Confocal Microscopy

Fluorescence microscopes are excellent tools to generate contrast in images of biological samples. However, they suffer from a major problem when you are imaging samples even slightly thicker than the z resolution limit. The problem is the out-of-focus blur. This happens because the system is unable to capture light from precisely the focal plane of the objective. Some out of focus light (light from planes below and above the focal plane) inadvertently creeps in and introduces noise. This results in blurred images.

There are few ways to overcome this problem and get sharper, in-focus images. This helps the microscope to perform precise optical sectioning. The most widely used and probably the gold-standard in optical sectioning is confocal microscopy. 

How does the confocal microscope reduce out-of-focus-blur?

A very basic understanding can be had from the description below.

Let us take a laser that emits at the excitation wavelength of the fluorophore of interest. The laser beam is reflected by a dichroic mirror towards an objective. See Figure 29. The beam is a made to fill the back aperture of the objective. This creates a focused, diffraction-limited spot of the excitation light at the focal point of the objective. If there are fluorophores in this focal volume, they will get excited and will emit fluorescence isotropically. A cone of this emitted light (that travels in the backward direction) is collected by the objective. You can see that we are using the epi-illuminaiton configuration here. The objective sends this fluorescence emission light out in a collimated beam shaft (remember Figure 2). Here the focused laser spot creates something similar to a point source of light at the focal point of the objective. This means the beam comes out collimated and is directed towards the dichroic mirror.

Figure 29: Measurement of fluorescence intensity excited using a diffracted limited laser spot. Excitation light is shown in violet and emission light is shown in green.

The dichroic mirror is designed to reflect the excitation wavelength (from the laser) and transmit the emission wavelength (from the fluorophores in the sample). The dichroic mirror hence transmits this collimated beam shaft of emitted light. A tube lens further down focuses this light to its focal point. If we now place here an emission filter and a photodetector we can measure the intensity of emitted light from the focal spot of the objective. Note that the objective-tubelens combination is acting like an infinite tube length microscope and is projecting a magnified image of the focal spot of the objective onto the detector plane. 

The detector measures the intensity of the light emitted from the focal spot. However, we have out of focus light which also gets detected by the detector.

The solution to this problem is a pinhole. The pinhole should be kept exactly at the focal point of the tube lens. This is very important. The detector is placed after the pinhole. The detector used in most common implementations of a confocal microscope is a Photo Multiplier Tube (PMT).  This is shown in Figure 30.

Let us now consider what happens to the light coming from the focal spot (Figure 30). If the light comes from a point source placed at the focal spot (refer Figure 2) it comes out collimated through the back aperture of the objective. Now a collimated beam of light enters the tube lens. The tube lens focuses the light to its focal point. The pinhole is placed exactly here. Almost all of the emission light that originates from the focal spot passes through the pinhole and is detected by the PMT.

Figure 30: Confocal Microscope-Light Coming from the focal plane of the sample. Violet shows excitation light and green shows emission light.

Now let us consider the out of focus light. First, the light that comes from a point at a distance shorter than the focal length of the objective. Look at Figure 8 from left to right. If the light comes from a point source placed at a distance shorter than the focal length, the light exits the lens slightly diverging. This scenario for a confocal microscope is shown in Figure 31.

Note that when we use high NA objectives the out of focus light it is not produced over large distances away from the focal point but probably a couple of micrometers above and below the focal point. This is the range of distances over which the objective collects fluorescence light. Beyond these distances, the intensity of excitation is very low to excite any detectable fluorescence. Hence the divergence of the beam as the light exits the lens is not very large.

This divergence, however, is significant enough to have an influence on how the beam exits the tube lens as is seen in Figure 31. Lets us consider Figure 6. Look at this figure from right to left. For that matter, you will see the same thing in Figure 4 and Figure 5 when looking at them from right to left. (Remember lenses are reciprocal systems, a ray diagram is true in both directions).

When a diverging beam enters a lens, it gets focused at a distance that is longer than the focal length of the lens. Now this means the light is not focused all the way down when it hits the pinhole (Figure 31). The pinhole hence blocks the light coming from a point on the sample that is at a distance shorter than the focal length of the objective. 

Figure 31: Confocal Microscope-Light Coming from a plane above the focal plane of the sample. Violet shows excitation light and green shows emission light.

