"Polarized" or "polarizing" opinions? - polarizing meaning

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Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

One note of caution; this description is only a first approximation. All that we have said is true as long as the size of the mirror is small compared to its radius of curvature. Another way of saying this is to limit ourselves to rays of light that lie close to the optic axis. The focal length of a spherical mirror is one half the radius of curvature of the mirror, f = R / 2 This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

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Curved mirrors can produce all sorts of images. We will restrict our attention to spherical mirrors. Mirrors that reflect on the inside of the spherical surface are called concave mirrors; they will cause parallel light to converge on a point. Mirrors that reflect on the outside of the spherical surface are called convex mirrors; they will cause parallel light to diverge as if coming from a common point. Figure 18.5 shows a cross sectional view of both a convex mirror and a concave mirror. The axis of symmetry is known as the optic axis; the axis of symmetry will pass through the center of curvature of the mirror. The optic axis will be a useful reference line throughout our study of image formation. Figure 18.D Curved "fun house" mirrors produce strange and unusual images. Cylindrical mirrors can even "decode" strange pictures and turn them into recognizable figures. Figure 18.5 A line perpendicular to a spherical mirror is called the optic axis. The optic axis passes through the center of curvature of the mirror and the focal point. The optic axis is an axis of symmetry. Rays of light from an object that is infinitely far away are parallel by the time we see them. Such parallel rays, after reflecting from a concave (or converging) spherical mirror, are bent so they converge on a single point. They pass through that point and then diverge from that point. After reflecting from a convex (or diverging) spherical mirror, such parallel rays are bent so they diverge as if they had come from a single point. If our eyes intercept these rays after their reflection they will look exactly as if they had originated from this point. For both mirrors, this point from which the light seems to have originated is called the focal point and is labeled by a capital letter F. The distance from the mirror to the focal point is the focal length and is labeled with a small letter f. We will adopt the convention that the focal length is positive for a concave mirror (f > 0) and is negative for a convex mirror (f < 0). These ideas are illustrated in Figure 18.6. Figure 18.6 Rays of light parallel to the optic axis are focused at a single point by a spherical mirror. This point is called the focal point of the mirror. The distance from the focal point to the mirror is the focal length. One note of caution; this description is only a first approximation. All that we have said is true as long as the size of the mirror is small compared to its radius of curvature. Another way of saying this is to limit ourselves to rays of light that lie close to the optic axis. The focal length of a spherical mirror is one half the radius of curvature of the mirror, f = R / 2 This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point. [Prev Section] [Next Section] [Table of Contents] [Chapter Contents]

Jiang, S. et al. Distinguishing adjacent molecules on a surface using plasmon-enhanced Raman scattering. Nat. Nanotechnol. 10, 865–870 (2015).

Neuman, T. T. et al. Coupling of molecular emitters and plasmonic cavities beyond the point-dipole approximation. Nano Lett. 18, 2358–2364 (2018).

Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

Imada, H. et al. Single-molecule investigation of energy dynamics in a coupled plasmon-exciton system. Phys. Rev. Lett. 119, 13901 (2017).

Liu, P., Chulhai, D. V. & Jensen, L. Single-molecule imaging using atomistic near-field tip-enhanced Raman spectroscopy. ACS Nano 11, 5094–5102 (2017).

Figure shows the direction of the induced dipole moment vector represented by the blue arrow for different vibrational symmetries and tip positions. The tip positions are at the phenyl ring (a) and (c) and in between two phenyl rings b, as indicated by the red dot. The dipole moment vector was obtained using pρ(ωL± ων) = αρσEσ(ωL) where pρ, Eσ, ωL and ων are the dipole moment vector, incident electric field vector of the plasmonic nanocavity, frequency of the laser and the frequency of the vibrational mode respectively, and αρσ is the Raman tensor given by equation 1 in the main text. The cartesian coordinate axes x, y, and z are represented by the subscripts ρ and σ. The vector of the plasmonic nanocavity at three different tip positions (red dot) is represented by the grey arrow.

