System analysis of a tilted field-widened Michelson interferometer for high spectral resolution lidar

Dong Liu; Chris Hostetler; Ian Miller; Anthony Cook; Johnathan Hair

doi:10.1364/OE.20.001406

1. Introduction

Atmospheric aerosols have significant impacts on climate forcing, hydrological cycle, and air quality and better measurements are needed to more accurately characterize aerosol optical and microphysical properties required to model and forecast these impacts. With robust calibration, simplified inversions that with no reliance on a priori assumptions, high spectral resolution lidars (HSRLs) [1–3] are becoming increasingly important in current aircraft and future space-based aerosol remote sensing applications.

The HSRL technique relies on spectral discrimination between scattering from molecules and aerosol particles. In the HSRL technique, the transmitted laser pulse is single-mode and spectrally narrow. The backscatter from aerosol particles (Mie scattering) is virtually the same bandwidth as the transmitted laser pulse. The backscatter from molecules (Cabannes-Brillouin scattering) is Doppler and pressure broadened by a few GHz. The spectral distribution of the backscattered signal in the HSRL is a linear combination of the scattering spectra from both aerosol particles and molecules [4, 5], as are shown in Fig. 1(a) . The aerosol scattering usually has a full-width half-maximum (FWHM) smaller than 100MHz while the FWHM of the molecular scattering is about 3.0GHz [4, 6]. The ultimate goal of the spectral filter in HSRL is to discriminate the two return signals. The discrimination can be accomplished by splitting the returned signal into two optical channels: the molecular backscatter channel, which contains the least aerosol signal (as is shown in Fig. 1(b)), and the total backscatter channel, which passes all frequencies of the returned signal. Notice, Fig. 1(b) is only for illustration, the spectrum of the filtered signal can be of any shape provided the aerosol signal is adequately filtered. When the spectral discrimination is accomplished, the profiles of aerosol extinction, backscatter coefficients, and extinction-to-backscatter ratio, $S_{a}$ can be derived [7–9].

Fig. 1 Schematic diagram for an HSRL return spectra, (a) is the input backscatter signal in HSRL, and (b) is a set of possible output signal in the molecular backscatter channel.

Download Full Size | PDF

Many filters have been developed to obtain the required spectral discrimination. Fabry-Perot interferometric filters [10, 11] are simple and can be tuned to any wavelength, but they have limited angular acceptance. Atomic and molecular absorption filters are robust and stable, and can obtain high quality spectral discrimination [4, 12, 13]; however, absorption filters are not photon efficient and there are no absorption lines at many convenient laser wavelengths. Field-widened interferometers [14, 15] are of high efficiency and can be built to any desirable laser wavelength. Applications, such as measuring Doppler linewidths [15], demonstrate they can be adopted as the interferometric spectral filter for HSRL system.

A compact, robust, quasi-monolithic tilted field-widened Michelson interferometer (MI) is under development as the spectral discrimination filter for a second-generation HSRL(HSRL-2) at National Aeronautics and Space Administration (NASA) Langley Research Center (LaRC). The MI consists of a cubic beam splitter, a solid arm and an air arm. Piezo stacks connect the air arm mirror to the body of the interferometer allowing the interferometer to be tuned within a small spectral range. The widened field of view makes the optical path difference (OPD) of the filter vary slowly with incident angle and allows the collection of light over a large angle. In this paper, the system performance is analyzed over several types of system imperfections, such as cumulative wavefront error, locking error, reflectance of the beam splitter and anti-reflection coatings, system tilt, and depolarization angle. The requirements of each imperfection for good interferometer performance are obtained.

This paper is constructed as follow: Section 2 makes a detailed description of the field-widened Michelson interferometric spectral filter and provides the definition of the Transmission Ratio that can be used to evaluate the performance of the Michelson spectral filter in HSRL; Section 3 shows the principle of the prototype tilted field-widened MI for HSRL; the system performances of the interferometer are analyzed over different system imperfections in Section 4, which is followed by discussions in Section 5, and finally, some conclusions are given in Section 6.

2. Field-widened Michelson spectral filter for HSRL

2.1 Michelson interferometer

The MI is one of the best known interferometers in optical testing and has been adopted for many applications [16, 17]. It consists of a 50/50 beam splitter and two arms with a mirror at each end. The input beam is directed into the system and produces two outputs, as are indicated by Output I and II in Fig. 2 .

Fig. 2 Ray diagram of Michelson interferometer.

Download Full Size | PDF

When the incident angle $θ_{0}$ is zero, the above shown interferometer is an ordinary Michelson interferometer and the irradiance $T_{I}$ at the Output I can be expressed as

T_{I} = 2 I_{0} t_{0} r_{0} [1 + \cos 2 π W]

where,

I_{0}

is the irradiance of input beam,

\sqrt{t_{0}}

and

\sqrt{r_{0}}

are the absolute transmittance and reflectance coefficients of the beam splitter and

W

is the OPD of the two arms in the unit of wavelength.