Lets us now consider the situation when the objective collects light from a point that is at a distance longer than the focal length of the objective. Look at Figure 6 again but this time look at it from left to right. You will see the same thing in Figure 4 and Figure 5 as well when looking at them from left to right. If the light comes from a point source placed at a distance longer than the focal length of the objective, the light exits the lens converging. Again, in a confocal microscope, this light is collected only from very short distances away from the focal point. So the light out of the objective is only slightly converging. This scenario is shown in Figure 32.

Now a converging beam is passing through the tube lens. Lets us consider Figure 8. Look at this figure from right to left. When a converging beam enters a lens, it gets focused at a point that is at a distance shorter than the focal length of the lens. Now this means the converging light in the confocal microscope is focused before the pinhole by the tube lens. After being focused as light propagates further it diverges and the pinhole blocks this light as well.

Figure 32: Confocal Microscope-Light Coming from a plane below the focal plane of the sample. Violet shows excitation light and green shows emission light.

So the combination of the objective, tube lens, and the pinhole blocks most of the out of focus light allowing only in-focus light to be detected by the PMT.

We have however collected intensity information from only a single diffraction limited spot. This does not create an image. To create an image, we need to scan the laser beam in a raster pattern and collect intensity information from many points across the XY plane. This creates a 2D array of intensity information stored in the memory of the computer. This array is then converted into an image.

How is the point scanning confocal microscope system configured?

The block diagram of one of most common implementations of a point scanning confocal microscope is shown in Figure 33. The diagram gives a more detailed description of the components involved in making a confocal microscope work. 

Figure 33: Block Diagram of a Point Scanning Confocal Microscope

The excitation light for a point scanning confocal microscope comes from a laser combiner. The laser combiner has a bank of lasers with different wavelengths. The different wavelengths are required to excite fluorophores with different excitation wavelengths.  These wavelengths are combined into a single collinear beam using a set of mirrors and dichroic mirrors. This combined laser beam is passed through an Acousto Optic Tunable Filter (AOTF). AOTF is controlled electronically through the software that runs the confocal microscope. The AOTF serves two functions:

  1. Select a wavelength or a set of wavelengths for excitation
  2. Control the intensity of each wavelength independently

The light out of the AOTF is launched into a single mode optical fiber. This arrangement helps the laser combiner to be separated from the scan head and imparts flexibility in placing the laser combiner with respect to the scan head. The other end of the optical fiber is connected to the scan head.  The excitation light diverges out as it exits the optical fiber. A collimating lens is used to collimate this light. The collimated output is directed towards the dichroic mirror.

The dichroic mirror reflects the excitation light towards the scanning mirrors. The scanning mirrors scan the excitation wavelength/s in X and Y dimensions in a raster pattern. The scanning mirrors are controlled electronically through the sofware of the confocal microscope.

Before directing towards the objective, the light excitation light out of the scanning mirrors is passed through a set of two lenses called the scan lens and the tube lens. The tube lens sits in the body of the microscope and the scan lens inside the scan head. These two lenses have the following functions:

  1. Together they act like a telescope to expand the laser beam from about few mm to about 1 cm. This is to ensure that the back aperture of the objective is overfilled. This, in turn, ensures that the full Numerical Aperture of the objective is available and maximum possible resolution is attained.
  2. Relay the image of the scanning mirrors to the back aperture of the objective. This ensures a proper scanning of the excitation laser spot across the defined ROI on the sample plane.

As described before the objective focuses the excitation light onto the sample. The fluorescence emission (in the backward direction) is collected by the objective. The fluorescence emission traces back the path taken by the excitation light all the way till the dichroic mirror. The dichroic mirror transmits the emission light. A tube lens (of the scan head) focuses the emission light onto the pin hole. The light out of the pinhole is filtered by an emission filter and directed towards the PMT.

The PMT produces a current that is proportional to the number of emission photons detected by it. A Trans Impedance Amplifier (TIA) is used to convert this current into a voltage and to amplify the voltage to appropriate levels. An Analog to Digital Converter (ADC) is used to convert the analog voltage signals into digital signals. Computers can only read digital signals. The intensity information in digital format is read by the computer and it converts this information into an image.

How does a confocal microscope generate an image?

Widefield fluorescence microscopes use imaging detectors like CCD or sCOMOS cameras for generating images. Cameras have an array of detectors in them which corresponds to the pixels we see in the images they generate. If a camera is specified as 5 Mega Pixel, then the camera chip has 5x10^6 independent detectors. When the lens system of the widefiled microscope projects an image onto the detector plane, such a camera can generate a digital image with 5x10^6 pixels.

