focal length of concave mirror experiment class 10

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CBSE Class 10 Science Lab Manual – Focal Length of Concave Mirror and Convex Lens

December 2, 2022 by Veerendra

EXPERIMENT 4(a)

Aim To determine the focal length of concave mirror by obtaining the image of a distant object.

Materials Required A concave mirror, a mirror holder, a small screen fixed on a stand, a measuring scale and a distant object (a tree visible clearly through an open window).

• Fix a concave mirror in the mirror holder and place it on the table near an open window. Turn the face of mirror towards a distant object (suppose a tree).
• Place the screen, fitted to a stand, infront of the concave mirror. Adjust the distance of screen, so that the image of the distant object is formed on it as given in the figure below. We can infer from the figure that a clear and bright image could be obtained if the distant object (a tree), is illuminated with sunlight and the screen is placed in the shade. A bright image of the Sun could also be obtained, if the sunlight is made to fall directly on the concave mirror.

• Measure the horizontal distance between the centre of the concave mirror and the screen with the help of a measuring scale.
• Record the observations in the observation table.
• Repeat the experiment two more times by obtaining the images of two different distant objects and measure the distance between the concave mirror and the screen in each case. Record them in the observation table.
• Find the mean value of the focal length for all the observations for different objects.

Observations And Calculations Least count of scale used = …………. mm = …………. cm

Focal length for first object (f 1 ) = ………… m Focal length for second object (f 2 ) = ………….. m Focal length for third object (f 3 ) = …………. m Mean focal length = \(\frac { { f }_{ 1 }+{ f }_{ 2 }+{ f }_{ 3 } }{ 3 }\) = ………… m

Result The approximate value of focal length of the given concave mirror is ………… m.

Precautions

• Concave mirror should be placed near an open window through which sufficient sunlight enters, with its polished surface facing the distant object.
• There should not be any obstacle in the path of rays of light incident on the concave mirror.
• If the image of the Sun has to be formed, then it should be focused on the screen only. The image of the Sun should never be seen directly with the naked eyes. Sunlight should never be focused on any part of the body as it can burn it.
• In order to obtain a sharp and clear image of the distant object on the wall/ground, it must be ensured that the object is well illuminated, so that amount of light incident on the concave mirror is sufficient to produce a well illuminated and distinct image.
• The measuring scale should be parallel to the base of both the stands.
• The mirror holder along with the mirror should be kept perpendicular to the measuring scale for precise measurements.

Sources of Error

• The measuring scale may not be parallel to the base of both the stands.
• The mirror holder, along with the mirror, may not be kept perpendicular to the measuring scale.
• Least count of measuring scale may not be correctly noted.
• Measurement of distance from pole may not be made accurate.

Viva – Voce

Question 1. How will you distinguish between a concave and a convex mirror? [NCERT] Answer: A concave mirror is the spherical mirror with inward curved reflecting surface, whereas a convex mirror is the spherical mirror with outward curved reflecting surface. Concave mirror forms a sharp image, whereas a convex mirror cannot form a sharp image of the distant object.

Question 2. To determine the focal length of a concave mirror, a student focuses a classroom window, a distant tree and the Sun on the screen with the help of a concave mirror. In which case will the student get more accurate value of focal length? [NCERT] Answer: Student will get more accurate value of focal length in the case of Sun.

Question 3. What will be the nature of the image formed by a concave mirror for a distant object? [NCERT] Answer: The nature of the image formed by a concave mirror for a distant object is real and inverted.

Question 4. In reflector type solar cookers, special concave (parabolic) mirrors are used. In such cookers, what should be the preferable position of food vessel for cooking? [NCERT] Answer: In reflector type solar cookers, the preferable position of food vessel should be at focus of the concave mirror.

Question 5. What type of mirror is used in torch? Give reasons. [NCERT] Answer: In torch, concave spherical or parabolic mirror is used because when the bulb (source) is kept at the focus of a concave mirror, parallel beam of light is obtained which travels a large distance.

Question 6. What type of mirror is used as a shaving mirror or in vanity boxes? [NCERT] Answer: Concave mirror is used as a shaving mirror or in vanity boxes, because when the object is placed between its focus and pole, the magnified, erect and virtual images of the object will be formed.