In a similar way, the irradiance $T_{I I}$ at the Output II is

\begin{matrix} \begin{array}{l} T_{I I} = I_{0} {t_{0}^{2} + r_{0}^{2} + 2 t_{0} r_{0} \cos [2 π W + 2 (δ_{T} + δ_{R})]} \\ = I_{0} (t_{0}^{2} + r_{0}^{2}) {1 + \frac{2 t_{0} r_{0}}{t_{0}^{2} + r_{0}^{2}} \cos [2 π W + 2 (δ_{T} + δ_{R})]} \end{array} & , \end{matrix}

where,

δ_{T}

and

δ_{R}

are the phase shifts of transmittance and reflectance of the beam splitter, respectively and usually

δ_{T}

+

δ_{R}

=

π

/2.

Output II is usually inaccessible as it is blocked by the input beam, and Output I is the one that is generally used. In optical wavefront testing, the OPD parameter $W$ is usually obtained from the interferogram at output I.

Above is the case that the input beam is monochromatic, but if it is not, the frequency dependence of all the parameters should be considered. So the irradiance at Output I and II can be respectively

T_{I}' = \int_{- \infty}^{\infty} 2 I (ν) t_{0} (ν) r_{0} (ν) [1 + \cos 2 π W (ν)] d ν

and

T_{I I}' = \int_{- \infty}^{\infty} I (ν) {t_{0}^{2} (ν) + r_{0}^{2} (ν) + 2 t_{0} (ν) r_{0} (ν) \cos [2 π W (ν) + 2 (δ_{T} (ν) + δ_{R} (ν))]} d ν,

here,

ν

is the frequency and all other parameters have the same meaning as the monochromatic condition. For simplification, the frequency dependence is omitted in the following derivation, but we should keep it in mind when doing real analysis.

2.2 Field-widened Michelson spectral filter

In optical wavefront testing, the OPD of the MI is usually set to be zero or near zero, which reduces the requirement for the monochromatism of the input beam. For a theoretically perfect MI with zero OPD, the output signal will not vary with incident angle of the input beam and the system could collect a full hemisphere of radiation. Yet having zero OPD, the system would not produce any useful information about the incident beam. So as a spectral filter, it should have an OPD matched to the spectrum of the input signal. The OPD determines the spectral filter’s Free Spectral Range (FSR), and a stable spectral filter requires a fixed OPD.

As is in Fig. 2, if the beam splitter is symmetrical, the OPD between the two arms will be [15]

W = 2 n_{1} d_{1} \cos θ_{1} - 2 n_{2} d_{2} \cos θ_{2},

where

{\begin{cases} n_{0} \sin θ_{0} = n_{1} \sin θ_{1} \\ n_{0} \sin θ_{0} = n_{2} \sin θ_{2} \end{cases} .

and here,

d_{1}

and

d_{2}

are the arm lengths,

n_{i}

and

θ_{i}

are refractive indices and the angles of incidence/refraction in the corresponding media, respectively. Then the OPD of the MI will be

W = 2 [n_{1} d_{1} {(1 - \frac{\sin^{2} θ_{0}}{n_{1}^{2}})}^{1 / 2} - n_{2} d_{2} {(1 - \frac{\sin^{2} θ_{0}}{n_{2}^{2}})}^{1 / 2}] .

The HSRL system collects the backscatter signals with a large aperture telescope, which helps to enable high efficiency. The HSRL telescope field of view has been set such that it is slightly larger than the divergence of the laser transmitter, whereas the laser transmitter divergence has been set large enough to meet eye safety requirements. This requires the spectral filter also have large field of view, which means that the spread of incident angles may be relatively large (e.g., approximately 1° full-angle for the LaRC HSRL-2 instrument).

Expanding $\sin θ_{0}$ we get

\begin{matrix} \begin{array}{l} W = 2 (n_{1} d_{1} - n_{2} d_{2}) - \sin^{2} θ_{0} (\frac{d_{1}}{n_{1}} - \frac{d_{2}}{n_{2}}) \\ - \frac{\sin^{4} θ_{0}}{4} (\frac{d_{1}}{n_{1}^{3}} - \frac{d_{2}}{n_{2}^{3}}) - \frac{\sin^{6} θ_{0}}{8} (\frac{d_{1}}{n_{1}^{5}} - \frac{d_{2}}{n_{2}^{5}}) \dots \dots \end{array} & , \end{matrix}

and we can find that, the OPD is power series of the sine squared incident angle. In order to enlarge the field of view, we can let the second term be zero, that is

d_{1} / n_{1} - d_{2} / n_{2} = 0.

Then this system would be independent of incident angle to third order and have an OPD between the two arms as

W = 2 (n_{1} d_{1} - n_{2} d_{2}) - \frac{\sin^{4} θ_{0}}{4} (\frac{d_{1}}{n_{1}^{3}} - \frac{d_{2}}{n_{2}^{3}}) - \frac{\sin^{6} θ_{0}}{8} (\frac{d_{1}}{n_{1}^{5}} - \frac{d_{2}}{n_{2}^{5}}) \dots \dots

where the 4^th and higher terms can be omitted when

θ

is small. Note that one can obtain a super field-widened Michelson filter by adding more glasses [18].