However confocal microscopes use single pixel detectors like PMTs. Single pixel means they donot have an array of detectors but only a single detector. So how are these detectors generate images?

The fact is in a confocal microscope the image is not generated by the detector. It is generated by a computer that has a certain amount of memory and processing power. The image is constructed by the computer using a set of intensity information sequentially gathered from the PMT and sequentially stored in the memory of the computer. 

The laser spot is scanned in the raster pattern and spot moves across the XY plane. As the laser spot steps from one spot to next, the output of the PMT changes every set time period. This time period is the Dwell Time-the time it takes the scanning system to move the laser spot from one pixel to the next on the sample plane. The ADC is programmed to take a reading of the PMT voltage (through TIA) every time period. So every successive time period (dwell time) the ADC spits out a digital value. This value corresponds to the intensity of fluorescence emission from the position of the laser spot at that particular time point.

Before you start imaging you need to tell the confocal software how many pixels you need in your image. If you say you need 512x512 pixels, the computer allocates a memory location in the form of an array of size 512x512. Each memory location has an address ([0,0] to [511,511]). Let us assume that an ROI is scanned. The scanning system positions the laser on the first spot (let's say on the top left of the ROI). The ADC reads the PMT output and generates a digital value. This corresponds to the intensity of fluorescence from the first spot. The computer stores this value in the array at memory location [0,0]. The laser spot is then moved right to the next spot. The ADC reads the value and this is stored in memory location [0,1], then the next one onto [0,2] and so on till memory location [0,511] is filled. This completes scanning of the first row. Now the laser spot is moved to the next row and the process repeats. Starting from [1,0] the memory locations are progressively filled till [1,511].

So as the laser beam progressively moves across the sample plane the memory locations get progressively filled up. This happens until the laser spot reaches the last point in the ROI on its bottom right. This fills the last cell in the array with address [511,511] and created a completely filled memory array. This array of intensity information is called a bit map. This bitmap is converted into an image in a given format like JPEG, PNG, TIF etc. or custom formats proprietary to confocal microscope manufacturers. This image can be displayed on a computer monitor or stored.

This whole process requires precise synchronization between the scanning system and the ADC. This is done by the control electronics which works under the command of the confocal software. Modern confocal microscopes donot step the laser spot across the XY plane. The laser spot instead is moved continuously in a raster pattern. The ADC simply reads periodically (every dwell time) from the PMT.

How does a confocal microscope generate 3D images?

To generate a 3D image, a 3D array is defined with the number of pixels in X, Y and Z directions specified. After the image of one plane is scanned, the relative position between the objective and the sample is changed by the Z step size. This is done by moving the objective or moving the sample placed on a Z stage. Z step size is calculated from the number of pixels defined in the Z direction and thickness of sample needed to be imaged. This process is repeated for multiple planes and a raster scan performed and an image generated for each plane. The number of image planes equals the number of pixels in the Z direction. The end result is a 3D array of intensity information. This array can be volume rendered or projected onto a single plane for visualization. 

 

Getting the Best Out of sCMOS Cameras

[Credits: Amit Cherian and Manoj V Mathew
Experiments were conducted at the Central Imaging and Flow Cytometry Facility (CIFF), National Center for Biological Sciences (NCBS), Bangalore, India.]

CMOS camera technology is evolving rapidly. The latest generation of sCMOS cameras performs very similarly to their CCD counterparts in terms of sensitivity and noise levels. They are much better than CCDs in terms of speed of operation as well as chip size. Also taking into account the lower costs of sCMOS cameras, they are increasingly becoming the detector choice for most biological microscopes.

We decided to test few of the parameters of various sCMOS cameras.

Camera:1 Hamamatsu Flash 4 V2 (C11440)

The first camera we got was the Hamamatsu Flash 4 V2 (C11440). See Figure B1.

Figure B1: Hamamatsu Flash 4 V2 sCMOS camera top and back views.

This is a very popular camera in the market and is widely used for biological imaging. This camera features a 2048x2048 chip that operates at max 100fps at full frame.

