1. Introduction

Optical testing is crucial in applications using manufactured optical elements. The tests are highly dependent on the element to be tested and its applications, the latter conditioning the required resolution and dynamic range. Of particular importance is the measurement of the wavefront introduced by an optical element, i.e., the spatial phase induced on an optical wave propagating in the element or being reflected by its surface.

The technique of choice for spatial-phase measurements is interferometry. The interference between two mutually coherent sources is sensitive upon their optical phase difference. Extraction of the phase difference between a reference optical wave and a test wave whose phase has been modified by an optical element under test allows a direct measurement of the phase induced by the optical element. A large variety of interferometric techniques are known, with respective advantages in terms of sensitivity, resolution, and implementation [1].

While wavefront measurements are routinely achieved using interferometry, other approaches to wavefront measurements exist. A particularly simple concept for wavefront measurements is based on the transport-of-intensity equation (TIE), as initially developed by Teague [2–4] and other authors [5–7]. This equation describes the evolution of the intensity of a propagating optical wave. Under Fresnel diffraction, the intensity evolution in the longitudinal direction is linked to the spatial intensity and phase of the wave. This can be used in two different ways: First, the TIE predicts the intensity modulation of an optical source when phase variations are induced, for example, by a manufactured optical element. This has some applications in optical system design, where the intensity of a coherent optical source after different propagation distances is of interest. Secondly, the TIE can be used for wavefront reconstruction, using as experimental data the intensity of an optical source measured after various amounts of Fresnel diffraction. Advantageously, only two measurements of the spatial intensity of the optical wave in two closely spaced planes perpendicular to the direction of propagation are needed to reconstruct the spatial phase of the wave by solving a second-order differential equation, i.e., with a non-iterative deterministic algorithm. Phase reconstruction based on the TIE is particularly interesting in experimental situations where interferometric techniques would perform poorly because of the requirements they put on the spatial and temporal coherence of the optical source, or because of sensitivity and resolution issues. Applications of the TIE include the reconstruction of the phase of an x-ray beam after propagation through an object [8], imaging with an electron microscope [9], optical microscopy [10], and the measurement of the wavefront of broadband incoherent sources distorted by propagation through the atmosphere [5]. Uniqueness of the reconstructed phase of a coherent field was shown in the case when the spatial intensity is strictly positive [6]. The requirement for closely spaced planes can be removed, however, at the expense of iterative retrieval algorithms [11]. Phase reconstruction based on intensity-only measurements after quadratic temporal or spectral phase modulation (the formal equivalent of Fresnel diffraction in the one-dimensional spectral or temporal domain) has also been a fruitful concept for the measurement of the electric field of short optical pulses using a one-dimensional version of the TIE in the spectral domain [12] and for the characterization of nonlinear phase shifts using a one-dimensional version of the TIE in the temporal domain [13].

This article examines the use of the TIE for optical testing of a flat surface, i.e., to measure the spatial phase introduced on a collimated optical wave by propagation into or by reflection on an optical element under test. A particular aspect of phase retrieval in such an application is that the TIE can be simplified significantly provided that the intensity propagating in (or being reflected by) the optical element under test be made uniform. Such an assumption allows a direct and fast Fourier inversion of the experimental data, as was previously shown by Nugent et al. [8]. The capabilities and noise properties of such a wavefront-reconstruction technique have been studied previously, with examples linked to microscopy [14, 15]. In this work, we present a quantitative evaluation of the influence of the experimental parameters, such as noise and distance between object and detection plane in the context of phase reconstruction for optical testing, particularly for measuring high-frequency surface modulations. The following sections (1) develop the necessary formalism; (2) introduce various properties of this approach; and, finally, (3) experimentally demonstrate this concept by characterizing the surface modulation introduced by magnetorheological finishing (MRF) on the surface of Nd:YLF laser rods [16,17]. The quantification of surface modulation based on this formalism is used for performance evaluation of laser rods for the pump laser of the optical parametric chirped-pulse-amplification front end of the OMEGA EP Laser Facility [18, 19].

