Raman spectrum of CCl4 with 532-nm excitation at different spectral resolutions.			

Therefore, spectral resolution can be defined in many different ways and, thus, it is advisable to specify exactly how a spectral resolution was or should be determined. Comparing actual measurement results under identical measurement conditions is certainly one of the best ways to illustrate this. It should also be noted that with few exceptions the natural linewidths of Raman lines are typically larger than 3 rel. 1/cm. Taking the Nyquist criterion into consideration, a resolution in the range of 1 rel. 1/cm should be sufficient for the majority of samples. 

When considering the spatial resolution of a confocal Raman microscope one may distinguish between the lateral (

) resolution and the depth resolution. However, in either case, it is possible that the limitations arise because of one of the five following points: 					

basic physics (that is, diffraction limit)

limited positioning accuracy of the mechanical components used

limitations because of the optical components used in the beam path (that is, beam distortion)

nondiffraction limited sample illumination

In terms of the positioning system, it is important to differentiate the single step accuracy of a stepper motor, the positioning reproducibility, and linearity. The reproducibility and linearity are the key factors for an imaging system since this allows a line by line imaging. If, for example, a straight line is imaged, then this line will only be imaged with the accuracy of the positioning reproducibility even though the step size might be much smaller. Figure 3 illustrates this effect with Figure 3a showing insufficient positioning reproducibility and Figure 3b showing sufficiently high positioning reproducibility. 

Figure 3: A straight line imaged with (a) insufficient positioning reproducibility and (b) sufficient positioning reproducibility.			

The necessary positioning reproducibility should therefore be several factors better than the smallest imaged object, or if the system should allow the best physically possible resolution, several factors (that is, 10×) better than the diffraction limit. 

Using high-quality components and positioning systems should allow imaging approaching the limits of physics. 

Based on the diffraction theory of Ernst Karl Abbe, Lord Rayleigh defined the diffraction limit in 1896. Rayleigh thereby quantified the minimal distance at which two point light sources can be identified. In this case, one of the sources is located at exactly the distance of the first minimum of the Airy function (point spread function) of the other one. This distance (

) can be expressed as a function of the wavelength emitted (λ) and the numerical aperture (NA) of the objective used as: 

The equation 1.22 λ/NA is often also found in this context, but this describes the distance between the two first minima of the Airy function and not the diffraction limit as described by Lord Rayleigh. Using the definitions of the Airy function the resolution can be easily derived by how much the intensity between the two emitting points has to decrease according to the Rayleigh criterion. Please note that the distance is different for other criteria of diffraction such as the Abbe criterion or the Sparrow criterion. 

To derive the resolution of a microscope using either of those criteria it would be necessary to have a variety of very small, well scattering objects (small compared to the size of the Airy disk; for example individual TiO

 particles) at varying distances. Following high-resolution Raman imaging, one would then have to analyze the signal intensities and based on the detectable distances between the spots one would have to derive the resolution. 

An easier way to determine the lateral resolution power of a system uses the FWHM of small objects. Based on the Airy function the relation between the FWHM of the light emitted from an object and the diffraction limited distance (

Because the measured signal is always a convolution between the object size, the emission characteristics, and the system function, the objects measured in such experiments also have to be small compared to the Airy disk. Mathematically speaking, the object's size should approach a delta function for the convolution. Such small objects naturally have only limited material which scatters and thus one may expect small Raman signals emitted by them. Carbon nanotubes (CNTs) on the contrary show a large Raman signal while being very small in diameter (typically ranging from subnanometer to a few nanometer range) and comparatively long (up to a few micrometers typically). These samples are thus the ideal probes to check the lateral resolution of a confocal Raman microscope. For a 532 nm excitation laser and an objective with an numerical aperture (NA) of 0.9 one should therefore be able to obtain a FWHM across CNTs of about 301 nm. 

Figure 4a shows the integrated intensity of the G-band on a sample of CNTs on a Si substrate. The scan range was 1.5 µm × 1.5 µm with 50 × 50 points and an integration time of 23 ms per point was used. An alpha300R system (WITec GmbH) was used for the measurement in combination with a fiber-coupled, frequency-doubled Nd:YAG laser (532 nm emission), a Zeiss 100× NA 0.9 objective, and a 50-µm multimode fiber acting as the pinhole for confocal microscopy. For this microscope and objective this corresponds to a projected pinhole size of 500 nm in the focal plane. The spectrometer used was a UHTS300 (WITec GmbH) with a 600-grooves/mm grating (BLZ at 500 nm) combined with a back-illuminated CCD camera.  

Figure 4: Integrated intensity of the G-band of a carbon nanotube with (a) the cross section position marked in red and (b) the cross-sectional intensity along this line. The lateral resolution of the microscope system can be characterized by the FWHM and is about 272 nm.			

The red line in Figure 4a indicates where the cross section shown in Figure 4b was extracted from. It can clearly be seen that the FWHM is even narrower than the predicted minimum. 

The theory of confocal microscopy (for example, see reference 13) shows that the achievable resolution can further be decreased by a maximum of 1/√2. In confocal Raman microscopy this fact is rarely used to its limit, because it requires a strong reduction of the pinhole diameter, which in turn reduces the throughput. 

Depth resolution is the best proof of the confocality of a system. The instrument design has a key influence on the achievable resolution, but the pinhole or slit as well as the way the sample is illuminated also play a crucial role for the depth resolution. Therefore, these points are outlined below before the physical limit of the depth resolution is discussed. 

The sensitivity of a high-resolution Raman imaging system is crucial to the quality of the acquired information.