The camera has two data interfaces:

  1. Camera Link: 100fps at full frame
  2. USB 3 Link: 30 fps at full frame

The camera also provides two data transfer modes with different readout times:

  1. Standard Scan: 10ms read out time (full frame)
  2. Slow Scan:33ms read out time (full frame)

The slow scan offers lower read noise and is useful for generating high-quality image data trading off speed.

Exp:1 Speed Test using Camera Link Interface and Internal Trigger

We first tested the camera speed using the camera link interface and internal trigger. We used both the transfer modes (standard and slow) and tested the camera speed under various frame sizes without any binning. All experiments were conducted using 16 bit digitization. The Table T1 summarizes our observations.  

Table T1: Maximum speed of operation in FPS for various frame sizes using the camera link interface (PDF Format) .

The tests were performed using the Hamatusu HCI software as well as the Micromanager software by operating in the live mode. Figure B2 shows the configuration of the camera in the two modes (standard and slow) as seen from the HCI software control panel.

Figure B2: Configuration of Camera in the two modes standard (left) and slow (right)

Frame rates were read out from the indications on the live window. Both HCI and micromanager software data were consistent.

It can be seen from Table T1 that the maximum speed of 100fps at full frame size, as indicated in the specs sheet was indeed achieved. Figure B3 shows the screen shot of the HCI software indicating 100fps at 2048x2048 at 1.0037ms exposure. (Camera was operated with sensor covered with lid).

Figure B3: Screen shot of HCI software showing 100fps

The Table T1 also indicates the maximum FPS at various cropped frame sizes of the camera. Please note that in the sCMOS architecture the whole row is read out at once. The acquisition speed is hence independent of the column size. So for all the cropped frame sizes, we used the maximum column width of 2048. 

Also, note that maximum speed for a given frame size is achieved when the frame is centered on the chip. The reason for this can be understood by looking at the sCMOS architecture as shown in Figure B4. The sCMOS chip is divided logically into two (top half (rows 0 to 1023) and bottom half (rows 1024 to 2047)). Each half has its own data read out circuitry. If you center the frame, the task of reading out the top and bottom halves of the frame is split equally between the two data readout circuitry. This minimizes the time required to read out the frame. 

Figure B4: sCMOS internal architecture showing two parallel data read out circuitry. (Source: Hamamatsu)

In the experiments described above the row- offset was set such that the cropped frame is centered. The Table T1 indicates the row offsets required to center the frame.

The table also indicates the maximum and minimum exposure times over which the maximum fps is sustained for a given frame size. For the maximum frame size of 2048x2048, 100 fps is sustained for exposure times between 1.0037ms to 9.99ms. For 2048x8 the camera achieved a speed of 20,543fps at 38.9774us exposure time. 

Note that there was no increase in speed when the frame size was decreased from 2048x8 to 2048x4. The acquisition speeds were same for both the frame sizes. 

It may also be noted that in the internal trigger mode the camera does not allow exposure times below 1.0037ms for frame sizes 2048x256 and above.

A log-log plot of row size vs speed in FPS returned a straight line as seen in Figure B5. This is as expected theoretically. The acquisition time decreases linearly with the decrease in the number of pixels. Since the acquisition speed is inversely proportional to the acquisition time, the log-log plot of row size vs acquisition speed (in fps) will be linear.  

Figure B5. Log-Log plot of frame size vs speed of acquisition in fps for Camera Link Interface using standard and slow scan modes.

We also tested the maximum speed of operation for various bin sizes of the camera. The results are summarized in Table T2. Hamatsu Flash 4 V2 offers 3 bin modes: 1x1, 2x2 and 4x4. 

Table T2: Maximum speed of operation for various bin sizes for Camera Link Interface and Internal Trigger

As expected, binning decreases the speed of operation for a given row size compared to the situation where there is no binning.

We tested the variation in speed for change in the row offset for a given frame size. The results are shown in Table T3. Experiments were conducted using a fixed frame size of 2048x256 using the camera link interface. 

Table T3: Variation in speed as a function of row offset (PDF Version)

The plot of camera speed in FPS as a function of Row Offset is given in Figure B6.

Figure B6: Speed in FPS as a function of Row Offset

As described in figure B4 the camera chip is divided into two segments of 2048x1024 pixels. Each segment has its own data transfer circuitry. For a frame size of 2048x256, for row offsets below 768 (1024-256=768) the frame lies entirely in the first segment and only one data transfer circuitry is involved. Hence the speed is minimum and constant for row offsets 0 -768 as can be seen from Table T3.