a, STA of the molecule showing the different excitation wavelengths used as coloured lines according to the colours in (b) and (c), STA shown is the same as Fig. 2c of the main text. In (b) and (c), the labels correspond to the wavelength used for the particular spectrum. b, Wavelength dependence of the Raman spectrum with wavelength as x-axis. The spectra were clipped from 755 nm for the data taken using 720 - 739 nm pump and from 790 nm for the those taken using 742 nm and above to account for the long pass filter response. The small lines at the left part of each spectrum correspond to the position of the excitation laser per spectrum. c, Wavelength dependence of the Raman spectrum with wavenumber as x-axis. Raman data were taken using 1 V, 50 pA, 30 s exposure time, and 1 mW laser power. The spectra in (b) and (c) were shifted vertically for clarity.

Baiardi, A., Bloino, J. & Barone, V. A general time-dependent route to resonance-Raman spectroscopy including Franck-Condon, Herzberg-Teller and Duschinsky effects. J. Chem. Phys. 141, 114108 (2014).

Ramanscattering vs Rayleigh scattering

Figure shows the calculated molecular orbital of the CuNc at the x-y (a–c) and x-z (d–f) planes. For a centrosymmetric molecule like CuNc, HOMO-LUMO transitions are only allowed when there is a change in the parity as invoked by the parity rule. As viewed from the x-y plane, the molecular orbital show odd parity for the HOMO a, and an even parity for the LUMO b, c. This implies that HOMO-LUMO transitions with excitation along the x-y plane are parity allowed according to the Laporte rule. On the other hand, as viewed from the x-z plane, the molecular orbitals shows an odd parity for both the HOMO (d) and the LUMO e, f, suggesting that, HOMO-LUMO transitions with excitation along the z-axis are parity forbidden according to the Laporte rule.

Figure 18.6 Rays of light parallel to the optic axis are focused at a single point by a spherical mirror. This point is called the focal point of the mirror. The distance from the focal point to the mirror is the focal length. One note of caution; this description is only a first approximation. All that we have said is true as long as the size of the mirror is small compared to its radius of curvature. Another way of saying this is to limit ourselves to rays of light that lie close to the optic axis. The focal length of a spherical mirror is one half the radius of curvature of the mirror, f = R / 2 This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

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Raman effectexample

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Figure 18.D Curved "fun house" mirrors produce strange and unusual images. Cylindrical mirrors can even "decode" strange pictures and turn them into recognizable figures. Figure 18.5 A line perpendicular to a spherical mirror is called the optic axis. The optic axis passes through the center of curvature of the mirror and the focal point. The optic axis is an axis of symmetry. Rays of light from an object that is infinitely far away are parallel by the time we see them. Such parallel rays, after reflecting from a concave (or converging) spherical mirror, are bent so they converge on a single point. They pass through that point and then diverge from that point. After reflecting from a convex (or diverging) spherical mirror, such parallel rays are bent so they diverge as if they had come from a single point. If our eyes intercept these rays after their reflection they will look exactly as if they had originated from this point. For both mirrors, this point from which the light seems to have originated is called the focal point and is labeled by a capital letter F. The distance from the mirror to the focal point is the focal length and is labeled with a small letter f. We will adopt the convention that the focal length is positive for a concave mirror (f > 0) and is negative for a convex mirror (f < 0). These ideas are illustrated in Figure 18.6. Figure 18.6 Rays of light parallel to the optic axis are focused at a single point by a spherical mirror. This point is called the focal point of the mirror. The distance from the focal point to the mirror is the focal length. One note of caution; this description is only a first approximation. All that we have said is true as long as the size of the mirror is small compared to its radius of curvature. Another way of saying this is to limit ourselves to rays of light that lie close to the optic axis. The focal length of a spherical mirror is one half the radius of curvature of the mirror, f = R / 2 This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

The observed peaks in the STM-TERS correspond well to DFT calculated vibrational modes and the powder Raman spectrum taken at room temperature (see Table S1). STM-TERS spectrum was taken using 738 nm laser with 1 mW and 30 s exposure time at Vb = 1 V and It = 50 pA. Powder Raman spectrum was taken using a homebuilt Raman microscope with 532 nm excitation at 1 mW power and 30 s exposure time.