Figure 3 shows the incident angle dependence of OPD for an ordinary Michelson interferometer (blue star) and a field-widened one (pink diamond) that has the same original OPD (150mm) and works at the same wavelength (355nm). As is shown in Fig. 3(a), the OPD suffers a change of more than 60 $λ$ for the ordinary MI with the incident angle at 1 degree while the OPD of the field-widened MI is very constant over a large range of incident angle. Figure 3(b) shows a detail illustration of the incident angle dependence of the field-widened MI and the discussed field-widened MI encounters an OPD change of only about 0.068 $λ$ . For a 400mm aperture, 1mrad field of view telescope, the spectral filter should have at least 16mrad if the input beam aperture is 25mm. The 16mrad divergence angle, or about 0.92 degree, is too large for ordinary MI to act as spectral filter, but it will not pose a problem for a field-widened MI.

Fig. 3 Comparison of OPD incident angle dependence between ordinary and field-widened Michelson interferometers, (a) incident angle dependence comparison, (b) detailed illustration of the performance of the field-widened MI.

Download Full Size | PDF

2.3 Transmission ratio of the Michelson spectral filter

Figure 4 shows a schematic diagram of the field-widened Michelson spectral filter for spectral discrimination in HSRL system. The backscatter signal which contains backscatter from the aerosols and molecules is collected by the telescope, and then directed into the MI that gives out spectral discrimination signals at its two outputs. Here, $S_{m}$ and $S_{a}$ are molecular and aerosol backscatter signal components, respectively. One of the key specifications of the Michelson filter is to pass as much aerosol signal as possible at one output (denoted as the Aerosol channel, and the transmittance of which is $T_{A}$ ) while suppressing it at the other (denoted as the Molecular channel, and the transmittance of which is $T_{M}$ ). And we add another specification that the molecular backscatter signal is split equally between the two outputs of the Michelson filter and therefore known a priori for system calibration. Then there are four constants in the Michelson filter that are important in the calibration and information retrieval of HSRL system and as are shown in Fig. 4, they are:

Fig. 4 Spectral discrimination schematic diagram of a field-widened Michelson spectral filter in HSRL system.

Download Full Size | PDF

Transmission of molecular backscatter into the molecular channel:

T_{m - M} = \int S_{m} (v) * T_{M} (v) d v,

Transmission of molecular backscatter into the aerosol channel:

T_{m - A} = \int S_{m} (v) * T_{A} (v) d v,

Transmission of aerosol backscatter into the molecular channel:

T_{a - M} = \int S_{a} (v) * T_{M} (v) d v,

and Transmission of aerosol backscatter into the aerosol channel:

T_{a - A} = \int S_{a} (v) * T_{A} (v) d v,

where

S_{m}

and

S_{a}

are molecular and aerosol backscatter signal, respectively, and

T_{M}

and

T_{A}

are the transmittance of the molecular and aerosol channels, respectively.

In practice, it is convenient to define the Aerosol Transmission Ratio (ATR) $ℜ_{a}$ as

ℜ_{a} = \frac{T_{a - A}}{T_{a - M}},

and the Molecule Transmission Ratio (MTR)

ℜ_{m}

as

ℜ_{m} = \frac{T_{m - A}}{T_{m - M}} .

Then, a high ATR means the aerosol signal is well blocked in the molecular channel and the stabilization of MTR contributes to the calibration of the system. Then the system will have better determination of the lidar extinction and backscatter coefficients and be less dependent on the calibration with higher ATR and more stable MTR.

Note that, the outputs $T_{I}$ and $T_{I I}$ vary as cosine function of the OPD and have a $π$ relative phase shift. Theoretically, either channel can be the aerosol or molecular channel.

3. Tilted Field-widened Michelson Spectral filter for HSRL

A tilted field-widened MI is being developed as the spectral discrimination filter for the HSRL-2 system at NASA LaRC. It consists of a near 50/50 cubic beam splitter, a solid arm and an air arm, as is shown in Fig. 5 . The solid arm is made of fused silica ( $n_{1}$ = 1.4765, $d_{1}$ = 87.578mm) with one end optically contacted to the beam splitter and the other end finely polished and coated with 100% mirror. The air arm contains air ( $n_{2}$ = 1.00027, $d_{2}$ = 59.318mm) between the beam splitter and the other mirror, which is connected by piezo stacks to the body of the interferometer and permitted to be translated and angle tuned within a small range.

Fig. 5 Layout of the field-widened Michelson spectral filter.

Download Full Size | PDF

The beam splitter is 50mm×50mm×50mm and has anti-reflection (AR) coatings on three outside surfaces indicated by $A R_{A}$ , $A R_{B}$ and $A R_{C}$ as shown in Fig. 5. The AR coating greatly reduces the reflection of the beam splitter cube external surfaces and improves the efficiency of the interferometer since less light is lost. The input signal is expanded to a 25.4mm aperture and has a full divergence of 16mrad (0.92°). The entire MI is tilted at an angle $θ_{T i l t}$ such that the Output II signal is accessible. A locking technique is employed to tune the MI to match the laser output frequency. The piezo is driven to change the length and angle of the air arm until the brightest/darkest signals are obtained for locking laser.

The above field-widened MI has an FSR of about 2.0GHz, and its performance over field angle is shown in Fig. 6 . The deviation of the interferometer output intensity is unnoticeable when the input divergence is smaller than 2.0 degree. As the input full divergence is only about 0.92°, a large flexibility can be obtained for the tilt angle of the interferometer.

Fig. 6 Field-widened performance of the Michelson interferometer.