Question 7. Give the condition to hold a mirror for finding the focal length of the concave mirror. Answer: While holding the mirror for finding the focal length of the concave mirror, the aperture of the mirror must not be obstructed.

Question 8. Give the position of an object to obtain a virtual, erect and an image larger than the object using a concave mirror. Answer: The object should be placed between the focus and pole of the mirror.

Question 9. Why do we obtain blurred image from a concave mirror sometimes? Answer: The reason behind the blurred image is that the mirror is away from the object.

Question 10. How can we find the focal length of a concave mirror, when the image is obtained by using a concave mirror? Answer: Focal length can be found out by measuring the distance between the mirror and the screen.

Question 11. Why do we use a screen for obtaining an image from a concave mirror? Answer: Since, the image formed by the mirror is real, it can be obtained on a screen.

Question 12. Is the centre of curvature a part of a spherical mirror? Comment in support of your answer. Answer: The reflecting surface of a spherical mirror forms a part of a sphere. This sphere has a centre. This point is called the centre of curvature of the spherical mirror. So, it is not a part of the mirror as it lies outside its reflecting surface.

Question 13. Which surface of a shining spoon should be polished, to use the spoon as a concave mirror? Answer: The surface of the spoon which is bulged outward should be polished to use the spoon as a concave mirror.

Question 14. When the light of the Sun is directed on a sheet of paper with the help of a mirror, then the paper starts burning. Why? Answer: The light from the Sun is converged at a point, as the sharp, bright spot by the mirror. Actually, this spot of light is the image of the Sun on the sheet of paper. The heat produced due to concentration of sunlight ignites the paper.

Question 15. All the rays of light, after reflection from a concave mirror meet at a point infront of the mirror. Name this point. Answer: When parallel rays of light fall on a concave mirror along its axis, then the rays meet at a point infront of the mirror after reflection from it. This point is called the focal point.

Question 16. How does the size of the image vary as an object is moved from close to the pole to a large distance? Answer: Size of the image decreases as the object is moved away from the pole to a large distance. The image will be virtual for the object between the pole and the focus.

Question 17. State the importance of pole of a mirror. Answer: Pole is the mid-point of the mirror which makes the incident light to go at some angle on the other side of principal axis.

EXPERIMENT 4(b)

Aim To determine the focal length of convex lens by obtaining the image of a distant object.

Apparatus/Materials Required A thin convex lens, a lens holder, a small screen fixed on a stand and a measuring scale.

• Fix a thin convex lens on the lens holder and place it on the table same as that done in the case of concave mirror.
• Place the screen fixed to a stand on the other side of the lens. Adjust the position of screen by moving it back and forth in front of the convex lens to get a sharp and clear image of the distant object.

• Now, measure the horizontal distance between the centre of the convex lens and the screen with the help of a measuring scale.
• Repeat this experiment two more times by obtaining the images of two different distant objects and measure the distance between the convex lens and the screen and record them in the observation table.
• Find the mean value of the focal length for all the observations, for different objects.

Observation Table Least count of scale used = ………… mm = ………… cm

Calculations Focal length for first object (f 1 ) = ………… m Focal length for second object (f 2 ) = ………… m Focal length for third object (f 3 ) = ……………. m Mean focal length or approximate focal length of lens (f) = \(\frac { { f }_{ 1 }+{ f }_{ 2 }+{ f }_{ 3 } }{ 3 }\)

Result From the above observations and calculations, the approximate value of focal length of the given convex lens is ………. m.

• The principal axis of the convex lens should be horizontal, i.e. the lens should be placed vertically.
• There should be no obstacle in the path of rays of light from the distant object incident on the convex lens.
• The image of the sun formed by the lens should be focussed only on the screen. The image of the sun should never be seen directly with the naked eye or it should never be focussed with a convex lens on any part of the body, paper or any inflammable material as it can burn.
• Sometimes, the parallel rays of light originating from a distant object and incident on a convex lens may not be parallel to its principal axis. The image in such situation might be formed slightly away from the principal axis of the lens.
• The base of the stands of the convex lens and screen should be parallel to measuring scale. To determine the focal length, the distance between the convex lens and the screen should be measured horizontally.
• The principal axis of convex lens may not be in horizontal position.
• There may be some obstacle in the path of light ray, coming from the distant object.

