Each section of this paper uses an example from optical science to illustrate electronic publishing capabilities not typically used (or even available) in print media. We include color plots, a movie and a reader-interactive Java applet as well as standard line drawings. It is safer to save any pdf article, especially if it contains links, on your hard drive first (in source format from Netscape) and then to use Acrobat Reader to read it.

Color Plot

The paraxial theory of wave optics [1] predicts free-space light beams with Gaussian intensity profiles. The intensity of a Gaussian beam averaged over an optical period can be expressed as:

where I ₀ is the peak intensity at the focus and W ₀ is the beam waist at focus. The variable z represents the distance from the focus in the direction of the beam, and the beam waist as a function of z is given by $w\left(z\right)=w_0\sqrt{1+{\left(z/z_0\right)}^2,}$ where z ₀ is the Rayleigh range defined by z ₀ = π$w_0^2$/λ.

The intensity of this Gaussian beam on a plane parallel to the direction of propagation and passing through the focus is shown in Fig. 1 as a contour plot. The central color zone represents the region of intensities that are within 10% of the peak intensity, the next color zone represents the region of intensities that are within 10–20% of the peak intensity, etc.

 

Figure 1: Ten intensity zones of a Gaussian beam (see text) within one z ₀ longitudinally and 1.5w ₀ radially of focus (the horizontal direction represents the z direction). The color scheme resembles the optical spectrum for frequency, i.e., orange represents lower values of the intensity and violet represents higher values of the intensity.

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Linked Figure with Enlargement

Diffraction by structures with sub-wavelength scale features can be a key consideration in the design of modern optical devices. Ray optics fails to be reliable. When polarization effects are important the scalar approach to Maxwell’s equations is invalid. New methods are being developed to treat optical transmission, absorption and reflection by materials with realistic complex dielectric indices.

In Fig. 2 we show the calculated electric field [2] near to the surface of a grating that has subwavelength details. The grating is metallic with refractive index n = 1.5+6i. Solutions were obtained by using a finite-difference technique to solve numerically the time-harmonic Maxwell equations:

A sketch of a segment of the grating shows that the field is incident from the top (in the positive z direction). The grating is rectangular with (in wavelength units) a grating period of 1.6, groove width 0.8, and groove height 0.24.

The colored surface in Fig. 2 represents the calculated complex field. It is suspended in a box whose y axis matches the y axis of the grating sketch, and whose z axis is centered at the grating surface (z = 0).

 

Figure 2: Diffraction of an E-polarized plane wave normally incident on a metal grating [2]. The height of the colored surface shows the amplitude of the electric field and the color shows the phase of the field. This figure is linked (click in the blue box) to an enlarged version where the details of the E field are clearly visible. [Linked plot: 22 KB]

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Movie in Quicktime Format

If we send two short light pulses separated by a time interval T ₀ into an optically thin Doppler-broadened two-level medium, then the medium generates a third ‘echo’ pulse at a time T ₀ after the second pulse. This phenomenon is termed a photon echo [3] and the pulse sequence is sketched in Fig. 3.

 

Figure 3: Sequence of pulses for a photon echo.

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The dynamics of the photon echo can be explained with the aid of the Bloch (pseudo-spin) vector picture, where U, V, and W are the three components of the pseudo-spin S⃗ in a fictitious three dimensional space. In this picture the SchrÖdinger equation for an individual atom can be written in terms of S⃗ = (U, V, W) as:

where T⃗ = (-Ω, 0, Δ) is a ‘torque vector’ expressed in terms of the Rabi frequency Ω of the input pulse and the detuning Δ for the atom. Eq. (4) shows that the evolution of S⃗ is a rotation in the Bloch vector space.

The photon echo has its ideal form when the areas of the input pulses are π/2 and π, and the durations of the input pulses are much shorter than $T_2^{*}$ (the inverse of the Doppler width) and T ₀ > $T_2^{*}$, in which case the rephasing of the atomic dipoles causing the echo is easy to describe analytically. Fig. 4 contains a frame of a movie [4] that shows the dynamics of a non-ideal photon echo. In the movie individual Bloch vectors are shown precessing at different rates due to their differing detunings. The dipole moments of the atoms are shown by projecting these Bloch vectors on a U-V plane (here the upper surface of the bounding box). In the movie, the Bloch vectors start from the ground state (pointing vertically down) at time T = 0. A short time after an external π/4 pulse is applied they reach the partially dephased and partially excited state shown in the snapshot. After further dephasing, at time T = 5 (in units of $T_2^{*}$), an external π pulse is applied to invert the vectors. After this, precession continues. The movie shows the rather sudden ‘magical’ rephasing that corresponds to echo formation at exactly T = 10.

 

Figure 4: Snapshot of a frame of a movie showing formation of a non-ideal photon echo. To run the movie, click in the blue box. [Linked movie: 1.2 MB]

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Reader-Interactive Plot via Java Applet

The intensity of a plane wave on a screen behind an aperture consisting of N slits can be expressed in the Fraunhofer diffraction [5] limit (i.e., when the distance from the source of light to the aperture and the distance from the aperture to the screen are effectively infinite) as:

where I ₀ is the peak intensity for N = 1. The parameters α and β are given by:

where λ is the wavelength of the incident light, and d is the distance between the slits, b is the width of the slits and θ is the angle of observation, as shown in Fig. 5.

 

Figure 5: Schematic diagram of multiple slit diffraction.

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The intensity of light on the observing screen is plotted as a function of β in Fig. 6 for N = 3 and d/b = 4. Fig. 6 is also linked to an ‘interactive plot’ created by a Java applet [6], where the reader can choose the number of slits N and the ratio d/b and plot the corresponding diffraction pattern online.

 

Figure 6: Sample from a Java applet of a plot of intensity for Fraunhofer diffraction from an N-slit grating, as a function of β for N=3, d/b=4. The plot ranges from β = -2π to β = 2π, and the peak value shown is I ₀/N ². To activate the applet, click in the blue box. [Media 3]

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