Download Full Size | PDF

The field-widened MI has two advantages over the absorption filter to act as the spectral filter in HSRL system. First, the field-widened MI is more photon-efficient. The MI has only a few percent loss while the iodine vapor cell suffers typically 50-70% absorption depending on the concentration. The high efficiency helps to reduce laser power and/or telescope aperture on the lidar system, thereby reducing mass, size, risk and cost. Second, the field-widened MI can be built and optimized for a wide range of wavelengths while few laser wavelengths have suitable absorption filters.

4. System analysis

Theoretically, the above tilted field-widened Michelson spectral filter can provide at least a 323 or higher ATR and a 1.008 MTR for a set of input signal with molecular signal of 3.0GHz width and the backscatter ratio of 1.033. But in reality, imperfections of the interferometer, alignment, figures of each optical surface, glass inhomogeneity, parallelism between the two arms, dimensions of the components in the system, and coating errors do exist and will all degrade the transmission ratios of the system.

At this point, it will be necessary to analyze the degradations of these imperfections to the transmission ratios of the system. As there are many degradation factors, some of which have the similar effects and some which may be correlated, it is necessary to analyze them in categories. The effects of the figures of each optical surface, inhomogeneity of refractive index, imperfection of phase shift in the coatings and even the parallelism between the two arms, are to increase the cumulative wavefront error of the system. The detuning of the piezo stacks, and noise and systematic error in electro-optic feedback loop will cause locking error to the interferometer system. The cubic beam splitter is used to split the input signal equally into two beams and maintain the highest transmission efficiency, and these are obtained by the 50/50 coating between the two 45° beam splitter prisms and the AR coatings on the three working surfaces. The imperfection of these coatings will all affect the system transmission ratios. The interferometer is tilted with an angle to make the Output II accessible. Performance with titled angle should be analyzed to get the largest tilt permit.

This section analyzes the effects of imperfections and defines the requirements of each error source for the transmission ratios. As is indicated in Section 2, the field-widened MI aims to get a high ATR and a stable MTR. To evaluate the stabilization of the MTR, the MTR change is introduced as

Δ ℜ_{m} = \frac{M T R - M T R_{0}}{M T R_{0}} \times 100 %,

where

M T R

and

M T R_{0}

are the current and designed values of MTR, respectively. MTR change describes the change of MTR with respect to the design value. In each analysis, results are given for ATR

ℜ_{a}

and MTR change

Δ ℜ_{m}

over the specific category of imperfection. Good performance can be obtained when ATR is high and MTR change is small. Each parameter is analyzed within ordinary nominal range or range that can be easily attained.

4.1 Cumulative wavefront error

Figure 7 shows three cumulative output wavefronts which illustrate different types of wavefront error: (a) tilt, (b) defocus and (c) random distribution generated from the Peaks function [19]. These three wavefronts are selected as they are typical in a MI.

Fig. 7 Different cumulative wavefront error distributions, (a) tilt, (b) defocus, (c) random distribution.

Download Full Size | PDF

In order to analyze the effect of the wavefront over transmission ratios, all other parameters are assumed to be perfect. Then the ATR and MTR change over these different wavefronts are calculated from Eqs. (3), (5), (10)-(12). Notice, Eq. (3) and Eq. (4) are complementary if all other parameters are perfect. Figure 8 gives the ATR (a) and MTR change (b) of the interferometer over cumulative wavefront error. For each wavefront of different distribution, Root-Mean-Square (RMS) values up to 0.02 $λ$ with 0.002 $λ$ steps are taken. The blue dash line is the condition for tilt wavefront and the green circle and pink star are for defocus and peaks, respectively. We can find in Fig. 8 that the response of different distributed wavefronts coincide with each other very well. This means when other parameters are perfect, the distribution of wavefront has little impact on the transmission ratios and we can predict the transmission ratios from RMS value of the cumulative wavefront. In Fig. 8(a), the ATR decreases as the RMS value of the cumulative wavefront increases. The better cumulative wavefront RMS we can get, the higher ATR may be obtained. From Fig. 8(b) we can find that as the wavefront increases, the MTR is quite stable. The maximum change of the MTR is about 0.006% if the RMS value of the wavefront is kept within 0.02 $λ$ . The stabilization of the MTR will greatly help the calibration of the Michelson filter. And further studies show that, a wavefront with about 0.0265 $λ$ RMS value can produce a 100 ATR and a 50 ATR can be obtained with a 0.0410 $λ$ wavefront. The MTR deviations for both conditions are 0.01% and 0.025%, respectively.

Fig. 8 System responses over RMS value of wavefront, (a) aerosol transmission ratio, (b) molecular transmission ratio change.

Download Full Size | PDF

In an interferometric filter, different optical rays over the effective aperture will have different OPDs. The OPD distribution over the aperture can be comprehended as a path-length plus the wavefront reconstructed from the output fringes. The path-length component of the OPD distribution can be affected by the dimension error of each arm, dispersion of the refractive index, and thermal effects over the whole system. In the HSRL-2 system, the interferometer can be locked to the pulse laser, so some error in the path-length component will not contribute to the degradation of the transmission ratios if the system is perfectly locked. But mirror surface figures, glass inhomogeneity, anti-reflecting coating figure, etc, that can affect the interferometer cumulative wavefront error will contribute to degrading the transmission ratios of the system greatly. In order to get a high ATR, efforts should be taken to perfect the figures of mirror, coating surfaces and homogeneousness of glass.