Question 1. How will you distinguish between a convex and concave lens? [NCERT] Answer:

  Question 2. To determine the focal length of a convex lens, a student focusses a classroom window, a distant tree and the Sun on the screen. In which case, will the student is closer to accurate value of focal length? [NCERT] Answer: In the case of Sun, because it works as an infinite object and rays will be perfectly parallel to the principal axis.

Question 3. What is the nature of an image formed by a thin convex lens for a distant object? What change do you expect, if the lens were rather thick? [NCERT] Answer: The nature of an image is real, inverted and diminished. If lens becomes thicker, only focal length of lens decreases.

Question 4. You are provided with two convex lenses of same aperture and different thickness. Which one of them will be of shorter focal length? [NCERT] Answer: A thick convex lens has shorter focal length.

Question 5. |f we cover one-half of the convex lens, while focussing a distant object, in what way will it affect the image formed? [NCERT] Answer: If we cover one-half of the convex lens, there will be no change in the nature of lens, only intensity of the image formed decreases.

Question 6. Can this method be used to find the approximate focal length of the concave lens? [NCERT] Answer: No, this method cannot be used to find the approximate focal length of the concave lens, because it always forms a virtual image.

Question 7. Which type of lens is used by the watchmakers, while repairing five parts of a wrist watch? [NCERT] Answer: Watchmakers use convex lens, and to obtain enlarged image, they place the object between optical centre and focal length.

Question 8. Give the complete detail of the nature of image so formed in this experiment. Answer: The nature of image formed in this experiment is as follows:

Question 9. When a ray of light emerges out from a denser medium of the lens, how will it bend into the rarer medium of air? Answer: It bends away from the normal at the point of incidence on the interface.

Question 10. How will a ray of light falling on a denser medium of convex lens bend? Answer: It bends towards the normal at the point of incidence on a denser medium of lens.

Question 11. What happens to a ray of light when it passes through the optical centre of a lens? Answer: When a ray of light passes through the optical centre of a lens, it goes without bending.

Question 12. State whether the nature of image formed by a convex lens depends on the position of object. Answer: Yes, it forms virtual image only when placed between focus and optical centre and for all other positions, it forms real image.

Question 13. On what factor, does the ability of a lens to converge or diverge the light rays depend? Answer: It depends upon the focal length of the lens.

Question 14. If the lens used in the experiment is a plano-convex lens, then what is the radius of curvature of the plane surface? Answer: The radius of curvature of the plane surface of plano-convex lens is infinity.

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Focal length of a concave mirror – theory and experiment

Spherical mirror has two types – Concave mirror and Convex mirror . In this article, we’re going to discuss the focal length of a concave mirror . Though, the focal length of both types of mirrors has the same definition and units, but the signs of focal length are different for these two mirrors . Here, we have included all these key points. Again, there is an experiment to determine the focal length of a concave mirror. The lab data and conclusion for this experiment are also discussed here.

Define focal length of a concave mirror

The focal length of a mirror is defined as the distance of its focal point from its pole. The focal length of a concave mirror is negative in sign if we consider the traditional sign convention rule for the mirror and lens. By the statement a concave mirror of focal length f we mean that the distance of the focal point of the mirror from its pole is f .

Why is the focal length of a concave mirror negative?

Parallel rays coming from the object on the left side will converge at a point on the left side or negative side, after reflecting from the mirror. This point is nothing but the focal point. Thus, the focal distance is negative for a concave mirror.

How to find the focal length of a concave mirror?

Determining the focal length of concave mirror experiment with lab report and conclusion.

In this part, we’re going to discuss the experiment for determining the focal length of a concave mirror with a lab report and conclusion. The concept used is that the parallel rays coming from a large distance meet at a point which is the focal point of the mirror. Then the distance of that point from the mirror will be the focal length of the mirror.

Apparatus used

Ray diagram, experimental data (lab report).

Calculation:

Conclusion:.

The focal length of the concave mirror is 11 cm .