2. Transport-of-intensity equation

A scalar optical wave of wavelength λ₀ = 2π/k ₀ with complex amplitude E(x,y,z) = $\sqrt{I(x,y,z)}·\exp \left[i\phi \right(x,y,z\left)\right]$ is propagating along the z axis in an isotropic medium of index equal to 1. The paraxial differential equation describing the free-space propagation of this optical wave is

From this equation, and following Ref. [2–4], one can derive the equation

where ∇ stands for the divergence operator.

Equation (2) is commonly known as the transport-of-intensity equation (TIE). This equation links the longitudinal changes of the spatial intensity (i.e., the quantity ∂I/∂z) to the spatial intensity I(x,y,z) and spatial phase φ(x,y,z). It can, for example, be used to predict the intensity modulation due to phase-to-intensity conversion via Fresnel diffraction. For wavefront measurement of an optical wave of arbitrary intensity, the intensity of the wave is measured in two different planes (located at z ₀ and z ₀ + dz along the longitudinal propagation axis) in order to approximate the derivative ∂I/∂z by the finite difference [I(x,y,z ₀ + dz)-I(x,y,z ₀)]/dz, as described in Fig. 1(a). Equation (2) can be written fully with the measured intensities as

which is a second-order differential equation in φ. This finite-difference expression of the TIE can be numerically solved to yield the spatial phase φ(x,y,z ₀). Various methods for solving this equation have been described, for example, based on the development of the spatial phase on a base of Zernike polynomials [20] or a more general base of orthonormal functions [21].

 

Fig. 1. (a). Notations for the TIE. The intensity of the wave under test propagating along the z axis is measured as a function of x and y in two closely spaced planes at z ₀ and z ₀ + dz. (b) Algorithm for phase reconstruction from the measured intensity.

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In optical testing, the optical source is usually implemented by the user. An obvious choice for the intensity of the test source is that of a constant intensity, i.e., I(x,y,z ₀) = I ₀. This practical consideration allows a significant simplification of Eq. (3), which can be written as

Equation (4) is a Poisson equation that can be solved via Fourier transforms. Indeed, the Fourier transform of Eq. (4) is

where FT[φ] and FT[I] are the two-dimensional spatial Fourier transforms of φ and I, respectively. Equation (5) allows one to write

where IFT stands for the inverse two-dimensional spatial Fourier transform. The resolution of the TIE in the case of a uniform intensity was studied in Ref. [14] and applied to phase imaging using x-rays in Ref. [8]. In these references, a decomposition of the phase as a sum of spatial harmonics leads to a one-to-one correspondence between the harmonics of the measured intensity and the harmonics of the phase solution. The proportionality coefficient between harmonics is inversely proportional to the square of the harmonic number, as is expected from Eq. (6). The fact that the argument of the inverse Fourier transform in Eq. (6) is singular for k_x = k_y = 0 stems from the fact that no information on the piston (i.e., the non-spatially varying component of the phase) can be retrieved by the TIE. The piston is in most cases of no interest in optical testing because it is relatively defined with respect to a reference optical wave and does not induce spatial modulation of a propagating optical wave. The reconstruction algorithm is schematized in Fig. 1(b). The Fourier transform of ΔI = I(x,y,z ₀+dz)/I ₀-1 is calculated and divided by k ² _x+k ² _y, where this quantity is different from zero. An inverse Fourier transform of the obtained quantity leads to the phase under test after proper scaling.