Raman effectnotes

This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

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Figure shows the intensity plots of individual vibrational modes taken at 1 s integration time, Vb = 1 V, It = 50 pA and 1 angstrom / pixel step size (40 pixels × 40 pixels). A complete Raman spectrum is taken per pixel simultaneously with the STM topography also shown here. Apart from one peak (1211 cm-1, see Supplementary Table 1), all experimental TERS peaks can be assigned unambiguously to calculated resonance Raman vibrational modes. A table comparing the DFT and experimental results is shown in Supplementary Table 1. Three image profiles can be seen and their appearance are due to the symmetry of the assigned vibrational modes as discussed in the main text.

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A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

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Figure 18.5 A line perpendicular to a spherical mirror is called the optic axis. The optic axis passes through the center of curvature of the mirror and the focal point. The optic axis is an axis of symmetry. Rays of light from an object that is infinitely far away are parallel by the time we see them. Such parallel rays, after reflecting from a concave (or converging) spherical mirror, are bent so they converge on a single point. They pass through that point and then diverge from that point. After reflecting from a convex (or diverging) spherical mirror, such parallel rays are bent so they diverge as if they had come from a single point. If our eyes intercept these rays after their reflection they will look exactly as if they had originated from this point. For both mirrors, this point from which the light seems to have originated is called the focal point and is labeled by a capital letter F. The distance from the mirror to the focal point is the focal length and is labeled with a small letter f. We will adopt the convention that the focal length is positive for a concave mirror (f > 0) and is negative for a convex mirror (f < 0). These ideas are illustrated in Figure 18.6. Figure 18.6 Rays of light parallel to the optic axis are focused at a single point by a spherical mirror. This point is called the focal point of the mirror. The distance from the focal point to the mirror is the focal length. One note of caution; this description is only a first approximation. All that we have said is true as long as the size of the mirror is small compared to its radius of curvature. Another way of saying this is to limit ourselves to rays of light that lie close to the optic axis. The focal length of a spherical mirror is one half the radius of curvature of the mirror, f = R / 2 This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

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StimulatedRamanscattering

Figure shows the STM topography (a) and (b) of the CuNc molecule and the DFT calculated molecular orbitals (c)–(e). The STM topography at HOMO a, was taken at Vb = -1.85 V and It = 10 pA while LUMO image b, was taken at Vb = 0.8 V and It = 10 pA. The STM image of the HOMO level is identical in shape to the DFT calculated HOMO c, Meanwhile, the DFT calculated LUMO is degenerated into LUMOα and LUMOβ ((d) and e, respectively). Considering that the STM can image the degenerated LUMO, of the molecule simultaneously, the STM image the STM image correspond well to the spatial distribution of the density ρ = |LUMOα|2 + |LUMOβ|2 shown in f. Since the STM image of CuNc on NaCl/Ag(111) mimics the DFT calculated molecular orbitals of a free molecule, we can argue that the NaCl is effective in electronically decoupling the CuNc molecule from the Ag(111) surface.