A concave mirror has a focal length of 20 cm. Find its radius of curvature. If an object is placed at 30 cm from the mirror, then find the image distance.

Now, using the mirror formula for concave mirror we get, {\color{Blue} \frac {1}{f} = \frac {1}{v} + \frac {1}{u}}

Then, from the above equation we get, the image distance as, v = 60 cm .

This is all from this article. If you have any doubts on this topic you can ask me in the comment section!

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• Determination of Focal Length of Concave Mirror and Convex Mirror

What are Convex and Concave Mirrors?

A curved mirror in which a reflective surface bulges out towards the light source is known as a convex mirror. The convex mirror reflects the light outwards and so it is not used to focus light. As the object comes nearer to the mirror, the size of the object gets larger until it reaches its original size. These mirrors are also known as diverging mirrors.

A concave mirror has a reflecting surface that caves inwards. The mirror also converges the light at one prime focus point; hence they are also called converging mirrors. They are applied to focus light. Depending upon the location of the object with respect to the mirror, the size of the image formed by the concave mirror varies. It can be real or virtual, inverted or erect and magnified, reduced, or be similar in size of the object depending upon the position.

Focal Length of Concave Mirror

This article will help you find the focal length of a concave mirror. Let’s look at the theory to obtain the image of a farther object.

Like a plane mirror, the concave mirror obeys the law of reflection of light.

Ray of light from an object – The rays of light emitted from a distant object, e.g., distant buildings or sun, are parallel to each other. When the parallel rays from the source fall on the concave mirror along the axis, reflect and meet at the point in front of the mirror, which is known as the mirror's principal focus.

At the focus of the mirror, a real, inverted, and very small image size is formed. 

Focal length – Focal length of the concave mirror is the distance between the pole P of the concave mirror and the focus F. By obtaining the Real image of the distant object, the focal length of a concave mirror can be determined, as shown in the di

Focal Length of Concave Mirror Formula

Let’s see the above-shown diagram, 

Focal Length – The space between the pole P of a concave mirror and the focus F is the focal length of a concave mirror. By obtaining the Real image of a distant object at its focus, the focal length of the concave mirror can be estimated as shown in the diagram.

The focal length of the convex mirror is positive, whereas that of the concave mirror is negative. The same can also be proved by using the mirror formula:

\[\frac{1}{f}\]=\[\frac{1}{v}\]-\[\frac{1}{u}\]

Let's see how

Since we know that an object is always placed at the left side or direction opposite the incidence ray of the mirror, the object distance will always be negative.

v = -v (Image distance is negative since images produced by concave mirrors are usually on the left side or direction opposite to the incidence ray)

Using mirror formula,

 \[\frac{1}{f}\]=\[\frac{1}{v}\]-\[\frac{1}{u}\]

Or \[\frac{1}{f}\]=\[\frac{u-v}{uv}\]

Or \[f=\frac{uv}{u-v}\]

Focal Length of Convex Mirror Using Convex Lens

A curved mirror in which the mirroring surface bulges towards the light source is known as the convex mirror. The light is reflected outwards in a convex mirror; therefore, they are not used to focus light. The convex mirror is also called a diverging mirror or fish-eye mirror.

The image created by a convex lens is erect and virtual since the focal point (F), and center of curvature (2F) are both imaginary points within the mirror that cannot be reached. As a result, the image formed by these mirrors cannot be projected on the screen as the image is inside the mirror. Hence the focal length cannot be determined directly. Initially, the size of the image is smaller than the object, but it gets larger as the object approaches the mirror. The diagram below shows the convex mirror.

(the image will be uploaded soon)

The focal length of a convex mirror can be determined by introducing the convex lens between the object and the convex mirror. With the help of a convex lens side by side with an object, an image can be obtained when the convex mirror reflects the rays along the same path, i.e. when rays fall naturally on the mirror. The space between the screen and the mirror is the radius of curvature, which is denoted by R.

By using the formula below, the focal length f of the convex mirror can be calculated.

\[F=\frac{R}{2}\]

R-Radius of curvature

A mirror with a reflecting surface facing outwards is a Convex mirror, whereas a mirror with a reflecting surface facing inwards is a Concave mirror. The coating of the Convex mirror is on the outside of the spherical surface while the coating of the Concave mirror is on the inside. 

