Complete presentation of the Gouy phase shift with the THz digital holography

Xinke Wang; Wenfeng Sun; Ye Cui; Jiasheng Ye; Shengfei Feng; Yan Zhang

doi:10.1364/OE.21.002337

1. Introduction

The well-known Gouy phase shift is that a converging spherical optical wave undergoes an axial phase change of $π$ in passing through the focal point. Since it was discovered, it always plays an important role in optics. In the principle of laser, it determines the resonant frequencies of transverse modes in laser cavities [1]. In optical coherence tomography, it enhances the dramatic contrast for sub-coherence length features of samples [2]. In parametric nonlinear optical microscopy, the Gouy phase shift causes the asymmetry of the coherent Anti-Stokes Raman Scattering signal [3]. And in attosecond optics, the Gouy phase gating technique improves the accuracy of phase-dependent pump-probe data [4]. Over the years, researchers have tried various theories to explain its origin, including geometrical properties of Gaussian beams [5], Heisenberg’s uncertainty relations [6, 7], Berry’s geometric phase [8], normal congruences of light rays as well as the principle of stationary phase [9].

To investigate the Gouy phase shift, all sorts of measurement methods have been proposed. In 2004, Chow et al. adopted the spatial mode interference-locking method to observe the Gouy phase evolution [10]. In 2004, Lindner et al. applied the stereo-above-threshold-ionization scheme to measure the carrier-envelope phase and study the influence of the Gouy phase shift on the focused laser pulses [11]. In 2006, Hamazaki et al. measured the Gouy phase shift by detecting the intensity profiles of a vortex beam with an asymmetric defect in passing through the focal point [12]. In 2012, Bon et al. used a wavefront sensor to obtain the optical path difference between the scattered field and the reference wavefront and retrieved the Gouy phase shift in the photonic jets [13]. With the maturation of the terahertz (THz) sensing and imaging technology, applications of THz techniques have been greatly developed in many fields. The complex amplitude of the THz electric field can be directly acquired by using the coherent electro-optic (EO) detection scheme. Therefore, some methods based on the THz detection have been proposed to observe the Gouy phase shift. In 1999, Ruffin et al. utilized the THz time domain spectroscopy (THz-TDS) system to observe the Gouy phase shift of a focused THz pulse [14]. In 2000, McGowan et al. used the reflection THz-TDS system to measure the Gouy phase shift of the scattered THz impulses from cylindrical and spherical targets [15]. In 2007, Zhu et al. used their improved THz-TDS system to measure the Gouy phase shift of a converging surface plasmon-polarizations THz beam on a stainless steel foil [16].

Since the THz focal-plane imaging technique was firstly proposed by Wu et al. [17], it has attracted a lot of attentions because it can directly acquire two dimensional (2D) coherent THz images of samples. From 2009 to now, we have improved this technique and achieved the imaging measurement with high resolution, high signal-to-noise ratio, and polarization information [18–20]. In this paper, the THz focal-plane imaging technique is adopted to implement a systemic presentation of the Gouy phase shift of a focused THz beam. A high density polyethylene lens (HDPL) is used to shape a converging spherical THz beam and its phase evolution is recorded by the Z-scan measurement of the HDPL. The three dimensional (3D) information of the Gouy phase shift with a single spectral component is presented. The Gouy phase shifts for different frequencies are compared. The influence of the Gouy phase shift on the THz polarization is also checked. In addition, the same experiment with another longer focal length HDPL is performed to analyze the effect of the f-number to the Gouy phase shift. To our knowledge, this is the firstly direct measurement of the complete field information, such as amplitude, phase, frequency dependence as well as polarization dependence of the Gouy phase shift, which is helpful for understanding and applying the Gouy phase shift.

