School of Instrument Science and Optoelectronic Engineering, Beihang University, Beijing 100191, China 2

State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China *Corresponding author: [email protected] Received 21 February 2014; revised 24 April 2014; accepted 16 May 2014; posted 29 May 2014 (Doc. ID 206917); published 26 June 2014

KTiOPO4 (KTP) crystal is a widely used nonlinear optical crystal, and it can meet high requirements for parallelism of crystal surfaces and the length in the light propagation direction in its application fields. In this paper, we present a method for measuring the thickness distribution of cuboid KTP crystals based on laser feedback interferometry (LFI). The accuracy of the measurements for the relative thickness distribution was 8.8 nm, and that of the absolute thickness can be improved by increasing the accuracy of the refractive index. This method is applicable to measurements of all light transmissive birefringent materials, and the results provide detailed instructions for crystal processing and polishing. © 2014 Optical Society of America OCIS codes: (140.1340) Atomic gas lasers; (260.1440) Birefringence; (260.3160) Interference; (120.6650) Surface measurements, figure. http://dx.doi.org/10.1364/AO.53.004195

1. Introduction

KTiOPO4 (KTP) crystal is widely used in the fields of laser frequency doubling [1–3], electro-optic modulators [4], and optical waveguide devices [5] as a nonlinear optical crystal for its excellent performance. When it is used as an amplitude modulator, the static phase retardation needs compensation. Wang et al. [6] reported that, when a quarter-wave plate is inserted between two KTP crystals that have exactly equal lengths in the light propagation direction, phase retardation compensation can be implemented if the two KTP crystals are perpendicular to each other in terms of the corresponding optical axis. The method meets high requirements for parallelism of crystal surfaces and length of the light propagation direction. Furthermore, when the crystal is used as a transmission element in the optical system, the 1559-128X/14/194195-05$15.00/0 © 2014 Optical Society of America

length of crystal in the light propagation direction has a direct impact on the optical path and the light polarization state, hence the thickness of crystal is an important parameter for optical system design. Mechanical stylus profilers [7] can measure thickness accurately, but the stylus may destroy the surface during tests; Ri and Muramatsu [8] realized non-contact thickness distribution measurements of transparent plates by using the sampling moiré method with an error ranging from 0.6% to 2.9% over a thickness of 3.5 mm, but the method is only applicable to isotropic materials with limited precision. In this paper, we propose a method to measure the thickness distribution of cuboid KTP crystal surfaces. The method is based on the laser polarization hopping phenomenon that is induced by optical feedback. Peek et al. [9] first reported that the steady-state intensity of a laser could be modified by introducing coherent optical feedback from an external surface. The physical basis is the interference of the back-reflected field with the standing wave 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS

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inside the laser resonant cavity. Hence, the optical feedback effect is also called self-mixing interference, and it has been studied hotly in the fields of displacement measurements [10,11], imaging vibration analyses [12], and microscopy [13]. 2. Experimental Setup

A semi-external cavity, single-mode, linearly polarized He–Ne laser was used as the light source, and the working wavelength was 632.8 nm. The ratio of gaseous pressure in the laser was He∶Ne 9∶1 and Ne20 ∶Ne22 1∶1. The laser cavity is made up of mirrors M1 and M2 with reflectivities of 99.8% and 98.8%, respectively. The cavity length was 150 mm. The initial polarization state of the laser was in the X direction, as shown in Fig. 1. The external cavity is made up of M2 and the feedback mirror ME with the sample KTP crystal between them. The external cavity length was 100 mm. ME has a reflectivity of 10% and is used to reflect laser beams back into the laser. A piezoelectric transducer (PZT2 ) drives ME to do reciprocating movements, and P is a polarizer that was used to separate the beam of different polarizations from the laser. In our experiment, the polarization direction of the polarizer was in the Z direction, as shown in Fig. 1. The beam can only reach and be detected by detector D2 when the polarization state of the laser is in the same direction with the polarizer polarization. Thus, D1 is used to detect the laser intensity and D2 is used to detect the polarization state of the light. The KTP crystal was cut into a cuboid along the x, y, z axis of the crystal cell, whose width, length, and height are denoted by a, b, c, respectively. The measurement beam passes through the x–o–z plane vertically with the x axis parallel to the X direction. 3. Theoretical Analysis

A model of the equivalent cavity of a Fabry–Perot interferometer was used to analyze the intensity transfer and polarization flipping in a laser with optical feedback from an external birefringent cavity.