As row offset is increased beyond 768, the second data transfer circuitry gets engaged and the speed increases almost linearly till a row offset of 896 (1024-(256/2)=896). For this row offset the frame is distributed symmetrically between the two segments and the speed is maximum. As the row offset is increased further the speed decreases almost linearly. For a row offset greater than 1024 the frame lies entirely in the second segment. The speed drops to the same minimum value as when the frame was entirely in the first segment. The same behavior was seen for both the scan modes.

Exp:2 Speed Test using USB Link Interface and Internal Trigger

We repeated the speed tests using the USB link interface and Internal Trigger. The results are shown in Table T4.

Table T4: Maximum speed of operation in FPS for various frame sizes using the USB link interface (PDF Format) .

Using the USB interface we achieved the max speed of 30fps at full frame as indicated in the manufacturer specifications. The difference in frame rates for the slow scan and standard scan modes was however minimal (mostly similar). For smaller frame sizes the frame rates increased to a maximum of 800fps at 2048x8 frame size. Like with the camera link interface there is no improvement in the speed while decreasing the frame size from 2048x8 to 2048x4.

The log-log plot of frame size vs speed produced a straight line for frame sizes larger than 2048x64 using the standard scan mode as shown in Figure B7.

Figure B7. Log-Log plot of frame size vs speed of acquisition in fps for USB Interface using standard scan

 

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Contributions:

  1. Manoj V Mathew: Conceptualized the ideas, designed the experimental setups, and wrote the article.  
  2. Amit Cherian (NCBS, Bangalore): Setup the equipment, conducted the experiments, tabulated and analyzed data and generated the figures.

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Basic Fluorescence Microscopy

Biological samples are quite complex structures with a lot of stuff in them. Looking for something very specific (for example looking at neurons inside C.elegans larvae) could be almost impossible many a times. Bright Field microscopy cannot reveal this.

Fluorescence microscopy comes in very handy in such situations. Here a component of interest is specifically labeled with a fluorescent molecule called a fluorophore (such as green fluorescent protein (GFP), DAPI etc:). Then by observing the fluorescence of the label the component of interest can be observed. Fluorescence Microscopy collects light only from the component of interest and not from other structures surrounding it (by filtering in the fluorescence wavelengths and filtering our all others).

Fluorescence is a quantum mechanical process that can be explained using the Jablonski Diagram shown in Figure 27.

Figure 27: Jablonski Diagram

The excitation process excites the ground state (S0) electron of the fluorophore to its lowest excited singlet state (S1). Excitation process happens very quickly in about few femtoseconds (fs). Since the S1 state is vibrationally broadened (has a number of vibrational states) excitation can occur over a range of wavelengths. This gives rise to the absorption spectrum of the fluorophore. Once the electron is in one of the higher vibrational levels of S1, it non-radiatively relaxes to the lowest vibrational level of the singlet excited state. This non-radiative decay happens in few picoseconds (ps).

The electron waits at the bottom of the singlet excited state for about few nanoseconds.  Fluorescence occurs when the electron returns to the ground state with the emission of a photon. The ground state is vibrationally broadened as well. The electron can return to any of these vibrational levels before non-radiatively decaying to the lowest vibrational level of the ground state.  This allows for a range of emitted photon energies (wavelengths), observed as the fluorescence emission spectrum.

Since the electron has lost some of its energy in non-radiative processes in the excited and ground states, the emitted photon has energy lower or in other words wavelength longer than the excitation photon. The difference between the peak excitation wavelength and the peak emission wavelength is called the Stoke's shift. 

The total time for the whole process (time for excitation+non-radiative decay+emission) is the fluorescnce lifetime. This is fs+ps+ns which is essentially ns. So, the fluorescence lifetime ends up being the time taken for the electron to decay from the bottom of the excited state to the ground state in the fluorescence emission process.

The basic ray diagram of a fluorescence microscope is given in Figure 28.

Figure 28: Fluorescence Microscope-Ray Diagram

Fluorescence Microscopy is all about separating the emission wavelength from all other wavelendths inlcuding the exciation light and any stray light. This is achieved with the help of the following optical components:

  1. Excitation Filter
  2. Dichroic Mirror
  3. Emission Filter

An epi-fluorescence configuration where the objective also acts as the condenser is used. The light from a broad-band intense light source like a Mercury Arc-Lamp is passed through the excitation filter. This filter transmits the excitation wavelength (band) towards the dichroic mirror. The dichroic mirror reflects the light onto the objective. After interacting with the sample the excitation light excites florescence in regions of the sample that has the specific fluorophore.  