Phase reconstruction using the TIE is particularly simple since it requires only intensity measurements. The phase of an optical element under test can be measured accurately and with good resolution over a surface limited only by the detection system (for example, the chip of the CCD camera used to measure the intensity). Because of the simple non-iterative nature of the reconstruction algorithm, fast update rates can be obtained. Also, the absence of a reference wave, as is common in interferometric testing, is beneficial in applications where the coherence of the source used for testing or environmental constraints such as vibration would not allow proper interferogram measurements. Equation (5) implies that, apart from k_x = k^y = 0, the power spectral density (PSD) of the phase [22] is directly given by the power spectral density of the measured intensity after scaling, following

The PSD is particularly important to characterize high-frequency surface modulations, such as those introduced by polishing techniques. Equation (7) implies that, for a given phase, the power of the harmonics of the intensity of the propagated wave increases significantly with the frequency of the modulations via (k ² _x +k ² _y)² and with the propagation distance via dz ².

3. Properties

The validity of Eq. (6) for phase reconstruction was tested numerically. The accuracy of the phase reconstruction for finite propagation distance dz and noise on the measured intensity is quantified. In particular, it is shown that the influence of the propagation distance dz on the reconstruction accuracy strongly depends on the frequency content of the phase under test. While smaller distances improve the reconstruction accuracy, they also make the diagnostic more sensitive to noise because the intensity variation induced by the phase under test is smaller. Therefore, the propagation distance must be chosen as a balance between the frequency content of the phase under test and the amplitude of the noise on the measured data. Such balance has previously been studied in general terms and demonstrated with test images [15]. An optical wave with constant intensity and a sinusoidal phase φ _test(x) = a cos(2π x/δ), where a and δ are the amplitude and period of the phase, respectively, is propagated along a distance using the kernel for Fresnel diffraction dz, i.e., exp[ikx ² dz/(2k ₀)]. The intensity of the obtained wave I(x,dz) is used for phase reconstruction using Eq. (6). The wavelength was chosen as λ₀ = 632.8 nm.

Figure 2 displays the reconstructed phase when the phase under test has a period equal to 0.2 mm and an amplitude equal to either 0.1 rad [Fig. 2(a)] or 1 rad [Fig. 2(b)]. One can see excellent reconstruction of these two sinusoidal phases, except for the case a = 1 rad, dz = 1 cm, where some discrepancy is visible. To understand this discrepancy, the quantity [I(x,dz)-I ₀]/(I ₀ dz) is plotted for dz = 1 mm and dz = 1 cm. While the measured intensity is sinusoidal in the first case, Fresnel diffraction induces higher harmonics in the second case. As Eq. (6) implies that there is a one-to-one correspondence between harmonics in the measured intensity and phase under test, a decrease in accuracy is induced.

 

Fig. 2. (a). Phase reconstructed for a 0.2-mm period and 0.1-rad amplitude for propagation distances of 1 mm and 1 cm; (b) phase reconstructed for a 0.2-mm period and 1-rad amplitude for propagation distances of 1 mm and 1 cm; (c) normalized intensity [I(x,dz) -I ₀]/(I ₀ dz) calculated for a 0.2-mm period and 1-rad amplitude for propagation distances of 1 mm and 1 cm. For short propagation distance, the sinusoidal phase leads to a sinusoidal intensity, while for longer propagation distances, higher harmonics are present in the intensity.

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A rms error metric between the test phase φ _test and the reconstructed phase φ _{reconstructed} has been defined to quantify the phase reconstruction accuracy following

The factor 2 in the denominator has been introduced so that the rms error is normalized to the amplitude of the phase under test in the case of a sinusoidal phase. For a sinusoidal phase, it is sufficient to choose an integration interval [x ₁, x ₂] equal to one period of the phase. The rms error takes for value 0.2% and 3.2% for a sinusoidal phase with a 0.1-rad amplitude and a 0.2-mm period reconstructed respectively after propagation dz = 1 mm and dz = 1 cm, and increases to 1.8% and 14.2% for a sinusoidal phase with a 1-rad amplitude and a 0.2-mm period reconstructed respectively after propagation dz = 1 mm and dz = 1 cm. Figure 3 displays ε _rms as a function of the amplitude and period of the sinusoidal phase under test for three different propagation distances. Short propagation distance [e.g., 1 mm as plotted in Fig. 3(a)] leads to accurate phase retrieval over a large range of periods and amplitudes. As the propagation distance is increased [e.g., 1 cm in Fig. 3(b) and 10 cm in Fig. 3(c)], phases with high spatial frequencies are not reconstructed as accurately. This is due to the fact that the finite difference of the spatial intensity is not an accurate description of the longitudinal derivative of the intensity for these phases.