The central idea is that the effects of the feedback cavity are converted into a reflectivity change of the laser mirror. The model can be expressed as [14]: E R11∕2 R21∕2 Eei2kL e2gL 1 − R2 Re1∕2 R11∕2 ei2kLl Ee2gL ;

(1)

where E is the laser intensity in the internal laser cavity, R1 and R2 are the reflectivities of the laser mirrors, Re is the reflectivity of the feedback mirror, L is the length of the laser cavity, l is the length of the feedback cavity, g is the gain coefficient of the laser, k 2π∕λ, and λ is the wavelength. The laser intensity in Eq. (1) is given by: I Io −

Re1∕2 R21∕2

1 − R2 cos2kl;

(2)

where I is the laser intensity with feedback and I o is the intensity without feedback. Equation (1) can be rewritten as: i2kl E R11∕2 ei2kL e2gL ER21∕2 1 − R2 R1∕2 e e 1∕2 i2kL 2gL R11∕2 Reff e e E;

(3)

1∕2 cos2kl and where Reff R2 21 − R2 R1∕2 2 Re Reff is the equivalent reflectivity of the laser mirror R2 and the feedback mirror Re. When the KTP crystal is inserted in the external cavity with the x axis parallel to the initial polarization direction of the laser, a geometric external cavity length can be split into two physical lengths, they are:

lx l nx − nair blz l nz − nair b;

(4)

where b is the length of crystal in the light propagation direction, nx 1.7635 is the index of refraction of the x axis, nz 1.8634 [15–17] is the index of refraction of the z axis, and nair is the index of refraction of air in the external cavity. Then, Reff can be expressed as: 4π l nx − nair b Rxeff R2 21 − R2 R21∕2 Re1∕2 cos λ 4π l nz − nair b : Rzeff R2 21 − R2 R21∕2 Re1∕2 cos λ (5) As effective gain of the laser is proportional to the equivalent reflectivity, the intensity of the laser is sinusoidally modulated. From Eq. (5), we can find the phase difference between x-light and z-light, which is: α

Fig. 1. Experimental setup. T, discharge tube; AD, A/D converter; DA, D/A converter. 4196

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4π jnz − nx bjfractional part 2δ; λ

(6)

where δ is the phase retardation of KTP crystal in the y axis direction. According to Eq. (6), the thickness of a

δ

Fig. 2. Computer calculations of intensity variations.

KTP crystal can be calculated by measuring phase difference α. When the external cavity is scanned by PZT2 , as shown in Fig. 2(a), in some cavity length, the tuning curve of z-light is at point A while the xlight is at point F; additionally, the effective gain of z-light is lower than 0 in the next moment, so the polarization state will flip to x-light. For the competition of the two orthogonal polarized states, only one polarized state can oscillate at a certain feedback cavity length. So, a complete tuning cycle is C-A-F-B-D, as shown in Fig. 2(b). The phase difference of the two tuning curves is: AB α × 2π: CD

(7)

And the phase retardation of KTP crystal is: δ

AB × π: CD

(8)

4. Measurement Results and Analysis

In Fig. 3, curve 1 is the displacement of PZT2, which represents the length changes of the external cavity; curve 2 is the light intensity detected by D1 ; and curve 3 is the polarization states detected by D2. It can be observed that the rising and falling edges of curve 3 correspond to the flipping point of curve 2. If the relationship between the length of PZT2 and time is linear, the phase retardation can be expressed as:

Fig. 3. Experiment curves.

E1 E2 F F × π 1 2 × π; T1T2 T3T4

(9)

where point E2 and E1 are a pair of isocandela points and E1 E2 denotes the time interval between the isocandela points. The definition also applies to points F i and T j , (i 1, 2; j 1, 2, 3, 4). However, PZT2 displayed nonlinearity in the experiment, and the nonlinearity was measured by a microchip Nd:YAG laser feedback interferometer [18] with a measurement accuracy of 1 nm over 10 s. The measurement result is presented in Fig. 4. Figure 4 shows that the PZT nonlinearity differs for the rising edge and falling edge of the driving voltage. The nonlinearity of PZT can be calibrated by averaging the displacement–voltage curve for the rising and falling edges. The linearity of the average displacement curve was calculated to be 0.025. So Eq. (9) can be rewritten as: E1 E F F π 1 2 × : 2 T1T2 T3T4

δ

(10)

Phase retardation of a quartz wave plate was measured repeatedly and the results are shown in Fig. 5. Figure 5 shows that the repeatability was 0.122 deg. A series of wave plates numbered as 1–7 were measured with our system and ellipsometry for comparison; the measurement accuracy of the ellipsometry was better than 0.1 deg, as shown in Table 1, and the error of laser feedback interferometry (LFI) was in the range of 0.12–0.49 deg. The KTP crystal we measured was polished, and the contour arithmetic mean deviation (Ra) of the surface was less than 1 μm according to the polish standard; the two light transmission surfaces were quite parallel to each other, if not, the feedback interferometry would be out of alignment. So the thickness difference of the surface was less than the length of one series (6.3 μm). Therefore, the flipping

Fig. 4. Piezoelectric transducer (PZT) nonlinearity. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS

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Fig. 6. Two-dimensional scanning translation stage.