For a Convex Mirror, the principal focus is behind, whereas, for a Concave Mirror, the principal focus is at the front. A point at which the reflected rays meet or appear to meet is the Principal focus.

To find the focal length of a Concave mirror:

The various ways to obtain the focal length of the concave mirror:

i)A spherical mirror whose reflecting surface is curved inwards and follows laws of reflection of light is a Concave mirror.

ii) The light rays that come in from a distant object are considered to be parallel to each other.

iii) The parallel rays of light will meet the point in the front of the mirror if the image formed is real, inverted, and small in size.

(Image will be uploaded soon)

iv) The image formed by the convex lens is real and can be obtained on the screen.

v)  the symbol ‘f’ is used to denote the difference between the principal axis P and the focus F of the concave mirror.  

To find the focal length of a Convex mirror:

The various ways to obtain the focal length of the convex lens:

The middle part of the convex lens is thicker and the edges are thinner.  This is known as a converging lens.

The refracted rays from the parallel beam of light converge on the other side of the convex lens.

The image would be real if the image is obtained at the focus of the lens,  inverted and very small.

‘f’ is the focal length which is the difference between the optical center of the lens and the principal focus.

As the image formed by the lens is real, the image can be obtained on the screen.

The procedure of determining the focal point of a Concave Mirror can be explained as follows: 

The distance between the selected object should be more than 50 ft.

The concave mirror placed on the mirror stand and the distant object should be facing each other.

The screen should be in front of the reflecting surface of the mirror and to be able to get a sharp image, adjustments should be made to the screen.

The distance between the concave mirror and screen can be determined by using a meter scale. The distance and focal length of the mirror will be the same as the given Concave Mirror.

To calculate the average focal length, we will have to repeat the above procedure three times.

The procedure of determining the focal point of a Convex Lens can be explained as follows: 

Arrange both the lens and the screen of them on the wooden bench.

The lens should be placed on the holder in such a way that it is facing a distant object.

Holder should be placed with the screen on the bench.

The position of the screen should be such that the sharp image of the distant object is obtained on it.

The difference between the two positions i.e. of the lens and of the screen has to be equal to the focal length of the given convex lens.

Shift the focus towards various other distant objects in order to calculate the focal length of the convex lens.

FAQs on Determination of Focal Length of Concave Mirror and Convex Mirror

1. What is a Relation Between Focal Length and Radius of Curvatur?

The relationship between the radius of curvature and focal length r is represented by r =2f

Consider a ray of light AB which is parallel to the principal axis and incident on a spherical mirror at point B. The normal surface at point B is CB, and CP = CB = R is the radius of curvature. The ray AB following reflection from the mirror will pass via F(concave mirror) or will appear to diverge from F(convex mirror) and obeys the law of reflection i.e., i = r.

From the geometry of the diagram,

∠BCP = θ = i

In D CBF, θ = r

∴BF = FC (because i = r)

If the hole of the mirror is small, B lies close to P, and therefore BF = PF.

=>FC = FP = PF

=>PC = PF + FC = PF + PF

=>R = 2 PF = 2f

=>f = R/2

Similar relation stands for convex mirror also. In obtaining this relation, we have believed that the aperture of the mirror is small.

2. Why is a Driving Mirror of Vehicles made of a Convex Mirror?

For all positions of an object, a convex mirror forms virtual, erect, and diminished image. As the image diminished in size, a broader field of view behind the vehicle is covered. The two qualities of the image formed by the convex mirror, viz. a wider and erect field of sight help the driver in driving the vehicle with ease.

3.What are the differences between Concave and Convex mirrors?

The main important differences between Convex and Concave mirrors are:

• Determination of the focal length of Concave mirror by obtaining the image of a distant object

To determine the focal length of a concave mirror by obtaining the image of a distant object.

Apparatus required.

• Concave mirror
• A meter scale
• A mirror holder
• A screen holder
• A concave mirror is defined as the spherical mirror whose reflecting surface is curved inwards and follows laws of reflection of light.
• When the object is at infinity, then real, inverted and highly diminished image is formed at focus.
• The light rays coming from a distant object can be considered to be parallel to each other.
• If the image formed is real, inverted and very small in size, then the parallel rays of light meet the point in the front of the mirror.