Notice, in a HSRL system, the transmitted laser pulse is single-mode and spectrally narrow enough that it is usually regarded as monochromatic. But Eqs. (1)-(2) could not be used here because when evaluating ATR, the involved aerosol backscatter signal has nearly the same spectral width with the laser source. So Eqs. (3)-(4) are used here to include the contribution of the spectral intensity of the aerosol signal.

4.2 Locking Error

An electro-optic feedback loop is employed in the frequency locking system to lock the interferometer to the laser. In the specific tilted field-widened MI, the frequency locking technique tunes the interferometer until it provides the darkest fringe at the Molecular channel and brightest at the other for locking beam. The locking beam is directly from the seed laser source and has a very small divergence. It is possible that there is error during the locking process, for instance, detuning of the piezo, and noise and systematic error in electro-optic feedback loop, etc. Then there will be a frequency displacement $Δ ν_{L o c k}$ between the MI and the frequency of the laser source. And the wavephase term of the output signals, as shown in Eqs. (1)-(4), will suffer from a shift of $2 π Δ ν_{L o c k} / F S R$ , where $F S R$ is the FSR of the MI. Figure 9 shows the responses over locking error in the system described in Section 3 and (a) is the ATR and (b) is the MTR change. In HSRL technique, the backscatter signal from aerosol particles (Mie scattering) is virtually the same bandwidth as the transmitted laser pulse (about 100MHz in the system discussed in this paper). The FSR of the system is about 2.0GHz with the parameters in Section 3. Responses with locking errors up to 0.06GHz maximum and 0.005GHz steps are given. The ATR decreases in Fig. 9(a) while the MTR slightly deviates from the nominal value in Fig. 9(b) as the locking error increases. But what is good is that an 85% ATR can be maintained and the MTR suffers a change smaller than 0.001% if the locking error can be kept smaller than 15MHz.

Fig. 9 System responses over locking error, (a) aerosol transmission ratio, (b) molecular transmission ratio change.

Download Full Size | PDF

4.3 Tilted Field-widened condition

The field-widened MI is specially designed that the OPD between the two arms varies slowly with incident angle. The divergence of the input beam determines the basic requirement on the field-widened angle. As is indicated above, the current field-widened MI needs to be tilted with a small angle to allow for sampling of the Output II signal. This will increase the requirement of the field-widened angle of the system. Figure 10 shows the responses of the interferometer over tilt angle, and (a) and (b) are the ATR and MTR change, respectively. The MTR change here means the MTR percentage change at the specific tilt. We can find in Fig. 10(a) that when the tilt angle is smaller than about 1 degree, the system can give out high ATR while the ATR decreases below 100 as the tilt increases from 1 degree to 2 degree. But the MTR is very stable all the time, as can be found in Fig. 10(b), and the change is smaller than 0.015% even when the interferometer is tilted with 2 degree. So considering the ATR, a tilt within 1 degree is a good choice. But the more the interferometer is tilted, the easier the reflected signal can be collected. Trade off may be taken when choosing the working tilt.

Fig. 10 System responses over tilt angle, (a) aerosol transmission ratio, (b) molecular transmission ratio change.

Download Full Size | PDF

4.4 Beam splitter coating

4.4.1 50/50 beam splitter ratio

An ideal beam splitter will have 50% reflectance and transmittance for both S- and P- polarizations. Practical beam splitters are not likely to have perfect reflectance/transmittance. The 50/50 beam splitter coating discussed in this analysis is assumed to have the same reflectance for both S- and P-polarization signals. As can be found in Eq. (1) and Eq. (2), the transmitting channel (Output I) of the Michelson with imperfect beam splitters has reduced peak transmission but theoretically perfect visibility while the visibility of the reflecting channel (Output II) is reduced to $\frac{2 t_{0} r_{0}}{t_{0}^{2} + r_{0}^{2}}$ , where $\sqrt{t_{0}}$ and $\sqrt{r_{0}}$ are the absolute transmittance and reflectance coefficients of the beam splitter, respectively and $t_{0} + r_{0} = 1$ if there is no absorption during the splitting. The intensities of the two output channels will be unbalanced if the transmittance and/or reflectance deviates from 50%. And this will lead to different responses when the different outputs are chosen to be aerosol/molecular channels.

Figure 11 shows the system responses when different outputs are chosen to be the aerosol channel: the blue stars are the condition for Output I while the pink squares are for Output II. We can find that when Output I is the aerosol channel, the performance of the interferometer decreases as the reflectance of beam splitter deviates from 50%. An ATR of about 75 is achieved when the reflectance is 45% while the MTR increases by about 2%. A 1% MTR change can be achieved if the reflectance deviation is smaller than 3.5% and the corresponding ATR is larger than 125.

Fig. 11 System responses over reflectance of 50/50 beam splitter, (a) is aerosol transmission ratio, and (b) is molecular transmission ratio change. Two curves are shown on each plot corresponding to different assignment of the “aerosol channel”.