Fig. 5. Measured repeatability.

point on curve 1 changes with the different thickness distribution. We measured the phase retardation of the surface by way of point-scanning; Fig. 6 shows the two-dimensional scanning translation stage with a turntable rotating along the y axis. The crystal had a length of c 6 mm and a width of a 5 mm; the laser spot diameter was 1.1 mm, and the stepping length was set to 1.2 mm in the X direction and 1.125 mm in the Z direction. A measurement result matrix of 4 × 5 was obtained. The measurement phase retardation was translated into multiples of one series to facilitate the calculation of thickness, and the result was: 2

0.2365 6 0.1939 6 δ δ0 6 0.2644 2π 6 4 0.2720 0.2889

0.1948 0.1721 0.2020 0.2145 0.2901

0.1596 0.2202 0.2123 0.2973 0.3365

3 0.2107 0.2322 7 7 0.2552 7 7. 0.3067 5 0.3443

Assuming φ is the total phase retardation of the sample, Eq. (6) can be rewritten as: φ

2π jnz − nx bj: λ

(11)

Then, δ0 is the fractional part of φ, and the integer part of φ (series of crystal) must be determined to calculate the thickness. The thickness of one series was: d0

λ 6.3343 μm: jnz − nx j

than 3 μ, whose accuracy is higher for thickness measurement. So the series of KTP crystal in the y axis direction was 1257. Substituting φ 1257 δ0 into Eq. (11), we can obtain the thickness distribution in the y axis direction as follows: 2

796371.3 6 796344.3 6 b6 6 796389.0 4 796393.8 796404.5

796344.9 796330.5 796349.5 796357.4 796405.2

796322.6 796361.0 796356.0 796409.8 796434.6

3 796355.0 796368.6 7 7 796383.1 7 7 nm; 796415.8 5 796439.6

Figure 7 shows the relative thickness distribution of surface x − o − z. Uncertainty of the refractive index of KTP crystal and the precision of laser feedback interference has an impact on the measurement of the relative thickness distribution. Differentiating Eq. (12) with respect to parameter (nz − nx ), we get: Δd0 −

λ × Δnz − nx : nz − nx 2

When refractive indices nz and nx have a precision of four decimal places, the maximum value of Δnz − nx is 5 × 10−5, and substituting Δnz − nx 5 × 10−5 to Eq. (13), Δd0 3.165 nm. The accuracy of the laser feedback interference system was 0.5 deg, and the error induced by the measurement system was:

(12)

The thickness b was measured to be 7.964 mm by a laser nanometer ruler [19], whose accuracy is higher Table 1.

Number

1

2

Comparison Results

3

4

5

6

7

LFI 24.60 36.31 41.68 56.24 126.11 137.20 146.56 Ellipsometry 24.48 35.82 41.25 55.90 126.42 137.57 147.04 Error 0.12 0.49 0.43 0.34 −0.31 −0.37 −0.48

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(13)

Fig. 7. Relative thickness of KTP.

0.5 × d0 Δd0 8.8 nm: 360

(14) 4.

Uncertainty of the absolute thickness is: P

Δd0 : d0 Δd0

(15)

As Δd0 ≪ d0, Eq. (15) can be rewritten as: P≈

6.

λ Δd0 − nz −nx 2 × Δnz − nx λ d0 jn −n j z

5.

7.

x

Δnz − nx 4.997 × 10−4 : jnz − nx j

(16)

From Eq. (16), it can be found that the uncertainty of absolute thickness is mainly determined by the accuracy of the refractive index and the birefringence difference. The measurement system error only has an effect on the relative thickness distribution, and the error of relative thickness was 8.8 nm. 5. Summary

Above all, we have achieved non-contact KTP crystal thickness measurements through a laser feedback interference system. The system is easy to collimate because of its simple structure, and it is applicable to measurements of all light transmissive birefringent materials. For KTP crystal, the accuracy of the measurement for the relative thickness distribution was 8.8 nm, and the uncertainty of the absolute thickness was 4.997 × 10−4. The accuracy of absolute thickness can be further improved by increasing the accuracy of the refractive index. The measurement results afford significant instructions for crystal processing and polishing, and the system can provide a significant guide to the industrial manufacturing process of crystal elements by greatly improving their precision.

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11.

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14.

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16.

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