Fluorescence is mostly isotropic. The fluorescence light that emitted in the backward direction is collected by the objective and relayed to the dichroic mirror. The dichroic mirror is designed to transmit the emission wavelength and it, in turn, relays the light to the emission filter. The dichroic mirror behaves differently for the excitation and emission wavelengths. It reflects the excitation wavelength and transmits the emission wavelength. The emission filter, filters out any residual excitation or stray light and transmits the emission wavelength towards the detector. 

The excitation light is way stronger than the emission light. The excitation filter should be designed to handle the entire power of the illuminaiton source usually in 100s of mWatts, of which few 10s of mWatts at the excitation band of the fluorophore is transmitted by the excitation filter . The fluorescence emission is usually in micro-watts, so the emission filter needs a very high transmission at the emission bandwidth and a sharp roll off outside this bandwidth.

In commercial fluorescence microscopes the three components are mounted in a single cube called the Filter Cube. This makes it convenient to change between fluorophores by changing the filter cube as a whole and not worry about individual components.

Configuring the fluorescence microscope in the epi-illumination configuration has a number of advantages:

  1. Objective doubles as the condenser. Alignment is easier. 
  2. Resolution depends on both Objective and Condenser NAs. Since an Objective usually has a high NA and since it is doubling as the condenser the resulting resolution is high.
  3. Easier to separate the excitation and emission wavelengths. Most of the intense excitation light is transmitted through the sample and only a small fraction is scattered back. Hence separating excitation and emission wavelengths in the backward direction (epi-configuration) is easier.

Like in case of bright field imaging we need to setup the epi-fluorescence microscope to achieve Köhler Illumination. There is a small problem here. The Aperture Diaphragm needs to be placed at the back focal plane of the condenser. Since the objective doubles as the condenser, placing it there means both the illumination and collection NAs are affected. Reducing the Aperture Diaphragm to decrease the NA of the condenser will lead to decrease in NA of the objective as it collects the emitted light. To overcome this problem Köhler Illumination in epi-illumination uses a configuration as shown in Figure 28.

Figure 28: Köhler Illumination in Epi-Illumination Configuration. Green is excitation path and red is emission path.

Figure 28 shows the schematic of an epi-fluorescence microscope that uses an arc lamp as the excitation source. Unlike in the case of Köhler Illumination for bright field imaging, the Aperture Diaphragm comes first in the light path and then the Field Diaphragm.

Consider a single point at the center of arc-gap. Light from this point is collimated by the Collector Lens and then focused by Lens B. Focal point of Lens B creates an image of that point on the arc-gap. As we discussed for Köhler Illumination in transmitted illumination configuration  the focal plane of Lens B is a conjugate plane of the light source. We can place the aperture diaphragm here. Lens C is placed at its focal distance from Aperture Diaphragm. This hence produces a collimated beam of light that is directed into the Field Lens.

The Field Aperture is kept at the focal distance away from the Field Lens. This creates an image of the Filed Aperture on the sample plane. The Field Lens focuses the collimated beam of light reflecting through the dichroic mirror onto the back focal plane of the objective. This creates an image of the arc-gap point at the back focal plane of the objective. The objective then collimates this light into a beam pencil to impinge on the sample plane. Like what we discussed earlier the different points on the light source create beam pencils at different angles that intersect at the focal point of the objective (acting as condenser). This creates the cone of light, condensing it on the sample plane.

Note that we have essentially moved the position of the Aperture Diaphragm from the back focal plane of the objective to a point before it. We can now control the NA of illumination without affecting the NA of light collection by the objective.

 

Microscope Magnification

The magnification of a microscope can be determined by using geometrical optics.

  1. Magnification of a Finite Tube Length Microscope

Let us first consider the simplest case of a microscope as shown in Figure 24. This is a finite tube length microscope that uses a single lens. Only the objective. The sample kept at a distance between 'f' and '2f'. As discussed earlier, for a single lens if an object is placed between f and 2f a real and magnified, but an inverted image is formed on the other side of the lens at a distance between 2f and infinity. Again for the reasons discussed before, we will consider an object shaped like an arrow.