Department of Chemistry, Faculty of Science and Institute for Chemical Reaction Design and Discovery (WPI-ICReDD), Hokkaido University, Kita-ku, Sapporo, Japan

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This work was financially supported in part by the Grant-in-Aid for Scientific Research (KAKENHI) nos 15H02025 (Y.K.), 15H03569 (N.H.), 17H04796 (H.I.), 17H05470 (H.I.), 17K18766 (H.I.), 18K14153 (R.B.J.), 16K21623 (K.M.) and 17K14428 (T.I.), and Grant-in-Aid for JSPS Fellows no. 15J03915 (K.M.) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. R.B.J. acknowledges the Special Postdoctoral (SPDR) program of RIKEN. Parts of the numerical calculations were performed with the aid of the HOKUSAI supercomputer system at RIKEN. We thank H. Ueba, K. Kimura and M. Balgos for helpful discussions, D. Miyajima and T. Aida for the vacuum purification and room temperature absorbance measurements and F.C. Catalan and R. Wong for carefully reading the manuscript.

Wu, S. W., Nazin, G. V. & Ho, W. Intramolecular photon emission from a single molecule in a scanning tunneling microscope. Phys. Rev. B 77, 205430 (2008).

Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

R.B.J., H.I. and N.H. designed the experiments. R.B.J. performed all experiments with contributions from H.I.; R.B.J., N.H. and H.I. analysed the experimental data, and K.M. provided DFT calculations of the vibrational modes. T.I., M.T. and T.T. performed the electric field simulations. B.Y. and E.K. prepared the Au tips. Y.K. planned and supervised the project. All authors contributed to the interpretation of the results and writing the manuscript.

18.2 Reflection from a Curved Mirror [Prev Section] [Next Section] [Table of Contents] [Chapter Contents] Curved mirrors can produce all sorts of images. We will restrict our attention to spherical mirrors. Mirrors that reflect on the inside of the spherical surface are called concave mirrors; they will cause parallel light to converge on a point. Mirrors that reflect on the outside of the spherical surface are called convex mirrors; they will cause parallel light to diverge as if coming from a common point. Figure 18.5 shows a cross sectional view of both a convex mirror and a concave mirror. The axis of symmetry is known as the optic axis; the axis of symmetry will pass through the center of curvature of the mirror. The optic axis will be a useful reference line throughout our study of image formation. Figure 18.D Curved "fun house" mirrors produce strange and unusual images. Cylindrical mirrors can even "decode" strange pictures and turn them into recognizable figures. Figure 18.5 A line perpendicular to a spherical mirror is called the optic axis. The optic axis passes through the center of curvature of the mirror and the focal point. The optic axis is an axis of symmetry. Rays of light from an object that is infinitely far away are parallel by the time we see them. Such parallel rays, after reflecting from a concave (or converging) spherical mirror, are bent so they converge on a single point. They pass through that point and then diverge from that point. After reflecting from a convex (or diverging) spherical mirror, such parallel rays are bent so they diverge as if they had come from a single point. If our eyes intercept these rays after their reflection they will look exactly as if they had originated from this point. For both mirrors, this point from which the light seems to have originated is called the focal point and is labeled by a capital letter F. The distance from the mirror to the focal point is the focal length and is labeled with a small letter f. We will adopt the convention that the focal length is positive for a concave mirror (f > 0) and is negative for a convex mirror (f < 0). These ideas are illustrated in Figure 18.6. Figure 18.6 Rays of light parallel to the optic axis are focused at a single point by a spherical mirror. This point is called the focal point of the mirror. The distance from the focal point to the mirror is the focal length. One note of caution; this description is only a first approximation. All that we have said is true as long as the size of the mirror is small compared to its radius of curvature. Another way of saying this is to limit ourselves to rays of light that lie close to the optic axis. The focal length of a spherical mirror is one half the radius of curvature of the mirror, f = R / 2 This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point. [Prev Section] [Next Section] [Table of Contents] [Chapter Contents]

Raman effectformula

Tip-enhanced Raman spectroscopy (TERS) is a versatile tool for chemical analysis at the nanoscale. In earlier TERS experiments, Raman modes with components parallel to the tip were studied based on the strong electric field enhancement along the tip. Perpendicular modes were usually neglected. Here, we investigate an isolated copper naphthalocyanine molecule adsorbed on a triple-layer NaCl on Ag(111) using scanning tunnelling microscope TERS imaging. For flat-lying molecules on NaCl, the Raman images present different patterns depending on the symmetry of the vibrational mode. Our results reveal that components of the electric field perpendicular to the tip should be considered aside from the parallel components. Moreover, under resonance excitation conditions, the perpendicular components can play a substantial role in the enhancement. This single-molecule study in a well-defined environment provides insights into the Raman process at the plasmonic nanocavity, which may be useful in the nanoscale metrology of various molecular systems.