2. Experimental setup

In this work, a THz balanced EO holographic imaging system [20] is utilized to measure the Gouy phase shift of a converging spherical THz wave, as shown in Fig. 1(a). The coming laser pulse with a 100 fs pulse duration, a 800 nm central wavelength and a 1 kHz repetition is divided into the pump beam and the probe beam for exciting and detecting the THz wave, respectively. The power of the pump beam is 890 mW and that of the probe beam is 10 mW. The pump beam firstly passes through a concave lens (L1) with a 50 mm focal length and illuminates a <110> ZnTe crystal with a 3 mm thickness. The THz beam, which is generated by the optical rectification in the ZnTe crystal, is collimated by a metallic parabolic mirror with a 150 mm focal length. The diameter of the THz beam is about 15 mm and the THz polarization is horizontal. A HDPL with a 50 mm focal length and a 50.8 mm diameter is used to shape a converging spherical THz wave. The THz wave is focused onto the sensor crystal (a <110> ZnTe crystal with a 3 mm thickness) for coherently imaging the THz light spot. The probe beam successively passes through a half wave plate (HWP) and a polarizer. Then, it is reflected onto the sensor crystal by a 50/50 beam splitter (BS). The initial probe polarization is vertical. The HWP and the polarizer are used to adjust the probe polarization for acquiring THz polarization information. The detailed principle and operation have been published in Refs [20, 21]. In the sensor crystal, the probe polarization carries the 2D THz information via the Pockels effect. The reflected probe beam is incident into the imaging unit of the system, which is composed of a quarter wave plate (QWP), a Wollaston prism (PBS), two lenses (L2 and L3), and a CY-DB1300A CCD camera (Chong Qing Chuang Yu Optoelectronics Technology Company). The unit achieves the balanced EO holographic imaging whose detailed principle has been published in Ref [19]. The CCD camera is synchronously controlled with a mechanical chopper to capture the image of the probe beam with a 2 Hz frame rate. The THz image is extracted by the dynamics subtraction technique [22, 23]. By changing the optical path difference between the THz beam and the probe beam, 128 THz images at different scan points are obtained. At each scan point, 50 frames are averaged to enhance the signal to noise ratio. The corresponding scanning time window is 17 ps.

Fig. 1 (a) THz balanced electro-optic (EO) holographic imaging system. The inset shows the Z-scan measurement of complex amplitude distribution around the focal point of the high density polyethylene lens (HDPL). (b) Normalized intensity distribution of the 1.3 THz component on the X-Z plane.

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To observe the Gouy phase shift of the converging spherical THz wave, a Z-scan measurement is implemented around the focal point of the HDPL, as shown in the inset of Fig. 1(a). At different positions, temporal images of the converging THz wave are obtained to build the evolution process of its focusing. Owing to the coherent imaging technique, both amplitude and phase of each spectral component can be accurately extracted by performing the Fourier transformation on the temporal THz signal at each pixel. Figure 1(b) shows the normalized intensity image of the 1.3 THz component on the X-Z plane. The position of the focal point is set as the original point. The scan range along the Z axis is from −22 mm to 28 mm and the scan resolution is 1 mm. In Fig. 1(b), the focusing process is clearly presented. The diameter of the focal spot is about 1 mm and the focal depth is about 14 mm. The transverse spatial resolution of the imaging system can reach 0.2 mm for the 1.3 THz component, which can satisfy the measurement requirement.