 

Fig. 3. (a). rms error obtained for a propagation distance dz = 1 mm; (b) rms error obtained for a propagation distance dz = 1 cm; (c) rms error obtained for a propagation distance dz = 10 cm. In the absence of noise, the reconstruction accuracy decreases for increasing frequency of the phase under test. Longer propagation distances lead to lower reconstruction accuracy.

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Investigation of the noise sensitivity was performed using additive noise uncorrelated from pixel to pixel. The standard deviation of the noise σ_noise was defined as a fraction of the initial intensity of the wave. The rms error is displayed in Fig. 4 for difference amounts of noise and propagation distances as a function of the amplitude and period of the phase under test. It can be seen that the noise mainly impacts the reconstruction of low amplitude phases because these test phases induce small changes in the propagating intensities that cannot be measured accurately in the presence of noise. For larger propagation distances, the longitudinal variations of the intensity are larger, which therefore increases the accuracy in the presence of noise.

 

Fig. 4. rms error obtained (a) for a propagation distance dz = 1 cm for additive noise of standard deviation 10⁻⁴, (b) for a propagation distance dz = 1 cm for additive noise of standard deviation 10⁻³, and (c) for a propagation distance dz = 10 cm for additive noise of standard deviation 10⁻³. For the sake of clarity, the rms error was limited to the maximal rms error in the absence of noise for each distance. In the presence of noise, the reconstruction accuracy decreases for decreasing amplitude of the phase under test.

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4. Experimental demonstration

The experimental demonstration of the technique studied in this article was performed by characterizing the spatial phase introduced by a Nd:YLF laser rod with one face polished by magnetorheological finishing (MRF) [16,17]. MRF is performed on one of the end faces of the rod in order to compensate the spatial phase distortion introduced by the rod; i.e., a spatial phase opposite to the spatial phase introduced by propagation in the rod is induced on the face of the rod so that the sum of the two phases is approximately zero. Proper cancellation of the phase is important for laser applications using these laser rods, and excellent performance of the wavefront correction has been reported [17]. Such wavefront correction is, however, only performed within some resolution limit of the MRF process itself, on the one hand, and the metrology used to apply this process, on the other hand. MRF polishing is performed by a fluid containing small magnetic particles. The fluid is injected between the surface to be polished and a sphere, and an electromagnet below the sphere is used to control the interaction between the fluid and the surface via the electromagnetic gradient. The shape of the polished surface is locally controlled by the time the surface is being polished at a given location. It is of interest to quantify the surface quality of the processed rods: for example, detect the magnitude and direction of the surface modulations introduced by the MRF process. While the available interferometers could provide a measurement of the wavefront introduced by the laser rod after processing, they did not have sufficient resolution to resolve the high-frequency surface modulations introduced by the MRF process. On the other hand, the available high-resolution profilometers could not provide surface information over a large aperture. It was found that the TIE offered significant advantages over other testing techniques and is very simple to implement. High-resolution description of the surface could be obtained over large apertures.