Download Full Size | PDF

Alternatively, when Output II is the aerosol channel, as are shown by the pink squares, the ATR gently increases as the reflectance of beam splitter deviates from 50% while the MTR change is similar to the condition of Output I except that it has a reverse slope. As is indicated above, the high ATR guarantees the aerosol signal to be well blocked in the molecular channel and the stabilization of MTR contributes to the calibration of the system. Comparing the results of the two configurations, we choose the output II as the aerosol channel because the ATR is large while the absolute MTR change is similar with respect to the Output I condition.

4.4.2 AR coating

The outside surfaces of the beam splitter are coated with AR coating to make the system more photon efficient. Perfect AR coatings will transmit the entire incident signal with no reflection. Practical AR coatings will have a non-zero reflectance such that part of the incident signal will be reflected and interfere with the primary signals. But due to the dramatic difference in the intensity, the reflected signals are likely to degrade the visibility of the fringes.

Assume an input beam with intensity $I$ is incident onto the interferometer, a portion of $r_{A R}$ of the irradiance will be reflected directly by AR coating $A R_{A}$ . The left $I (1 - r_{A R})$ will enter into the interferometer. This portion of beam will be split into two beams by the beam splitter and after traveling through the interferometer, two interfering outputs, $I_{C o h t, I}$ and $I_{C o h t, I I}$ , will be yielded, which are the primary output signals of the interferometer. As is indicated above, a $r_{A R}$ portion of the original light will be reflected every time it hits the AR coatings. The back reflected beams resulted by multi reflections of the AR coatings become stray beams and will gradually merge out of the interferometer, degrading the visibility of the fringe of the interferometer. Now the transmittances of the two channels are no longer perfect sine functions but are compressed sine functions with an additional background signal.

The outputs at Output I and Output II of the interferometer can then be expressed respectively as

T_{I}' = I_{C o h t, I} + I_{b k g d, I}

and

T_{I I}' = I_{C o h t, I I} + I_{b k g d, I I},

where

I_{C o h t, I}

and

I_{C o h t, I I}

are the primary coherent signals of the MI, respectively,

I_{b k g d, I}

and

I_{b k g d, I I}

are the back ground signals at Output I and Output II, respectively. The back ground signal at Output I

I_{b k g d, I}

is mainly resulted by the multi reflections in the MI while the one at Output II

I_{b k g d, I I}

contains one more source: the directly reflected beam from AR coating

A R_{A}

.

It is not difficult to calculate the background signal from the multi reflections of the AR coatings inside the MI. The basic method is described next. When a beam hits AR coating $A R_{B}$ , a portion $r_{A R}$ will be reflected and the (1- $r_{A R}$ ) portion will go through. The transmitted portion will oscillate in the air arm and output (1- $r_{A R}$ ) of its power in each round trip. Since there is air in the air arm and the M2 mirror has 100% reflectance, all the light in the air arm will finally emerge if the absorption of air is neglected. So $A R_{B}$ can be omitted when calculating the background signals. Then after the first trip of the input beam in the MI, two outputs of the interferometer will be produced. Back reflected stray beams, which are $r_{A R} / (1 - r_{A R})$ in intensity of the output signals, will be the input beams of the next trip. From now on, keep in mind we have two input beams in different traveling directions, from $A R_{A}$ and from $A R_{C}$ and an iterative process can be employed then. The background signals resulted from the multi reflections in the MI can then be obtained by subtracting the total output signals by the primary coherent signals following Eq. (13) at each output port.

The responses of the interferometer over the reflectance of AR coatings on the three working surfaces of the beam splitter are shown in Fig. 12 , and (a) and (b) are the ATR and MTR change, respectively. The reflectance of the three AR coatings is assumed to be the same. As can be found in Fig. 12(a), the AR coatings have remarkable effects over the ATR: it decreases to about 200, which is 62% of the design value, when the reflectance of AR coating increases to 0.25%.

Fig. 12 System responses over reflectance of AR coating, and (a) is aerosol transmission ratio, and (b) molecular transmission ratio change.

Download Full Size | PDF

4.4.3 Polarization angle

As is indicated in Subsection 4.4.1, practical 50/50 beam splitter coating is likely to have imperfect reflectance/transmittance. It is very difficult to make a polarization-insensitive beam splitter cube. The field-widened MI discussed in this paper is polarization sensitive and is designed to work with S-polarization light. The beam splitter gives a near 50% reflectance for S-polarization light but a very low reflectance (about 3%-8%) for P-polarization. It is very possible that the polarization of the input signal has an angle with respect to the beam splitter of the MI. Studies have been carried out on the response of the interferometer over the polarization angle and the results are shown in Fig. 13 . As is shown in Fig. 13(a), the ATRs with different beam splitter reflectance for S- and P- polarizations are given. Parts of the plots are expanded for better visualization. We can find that the ATR response is firstly determined by reflectance of S-polarization and then the polarization angle. Larger deviation from 50% of the S- and P-polarization reflectance and larger polarization angle tend to produce higher ATR but also higher MTR change. Through the −4% to 4% deviation range, the ATR does not change much though a little higher with larger polarization angle. Figure 13(b) shows the MTR changes for the same series of S- and P-polarization reflectance. Unlike the conditions for ATR, we can get a better MTR if we get lower S- and P-polarization reflectance and smaller polarization angle.