3. Results

3.1 Three dimensional information of the Gouy phase shift

Utilizing the THz balanced EO imaging system, 3D information of the Gouy phase shift can be obtained in the focusing procedure. In addition, the optical path of the THz beam does not change during the Z-scan measurement, so the linear phase shift $k L$ ( $k$ is the wavelength vector and $L$ is the total optical path of the THz beam) can be effectively avoided to more clearly observe the Gouy phase shift. The experimental results are shown in Figs. 2(a)-2(d). Figures 2(a)-2(c) present the unwrapped 2D phase distributions of the 1.3 THz component at $Z=-22 mm$ , $0 mm$ , and $28 mm$ , respectively. The corresponding observation planes are marked with the white dashed lines in Fig. 2(d). In Figs. 2(a)-2(c), the unwrapping processes start from the original points (X = 0 mm and Y = 0 mm). The colorbar marks the different phase values with different colors. When the intensity is less than 0.2 on each normalized THz intensity image, the color of the pixel is set as gray on its phase map to filter the uncertain phase noise. Therefore, the phase distributions are valid in the circular regions in Figs. 2(a)-2(c). Figure 2(a) is the phase map before the focal point. It exhibits a quasi-Gaussian distribution which is induced by the HDPL. Figure 2(b) is the phase map on the focal point, which presents an almost plane distribution. Figure 2(c) is the phase map behind the focal point, which presents an inverse Gaussian distribution. The phase evolutions presented in Figs. 2(a)-2(c) are in agreement with the focusing properties of a Gaussian light beam [1]. At different Z axial positions, the central line of each phase map is extracted to build the longitudinal phase distribution on the X-Z plane, as shown in Fig. 2(d). It explicitly displays that the phase variation as the THz beam passes through the focal point. As the THz beam progressively closes to the focal point, the phase along Z-axis gradually decreases from 1.5 to 0. When the THz beam departs from the focal point, the phase continues to decrease from 0 to −1.5. Because the linear phase shift can be neglected in the measurement, the phase variation originates from the transverse spatial confinement induced by the HDPL [7, 24], namely the Gouy phase shift.

Fig. 2 Experimental and simulated three dimensional information of the Gouy phase shift for 1.3 THz radiation. (a)-(c) are the unwrapped transverse phase distributions at Z = −22 mm, 0 mm and 28 mm, respectively. (d) is the longitudinal phase distribution on the X-Z plane. (e)-(g) are the simulated transverse phase maps at Z = −22 mm, 0 mm, and 28 mm, respectively. (h) is the simulated longitudinal phase map.

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It has been pointed out that when the optical system has a small numerical aperture, the Gouy phase shift can be simulated with the scalar diffraction theory [9, 25]. In the experiment, the diameter of the incident THz beam is about 15 mm and the focal length of the HDPL is 50 mm, so it satisfies this condition. To further understand the evolution of the Gouy phase shift, the Fresnel diffraction integral is applied to exhibit the phenomenon. For simplification, we consider that the collimated incident THz beam has a Gaussian amplitude and a plane phase distribution. It can be expressed as

U_{1} (x, y) = \exp [- \frac{(x^{2} + y^{2})}{w^{2}}],

where

(x, y)

are the spatial coordinates on the incident plane,

w

is the radius of the light spot. Here, only the complex amplitude of the THz beam is considered, so its time factor

\exp (- i ω t)

is not included in Eq. (1). Under the thin element approach, the THz field is attached by a pure phase distribution after passing through the HDPL and can be written as

U_{2} (x, y) = \exp [- \frac{(x^{2} + y^{2})}{w^{2}}] \exp [- j \frac{k}{2 f} (x^{2} + y^{2})],

where

f

is the focal length of the HDPL. The propagation of the converging spherical THz field can be expressed as [26]

U_{z} (x_{z}, y_{z}) = \frac{1}{j λ z} \iint_{\infty} U_{2} (x, y) \exp {j \frac{k}{2 d} [{(x - x_{z})}^{2} + {(y - y_{z})}^{2}]} d x d y,

where

λ

is the wavelength,

z

is the propagation distance, and

(x_{z}, y_{z})

are the spatial coordinates on the propagation plane. Because the optical path of the THz beam is not changed in the experiment, the linear phase shift is not considered here [27]. To compare with the experimental results, the propagation distance

z

is also adjusted from −22 mm to 28 mm around the focal point and its resolution is 1 mm. Figures 2(e)-2(h) present the simulation results. Figures 2(e)-2(g) manifest the cross sections of the 3D phase map at Z = −22 mm, 0 mm, and 28 mm, respectively. Figure 2(h) shows the longitudinal phase distribution on the X-Z plane. It can be seen that the simulation results are compatible with the experimental results well, which demonstrates that the physical principles used in the simulation are enough to explain the Gouy phase shift in the experiment. The result also accords with the simulated one obtained by using the modified McCutchen’s method in Ref [27]. Here, only the phase modulation of the HDPL and the scalar diffraction of the focused THz field are taken into account, thus it can be ensured that the measured Gouy shift phase is attributed to the transverse spatial confinement of the THz field. As described in Refs [7, 24], the HDPL induces the nonvanishing transverse wave vectors

k_{x}

and

k_{y}

of the THz field. The transverse wave vectors bring an additional accumulated phase along the propagation direction in the diffraction, which results in the Gouy phase shift.