The experimental test setup is presented in Fig. 5(a). The test source is a beam from a He:Ne laser that has been expanded and collimated. This source is reflected onto the surface under test (either the rod’s face that has received the MRF processing or the flat reference face) at quasi-normal incidence. The surface under test is reimaged onto a video camera with two lenses of focal length f = 12 cm separated by the distance 2f, and the distance from the surface to the first camera is f. The camera is set on a translation stage in order to modify the virtual distance dz between the image of the surface and the camera. The image measured from the camera is acquired by a 14-bit frame grabber for numerical processing on a computer. The relative rms noise on the acquired images was measured to be σ_noise = 5 × 10⁻³, and the actual dynamic range on the measured intensity profiles is therefore only approximately eight bits. Images were typically recorded for a set of distances ranging from 12 mm to 36 mm. Phase reconstruction was in each case performed with only one measured image using Eq. (6). Figure 5(b) displays the evolution of the peak-to-valley of the measured phase with respect to the distance dz. While detection noise influences the measurement of the peak-to-valley phase, its effect is not significant over distances larger than 15 mm in our implementation. As shown below, consistent phase measurements have been obtained over a large range of distances. One should note that the phase introduced by the surface of the rod is half the measured phase since the measurement is double pass.

 

Fig. 5. (a). Layout for the characterization of the surface modulation of a laser rod using the TIE. The virtual distance between the image of the surface and the detector is changed by longitudinally translating the CCD camera used to spatially resolve the intensity. (b) Peak-to-valley of the reconstructed phase as a function of the distance dz.

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The animation of Fig. 6 shows the normalized intensity ΔI = I/I ₀ − 1 measured as a function of the distance between the image of the surface and the detection plane. The intensity modulation increases when the propagation distance is increased. Three measured intensities and reconstructed phases for dz = 12 mm, dz = 24 mm, and dz = 36 mm are plotted in Fig. 7. Good agreement between the retrieved phases is obtained, particularly as the propagation distance is increased. The rms error between the phases reconstructed at dz = 24 mm and dz = 36 mm is 0.04 rad. Finally, as a test of consistency, an optical wave with constant intensity and the reconstructed spatial phase in these three cases was generated and propagated by the corresponding distance. The normalized intensity of the calculated field after Fresnel diffraction is also plotted in Fig. 7 and shows excellent agreement with the measured intensity profiles. Increased accuracy is expected with the use of a scientific low-noise camera, which would have a dynamic range significantly higher than eight bits.

 

Fig. 6. Animation of the spatially resolved intensity as a function of the distance from the surface under test to the camera (0.78 MB). Intensity modulations increase when the propagation distance is increased. [Media 1]

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Fig. 7. The three columns represent the measured intensity (first column), calculated phase (second column), and calculated intensity (third column) for propagation distances equal to 12 mm (first line), dz = 24 mm (second line), and dz = 36 mm (third line).

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Plots of the two-dimensional power spectral density of the measured intensity are displayed for the reference face [Fig. 8(a)] and the face under test [Fig. 8(b)] for dz = 24 mm. The latter shows the presence of significant directionality in the surface variations, i.e., the spectral density takes significant values for k_x = 0. The PSD of the measured intensity summed along x is plotted in Fig. 8(c) for the propagation distances dz = 12 mm, 24 mm, and 36 mm. As noted in Transport-of-intensity equation, the PSD is increasing because of the dz dependence of the relation between the PSD of phase under test (which does not depend on the propagation distance) and the PSD of the measured intensity.

 

Fig. 8. (a). Two-dimensional PSD of the intensity measured for dz = 24 mm in the case of the reference face; (b) two-dimensional PSD of the intensity measured for dz = 24 mm in the case of the face with MRF processing; (c) PSD of the intensity integrated over k_x measured for dz = 12 mm, dz = 24 mm, and dz = 36 mm.

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5. Conclusions

The surface quality of manufactured optics can be evaluated using a simplified version of the transport-of-intensity equation. It was shown theoretically and experimentally that such an approach provides simple, accurate, and fast phase retrieval. Retrieval of surface modulations can be performed using one intensity measurement of an optical wave. Simulations have described the accuracy of this technique and the influence of noise. Experimental investigations of the phase introduced by the magnetorheological finishing process on laser rods have been conducted. Such an approach to the measurement of the surface quality of Nd:YLF rods is currently used for the OMEGA EP Laser Facility [18].