Fig. 13 System responses over polarization angle, and (a) is aerosol transmission ratio, and (b) molecular transmission ratio change.

Download Full Size | PDF

In fact, the system response over polarization angle follows the curve for the 50/50 beam splitter ratio error obtained in Subsection 4.4.1. The larger the reflectance deviates from 50%, the higher ATR and MTR change can we get. The reflectance of P-polarization is relatively smaller than S-polarization, so larger ATR and MTR change can be achieved with larger polarization angle. Trade off should be taken between better blocking of the aerosol signal at the molecular channel and more stable calibration constant MTR.

5. Discussions

According to the analysis results, the cumulative wavefront error and the AR coating of the beam splitter present the most significant effect in degrading the ATR and more care should be taken in controlling these two. A cumulative wavefront error of about 0.01 $λ$ RMS will probably provide a good response. This is a challenging requirement but might be achieved by some iterative fluid jet polishing cycles [20] on the air arm end mirror. An AR coating with 0.1% reflectivity seems to work well. Such reflectivity with some degrees of tilt will not be a problem but it will be a challenge if the coating figure is required to be very good. Fortunately, the requirement on coating figure is not quite high as a cumulative wavefront error is considered in the interferometer system.

Besides the cumulative wavefront error and AR coating, the tilt angle is another parameter that may degrade the ATR. The tilt performance of the current field-widened MI is determined by the design configuration and parameters. But the limit of the tilt depends on the distribution of field angles in the signal as the interferometer should be tilted enough that the input and output beams could be separated. The MI can be designed and fabricated to provide good response over the field angle of the signal.

The locking error may present some degradation to the ATR at the moment as locking error smaller than 10MHz has been demonstrated. But we find it is possible to obtain a locking error less than 1MHz, and we can raise the ATR from 92% of the design value to over 99% and the locking error can then be neglected.

Considering the reflectivity of the 50/50 beam splitter, the Output I has theoretically perfect visibility and can achieve better blocking of the aerosol signal. So the Output I is selected to be the molecular channel. The ATR keeps very high when the reflectance deviates from 50% while the MTR change remains below 1% with reflectance deviation smaller than 2.5%. Notice, the MTR is very stable over the cumulative wavefront error, locking error and AR coating though there is a 0.2% MTR change with a 0.1% AR coating reflectance. This will help considerably with the calibration and retrieval of the HSRL. The response for ATR is also encouraging because we can get enhanced ATR when the reflectance deviates from the ideal condition with Output II as the aerosol channel. It seems trade off should be taken as the MTR change increases with the reflectance deviation. But we choose to obtain as an ideal reflectance as possible because the ATR is enhanced at the cost that the powers of the two channels are unbalanced and this is not good for system calibration.

Regarding the polarization angle, the reflectance of 50/50 beam splitter determines the responses of the interferometer and the polarization angle contributes little. But of course, the smaller the polarization angle, the better performance of the interferometer. We can keep the polarization angle smaller than 0.5 degree and the degradation of the system performance can be neglected.

Notice, with each category analyzed, each parameter of the interferometer can be converted to the corresponding category. Tilt of the interferometer components, for instance, the piezo translation tilt, air arm mirror alignment and fabrication errors that can lead to a tilt effect to the total interferometer wavefront, can be accounted as the tilt wavefront error. The inhomogeneity of the glass components will result in random wavefront error and the detuning of the piezo can be regarded as locking error, and so on. Then, the performance of the interferometer can be predicted according to results above.

6. Conclusions and future work

The tilted field-widened MI is adopted to perform spectral discrimination in the HSRL system. The transmission ratios, which include aerosol transmission ratio (ATR) and molecular transmission ratio (MTR), are proposed to evaluate the performance of the Michelson spectral filter in HSRL. High ATR is desired as it means the aerosol signal is well blocked in the molecular channel while the MTR is hoped to be around unit and stable because equally splitting of the molecular signal helps to calibrate the HSRL system. System imperfections of the field-widened MI are analyzed in categories and the requirement of each parameter that can produce good response are obtained. What should be noticed is that, the above analyses are mostly for one parameter each. There may be correlations between the discussed parameters of the interferometer and the performance of the entire field-widened MI may not be so good then. In fact, an ATR of about 30 is considered to provide adequate separation of the signals in the HSRL-2 instrument, 50 would be good and an ATR over 100 would be excellent. The correlations between the parameters and the requirements of each parameter that contribute to adequate, good and even excellent performance will be the future work.

Acknowledgments

The authors would like to express their appreciation to Dr. Pengwang Zhai, Science Systems and Applications, Inc., USA, and Dr. Zhaoyan Liu, National Institute of Aerospace, USA, for discussions and comments during the study and also to Mr. Richard Hare, Mr. Terry Mack, and Mr. David Harper for the kind help during the preparation of the paper.

This research was supported by NASA Postdoctoral Program at NASA Langley Research Center, administered by Oak Ridge Associated Universities through a contract with NASA.

References and links

1. C. Weitkamp and E. Eloranta, “High Spectral Resolution Lidar,” in Lidar (Springer Berlin / Heidelberg, 2005), pp. 143–163.