To more distinctly observe the Gouy phase shift, the central rows of Figs. 2(d) and 2(h) at X = 0 mm are extracted and plotted in Fig. 3(a). The blue circular dotted line is the experimental result and the red solid line is the simulated one. Both of them nicely manifest the expected Gouy phase shift between the front and the back of the focal point. Owing to the limit of the measurement range, the phase shift is little less than $π$ . There is a small oscillation at Z = −14 mm on the experimental curve, which may originate from the on-axis phase anomaly described in Ref [25]. or the image aberration of the HDPL. The well-known paraxial expression $- t a n^{- 1} (Z / Z_{R})$ of the Gouy phase shift is also used to confirm the experimental and simulation results, where $Z_{R} = π w_{0}^{2} / λ$ is the Rayleigh range, $w_{0}$ is the radius of the focal spot. The calculated result is given by the green dashed line in Fig. 3(a), which is consistent with the experimental and simulated results.

Fig. 3 Longitudinal (a) and transverse (b) distributions of the Gouy phase shift. The blue circular dotted line is the experimental result. The red solid line is the simulation result. The green dashed line is the calculated result by using the analytic expression of the Gouy phase shift.

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The THz balanced EO imaging system allows one to obtain not only the phase shift along the Z axis, but also the transverse phase difference between the THz fields before and after passing through the focal point. In Figs. 2(d) and 2(h), the column data at Z = −22 mm and 28 mm are selected out and their subtractions are shown in Fig. 3(b). The experimental (the blue circular dotted line) and simulation (the red solid line) results overlap each other well. In Fig. 3(b), it can be seen that the phase difference close to $π$ around the optical axis (X = 0 mm). This region approximately equates to the size of the focal spot. Beyond this region, the phase difference rapidly decreases and its curve is very similar to a Gaussian line shape. The phenomenon shows that the Gouy phase shift is just a special case of the transverse phase difference.

3.2 Gouy phase shifts for different spectral components

Because the measured THz pulse has a broad spectral range, it allows one to observe the Gouy phase shifts for different frequencies. The longitudinal phase maps at 0.7 THz, 1.0 THz, 1.6 THz and 1.9 THz are shown in Figs. 4(a)-4(d), respectively. It can be seen that there are the significant Gouy phase shifts with different frequencies, which are similar to Fig. 2 (d). To compare their discrepancies, the axial phase distributions and the transverse phase differences are extracted from Figs. 4(a)-4(d) and are shown in Figs. 4(e) and 4(f), respectively. In Fig. 4(e), it is clearly seen that the axial phase distribution for a higher frequency shows a steeper variation and its Gouy phase shift rapidly reaches to $π$ , while the axial phase distribution for a lower frequency exhibits a slower variation and its Gouy phase shift gradually tends to $π$ . As it is well-known, the Gouy phase shift is only obvious in the Rayleigh range [28]. The THz wave with a higher frequency has a shorter Rayleigh range and larger transverse wave vectors around the focal point, so the evolution of its Gouy phase shift is sharper. In Fig. 4(f), it is apparently demonstrated that the THz wave with a higher frequency has a larger transverse phase difference and a narrower central placid area, because its phase variation is faster and its focal spot is smaller.

Fig. 4 Gouy phase shifts for difference spectral components. (a)-(d) are the longitudinal phase maps at 0.7 THz, 1.0 THz, 1.6 THz, and 1.9 THz. (e) and (f) are the longitudinal and transverse distributions of the Gouy phase shifts extracted from (a)-(d).