Figure 24: Magnification of a Finite Tube Length microscope using camera as a detector

Lets us now look at what magnification is? Simply put, it is the ratio of : (Size of the Image/Size of the Object).

Let us take a closer look at Figure 24. 

(Size of the Image/Size of the Object)=(O'A'/OA)

Now let us consider the triangles COA and CO'A'. It is easy to see that these are similar triangles having the same angles.

Hene by property of similar triangles.

(O'A'/OA)=(CO'/CO):------------------------------------------------------(1)

This Implies:

Magnification of Finite Tube Length microscope=

(Distance of the image from the center of the Objective/Distance of the object from the center of the Objective)-----(1)

Now let us consider a hypothetical scenario where the inverted image of the arrow is moved or translated across the principal axis to the center of the lens as shown in Figure 25.

Figure 25: Magnification of a Finite Tube Length microscope using the camera as a detector. Image translated to the center of the objective.

A line that is drawn from the top of the object (arrow) and passing through the focal point (F) of the objective will intersect the top of the inverted image of the object.

Lets now look at the similar triangles AOF and FCA''. 

Again from the property of similar triangles (CA''/AO) = Magnification= (FC/FO)=(f/r)------------------------------- (2)

This implies:

Magnification of finite tube length microscope=

(Focal Length of the Objective/Distance of the object from the focal point)-----------------------------------(2)

As we had noted earlier for an object placed at the front of the objective a magnified image is formed at the back of the objective only if the object is kept between 'f' and '2f'. 

As the object moves towards f from equation (2) it is obvious that the magnification increases and from equation (1) it turns out that the image will be formed further away from the lens.

It can also be seen from equation (2) that as the object approaches 'f' the magnification approaches infinity which simply means that the beam comes out collimated and the image is formed at infinity. This is also true from equation (1) as the distance of the image from the objective approaches infinity, magnification approaches infinity.

It is now clear that for a finite tube lens microscope the magnification is not a constant, it changes with the distance of the object from the focal point.

Please note that when a manufacturer specifies a magnification for an objective for finite tube length configuration it is matched to a certain tube length. The tube length specifies the distance from the objective where the specified magnification is achieved. If a detector is fixed at this distance, the sample will come into focus only at a fixed distance away from the focal point of the objective.

If the tube length is increased, magnification increases and if the tube length is decreased, magnification decreases.

2.Magnification of an Infinity Corrected Microscope

Now let us consider an infinity corrected microscope as shown in Figure 26. The objective has a focal length of 'fo' and the tube lens has a focal length of 'fi'. Working of the infinity corrected microscope is described in detail in an earlier article.

Figure 26: Magnification of an infinity corrected microscope using the camera as a detector.

The object is kept at a focal distance ('fo') away from the objective and the image is formed at a focal distance ('fi') from the tube lens. An infinity path exists between the objective and the tube lens.

For ease of ray tracing, the tube lens is kept focal distance ('fi') away from the objective lens. 

From equation (1), magnification=(distance of image from lens/distance of object from the lens)

In an infinity corrected configuration image is formed a focal distance ('fi') away from the tube lens and the object is kept focal distance ('fo') away from the objective. 

It then turns out that-

Magnification of Infinity Corrected Microscope=(fi/fo)-----------------------(3)

=(Focal length of tube lens/Focal length of the objective)-------------------(3)

To have a positive magnification the tube lens focal length ('fi') has to be greater than the objective focal length ('fo'). This is mostly the case with most microscopes. They have an objective focal length of few mm and the tube lens focal length typically > 100mm. Please donot confuse the working distance of the objective and focal length of the objective here. The focal length will be much larger than the working distance of an objective.

So, now if magnification depends both on the objective focal length and the tube lens focal length what does it mean to say an objective has a magnification of say 60X. This, in fact, means that the objective will provide a magnification of 60X when coupled with a tube lens of a standard focal length supplied by the manufacturer for microscopes for which that particular 60X objective was designed. 

If you mate the objective with a microscope from another manufacturer or a different model by the same manufacturer for which this objective was not designed for you will not get 60X magnification: because the tube lens focal length is different. This may also lead to other problems like with aberration corrections which may not work well as they are matched to a certain tube lens.

In case you are building your own microscope and using one of the commercially available objectives, make sure you know for what tube lens focal length this objective was designed for.