2. S. T. Shipley, D. H. Tracy, E. W. Eloranta, J. T. Trauger, J. T. Sroga, F. L. Roesler, and J. A. Weinman, “High spectral resolution lidar to measure optical scattering properties of atmospheric aerosols. 1: theory and instrumentation,” Appl. Opt. 22(23), 3716–3724 (1983). [CrossRef] [PubMed]

3. M. Esselborn, M. Wirth, A. Fix, M. Tesche, and G. Ehret, “Airborne high spectral resolution lidar for measuring aerosol extinction and backscatter coefficients,” Appl. Opt. 47(3), 346–358 (2008). [CrossRef] [PubMed]

4. J. W. Hair, C. A. Hostetler, A. L. Cook, D. B. Harper, R. A. Ferrare, T. L. Mack, W. Welch, L. R. Izquierdo, and F. E. Hovis, “Airborne high spectral resolution lidar for profiling aerosol optical properties,” Appl. Opt. 47(36), 6734–6752 (2008). [CrossRef] [PubMed]

5. D. Bruneau and J. Pelon, “Simultaneous measurements of particle backscattering and extinction coefficients and wind velocity by lidar with a Mach-Zehnder interferometer: principle of operation and performance assessment,” Appl. Opt. 42(6), 1101–1114 (2003). [CrossRef] [PubMed]

6. C.-Y. She, “Spectral structure of laser light scattering revisited: bandwidths of nonresonant scattering lidars,” Appl. Opt. 40(27), 4875–4884 (2001). [CrossRef] [PubMed]

7. S. T. Shipley, D. H. Tracy, E. W. Eloranta, J. T. Trauger, J. T. Sroga, F. L. Roesler, and J. A. Weinman, “High spectral resolution lidar to measure optical scattering properties of atmospheric aerosols. 1: theory and instrumentation,” Appl. Opt. 22(23), 3716–3724 (1983). [CrossRef] [PubMed]

8. P. B. Russell, T. J. Swissler, and M. P. McCormick, “Methodology for error analysis and simulation of lidar aerosol measurements,” Appl. Opt. 18(22), 3783–3797 (1979). [PubMed]

9. A. Ansmann, M. Riebesell, and C. Weitkamp, “Measurement of atmospheric aerosol extinction profiles with a Raman lidar,” Opt. Lett. 15(13), 746–748 (1990). [CrossRef] [PubMed]

10. M. J. McGill and W. R. Skinner, “Multiple Fabry-Perot interferometers in an incoherent Doppler lidar,” Opt. Eng. 36(1), 139–145 (1997). [CrossRef]

11. A. Heliere, A. Lefebvre, T. Wehr, J.-L. Bezy, and Y. Durand, “The EarthCARE mission: Mission concept and lidar instrument pre-development,” in the 23rd International Laser Radar Conference (IEEE International, Nara, Japan, 2006).

12. P. Piironen and E. W. Eloranta, “Demonstration of a high-spectral-resolution lidar based on an iodine absorption filter,” Opt. Lett. 19(3), 234–236 (1994). [CrossRef] [PubMed]

13. J. W. Hair, L. M. Caldwell, D. A. Krueger, and C.-Y. She, “High-spectral-resolution lidar with iodine-vapor filters: measurement of atmospheric-state and aerosol profiles,” Appl. Opt. 40(30), 5280–5294 (2001). [CrossRef] [PubMed]

14. J. Ring and J. W. Schofield, “Field-compensated michelson spectrometers,” Appl. Opt. 11(3), 507–516 (1972). [CrossRef] [PubMed]

15. G. G. Shepherd, W. A. Gault, D. W. Miller, Z. Pasturczyk, S. F. Johnston, P. R. Kosteniuk, J. W. Haslett, D. J. W. Kendall, and J. R. Wimperis, “WAMDII: wide-angle Michelson Doppler imaging interferometer for Spacelab,” Appl. Opt. 24(11), 1571–1584 (1985). [CrossRef] [PubMed]

16. D. Malacara, Optical Shop Testing (John Wiley & Sons, Inc., New Jersey, 2007).

17. P. Hariharan, Optical Interferometry (Academic Press, 2003).

18. W. A. Gault, S. F. Johnston, and D. J. W. Kendall, “Optimization of a field-widened Michelson interferometer,” Appl. Opt. 24(11), 1604–1608 (1985). [CrossRef] [PubMed]

19. “Matlab Peaks Function,” (The Mathworks), http://www.mathworks.com/help/techdoc/ref/peaks.html.

20. http://www.lightmachinery.com/Fluid-Jet-Polishing.html.

System analysis of a tilted field-widened Michelson interferometer for high spectral resolution lidar

Abstract

1. Introduction

2. Field-widened Michelson spectral filter for HSRL

2.1 Michelson interferometer

2.2 Field-widened Michelson spectral filter

2.3 Transmission ratio of the Michelson spectral filter

3. Tilted Field-widened Michelson Spectral filter for HSRL

4. System analysis

4.1 Cumulative wavefront error

4.2 Locking Error

4.3 Tilted Field-widened condition

4.4 Beam splitter coating

4.4.1 50/50 beam splitter ratio

4.4.2 AR coating

4.4.3 Polarization angle

5. Discussions

6. Conclusions and future work

Acknowledgments

References and links

Cited By

Figures (13)

Equations (19)

Optics Express