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3.3 Polarization property of the Gouy phase shift

The THz EO balanced imaging system has the polarization measurement function [20], so the different polarization components of the THz field can be accurately acquired. In the experiment, the polarization property of the THz field passing through the focal point is also checked. At each scan position of the HDPL, the THz horizontal and vertical polarization components at the original point (X = 0 mm and Y = 0 mm) are obtained and their axial energy ratios with different frequencies are calculated by using the expression $(I_{hor} - I_{ver}) / I_{tot}$ , where $I_{hor}$ and $I_{ver}$ are the horizontal and vertical spectral intensities, $I_{tot}$ is the sum of them [20, 29]. Figure 5 shows the experimental results for 0.7 THz, 1.0 THz, 1.3 THz, 1.6 THz, and 1.9 THz, respectively. It can be seen that the energy ratios for different spectral components almost keep a constant 1 around the focal point (Z = 0 mm). Before and behind the focal point, the energy ratios present a slight drop due to the attenuation of the optical intensity and the noise of the system, but they are still above 0.9. It demonstrates that the THz polarization is basically not influenced by the Gouy phase shift in the experiment. It also indicates that the application of the scalar diffraction simulation is suitable.

Fig. 5 Axial energy ratios with 0.7 THz, 1.0 THz, 1.3 THz, 1.6 THz, and 1.9 THz components.

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3.4 Gouy phase shift with different focal lengths

To more completely investigate the Gouy phase shift, another HDPL with a 100 mm focal length is chosen to modulate the wave front of the incident THz field. By the Z-scan measurement of the HDPL, the evolutions of the intensity and the phase are recorded in focusing process. Figures 6(a) and 6(b) give the longitudinal intensity and phase maps at 1.3 THz. The diameter of the focal spot is about 1.7 mm and the focal depth is about 20 mm. The phase map shows the phase variation of the THz field passing through the focal point, which is very analogous with the phenomena shown in Fig. 2(d) and Figs. 3(a)-3(d). To compare with the above results, the axial phase distributions and the transverse phase differences for these two HDPLs are plotted in Figs. 6(c) and 6(d), respectively. It can be seen that the Gouy phase shift with a longer focal length HDPL has a slower phase variation and a lesser phase difference. The reason resembles the problem which has been explained in the Section 3.2. With a larger f-number, the converging spherical THz wave has a longer Rayleigh range and smaller transverse wave vectors in the Rayleigh range, which result in the retardation of the Gouy phase shift. The phenomenon is also consistent with the rules described by the analytical expression $- \tan^{- 1} (Z / Z_{R})$ and the vector diffraction simulation in Ref [30].

Fig. 6 Gouy phase shift with a 100 mm focal length HDPL. (a) and (b) are the longitudinal intensity and phase distributions of the 1.3 THz component on the X-Z plane. (c) and (d) show the axial phase distributions and the transverse phase differences with the 100 mm and the 50 mm focal length HDPLs.

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4. Conclusion

In a conclusion, we utilized the THz balanced EO holographic imaging system to present the 3D information of the Gouy phase shift of the converging spherical THz beam evolving through the focal point. The Fresnel diffraction integral is used to simulate this process and the simulation results agree well with the experimental ones, which show that the Gouy phase shift originates from the accumulated phase brought by the non-vanishing transverse wave vectors. The experimental results also show that the lower frequency and the larger f-number retard the evolution of the Gouy phase shift. In addition, the Gouy phase shift does not present an obvious impact on the THz polarization. These phenomena accord with the explanations of Feng [7] and Yi [24]. Based on the special advantages of the THz balanced EO imaging system, we give a comprehensive presentation of the Gouy phase shift. This work is valuable for understanding the propagation of the electromagnetic field and diffractive optics design.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.10904099, 11204188, 61205097, 91233202 and 11174211), the National High Technology Research and Development Program of China (No. 2012AA101608-6), and the Beijing Natural Science Foundation (No. KZ201110028035) and the Program for New Century Excellent Talents in University.

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Complete presentation of the Gouy phase shift with the THz digital holography

Abstract

1. Introduction

2. Experimental setup

3. Results

3.1 Three dimensional information of the Gouy phase shift

3.2 Gouy phase shifts for different spectral components

3.3 Polarization property of the Gouy phase shift

3.4 Gouy phase shift with different focal lengths

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (3)

Optics Express