Bhattarai, A. et al. Tip-enhanced Raman scattering from nanopatterned graphene and graphene oxide. Nano Lett. 18, 4029–4033 (2018).

Peer review information Nature Nanotechnology thanks Patrick (Z.) El-Khoury and the other, anonymous, reviewers for their contribution to the peer review of this work.

Ramanspectroscopy principle

Whiteman, P. J., Schultz, J. F., Porach, Z. D., Chen, H. & Jiang, N. Dual binding configurations of subphthalocyanine on Ag(100) substrate characterized by scanning tunneling microscopy, tip-enhanced raman spectroscopy, and density functional theory. J. Phys. Chem. C 122, 5489–5495 (2018).

Egidi, F., Bloino, J., Cappelli, C. & Barone, V. A robust and effective time-independent route to the calculation of resonance raman spectra of large molecules in condensed phases with the inclusion of Duschinsky, Herzberg–Teller, anharmonic, and environmental effects. J. Chem. Theory Comput. 10, 346–363 (2013).

Yang, B., Kazuma, E., Yokota, Y. & Kim, Y. Fabrication of sharp gold tips by three-electrode electrochemical etching with high controllability and reproducibility. J. Phys. Chem. C 122, 16950–16955 (2018).

The focal length of a spherical mirror is one half the radius of curvature of the mirror, f = R / 2 This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

Ding, S. Y., You, E. M., Tian, Z. Q. & Moskovits, M. Electromagnetic theories of surface-enhanced Raman spectroscopy. Chem. Soc. Rev. 46, 4042–4076 (2017).

A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

Jaculbia, R.B., Imada, H., Miwa, K. et al. Single-molecule resonance Raman effect in a plasmonic nanocavity. Nat. Nanotechnol. 15, 105–110 (2020). https://doi.org/10.1038/s41565-019-0614-8

Rays of light from an object that is infinitely far away are parallel by the time we see them. Such parallel rays, after reflecting from a concave (or converging) spherical mirror, are bent so they converge on a single point. They pass through that point and then diverge from that point. After reflecting from a convex (or diverging) spherical mirror, such parallel rays are bent so they diverge as if they had come from a single point. If our eyes intercept these rays after their reflection they will look exactly as if they had originated from this point. For both mirrors, this point from which the light seems to have originated is called the focal point and is labeled by a capital letter F. The distance from the mirror to the focal point is the focal length and is labeled with a small letter f. We will adopt the convention that the focal length is positive for a concave mirror (f > 0) and is negative for a convex mirror (f < 0). These ideas are illustrated in Figure 18.6. Figure 18.6 Rays of light parallel to the optic axis are focused at a single point by a spherical mirror. This point is called the focal point of the mirror. The distance from the focal point to the mirror is the focal length. One note of caution; this description is only a first approximation. All that we have said is true as long as the size of the mirror is small compared to its radius of curvature. Another way of saying this is to limit ourselves to rays of light that lie close to the optic axis. The focal length of a spherical mirror is one half the radius of curvature of the mirror, f = R / 2 This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

Cvraman effect

Miwa, K., Sakaue, M. & Kasai, H. Effects of interference between energy absorption processes of molecule and surface plasmons on light emission induced by scanning tunneling microscopy. J. Phys. Soc. Jpn 82, 124707 (2013).

Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

Miwa, K., Sakaue, M., Gumhalter, B. & Kasai, H. Effects of plasmon energetics on light emission induced by scanning tunneling microscopy. J. Phys. Condens. Matter 26, 222001 (2014).

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Lee, J., Crampton, K. T., Tallarida, N. & Apkarian, V. A. Visualizing vibrational normal modes of a single molecule with atomically confined light. Nature 568, 78–82 (2019).

Anti StokesRamanscattering

Chen, X., Liu, P., Hu, Z. & Jensen, L. High-resolution tip-enhanced Raman scattering probes sub-molecular density changes. Nat. Commun. 10, 2567 (2019).

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Swart, I., Sonnleitner, T. & Repp, J. Charge state control of molecules reveals modification of the tunneling barrier with intramolecular contrast. Nano Lett. 11, 1580–1584 (2011).

Chai, J.-D. & Head-Gordon, M. Systematic optimization of long-range corrected hybrid density functionals. J. Chem. Phys. 128, 084106 (2008).

[Prev Section] [Next Section] [Table of Contents] [Chapter Contents] Curved mirrors can produce all sorts of images. We will restrict our attention to spherical mirrors. Mirrors that reflect on the inside of the spherical surface are called concave mirrors; they will cause parallel light to converge on a point. Mirrors that reflect on the outside of the spherical surface are called convex mirrors; they will cause parallel light to diverge as if coming from a common point. Figure 18.5 shows a cross sectional view of both a convex mirror and a concave mirror. The axis of symmetry is known as the optic axis; the axis of symmetry will pass through the center of curvature of the mirror. The optic axis will be a useful reference line throughout our study of image formation. Figure 18.D Curved "fun house" mirrors produce strange and unusual images. Cylindrical mirrors can even "decode" strange pictures and turn them into recognizable figures. Figure 18.5 A line perpendicular to a spherical mirror is called the optic axis. The optic axis passes through the center of curvature of the mirror and the focal point. The optic axis is an axis of symmetry. Rays of light from an object that is infinitely far away are parallel by the time we see them. Such parallel rays, after reflecting from a concave (or converging) spherical mirror, are bent so they converge on a single point. They pass through that point and then diverge from that point. After reflecting from a convex (or diverging) spherical mirror, such parallel rays are bent so they diverge as if they had come from a single point. If our eyes intercept these rays after their reflection they will look exactly as if they had originated from this point. For both mirrors, this point from which the light seems to have originated is called the focal point and is labeled by a capital letter F. The distance from the mirror to the focal point is the focal length and is labeled with a small letter f. We will adopt the convention that the focal length is positive for a concave mirror (f > 0) and is negative for a convex mirror (f < 0). These ideas are illustrated in Figure 18.6. Figure 18.6 Rays of light parallel to the optic axis are focused at a single point by a spherical mirror. This point is called the focal point of the mirror. The distance from the focal point to the mirror is the focal length. One note of caution; this description is only a first approximation. All that we have said is true as long as the size of the mirror is small compared to its radius of curvature. Another way of saying this is to limit ourselves to rays of light that lie close to the optic axis. The focal length of a spherical mirror is one half the radius of curvature of the mirror, f = R / 2 This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point. [Prev Section] [Next Section] [Table of Contents] [Chapter Contents]

Jiang, N. et al. Nanoscale chemical imaging of a dynamic molecular phase boundary with ultrahigh vacuum tip-enhanced Raman spectroscopy. Nano Lett. 16, 3898–3904 (2016).

f = R / 2 This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors. Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed. Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2. Q: How are you able to see a virtual image? A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image. Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach? A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

Imada, H. et al. Real-space investigation of energy transfer in heterogeneous molecular dimers. Nature 538, 364–367 (2016).

Latorre, F. et al. Spatial resolution of tip-enhanced Raman spectroscopy – DFT assessment of the chemical effect. Nanoscale 8, 10229–10239 (2016).

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