• Fix a concave mirror in the mirror holder and place it on the table near an open window. Turn the face of mirror towards a distant object (suppose a tree).
• Place the screen, fitted to a stand, in front of the concave mirror.
• Adjust the distance of screen, so that the image of the distant object is formed on it as given in the figure below. We can infer from the figure that a clear and bright image could be obtained if the distant object (a tree), is illuminated with sunlight and the screen is placed in the shade. A bright image of the Sun could also be obtained, if the sunlight is made to fall directly on the concave mirror.
• When a sharp image of distant object is obtained, then mark the position of the Centre of the stand holding the mirror and the screen as (a) and (b), respectively (see Fig 3).
• Measure the horizontal distance between the Centre of the concave mirror and the screen with the help of a measuring scale.
• Repeat the experiment two more times by obtaining the images of two different distant objects and measure the distance between the concave mirror and the screen in each case. Record them in the observation table.
• Find the mean value of the focal length for all the observations for different objects.

Observations

Calculation

The approximate value of focal length of the given concave mirror = 29.93 cm.

Precautions

• Concave mirror should be placed near an open window through which sufficient sunlight enters, with its polished surface facing the distant object.
• There should not be any obstacle in the path of rays of light incident on the concave mirror.
• In order to obtain a sharp and clear image of the distant object on the wall/ground, it must be ensured that the object is well illuminated, so that amount of light incident on the concave mirror is sufficient to produce a well illuminated and distinct image.
• The measuring scale should be parallel to the base of both the stands.
• The mirror holder along with the mirror should be kept perpendicular to the measuring scale for precise measurements.

Class 10 Physics Practicals

• To study the dependence of current on the potential difference with graph
• Determination of equivalent resistance of two resistors when connected in series
• Determination of equivalent resistance of two resistors when connected in parallel
• Determination of the focal length of Convex lens by obtaining the image of a distant object
• Tracing the path of a ray of light passing through a rectangular glass slab for different angles of incidence
• Tracing the path of the rays of light through a glass prism

Class 10 Chemistry Practicals

• Finding the pH of the samples by using pH paper/universal indicator
• Studying the properties of acids and bases (HCl & NaOH) on the basis of their reaction
• Performing and observing the reactions and classifying them into: i. Combination reaction, ii. Decomposition reaction, iii. Displacement reaction and iv. Double displacement reaction
• Observing the action of Zn, Fe, Cu and Al metals on the salt solutions
• Studying the properties of acetic acid (ethanoic acid)
• Studying saponification reaction for preparation of soap
• Studying the comparative cleaning capacity of a sample of soap in soft and hard water

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To determine the focal length of a concave mirror by obtaining image of a distant object.

To determine the feral length of a concave mirror by obtaining image of a distant object.

Materials and Apparatus Required

A concave mirror, a metre scale, a small screen fixed on a stand and a mirror holder.

A spherical surface which is silvered or brightly polished so as to reflect light incident on its concave side is called concave mirror. The centre of the sphere of which the mirror is a part is called the centre of curvature. The central point P of the spherical surface of the mirror is called the pole of the mirror. The line joining the pole to the centre of curvature C is called the principal axis. The radius of the sphere of which the surface of the mirror forms a part is called the radius of curvature of the mirror.

The incident rays parallel to principal axis on a concave mirror, after reflection meet at a point on the principal axis. This point is called the principal focus or simply focus. It is denoted by F. A plane passing through the focus and normal to principal axis is called focal plane. The distance of focus from the pole of mirror is called focal length of the mirror. It is denoted by f.

i.e. ( 1 / f) = (1 / v) + (1 / u)

f = uv / (u = v)

where, v = distance of image from mirror u = distance of object from mirror f = focal length of the mirror

When the object is at infinity in front of a concave mirror, its image is formed at the focus of the mirror.

A distant object (a tree or a distant building or an electricity pole) can be considered as an object at infinity, and its image will be formed at the focus of the mirror. The image formed is real, inverted and very small in size.

Since the image formed by the mirror is real, it can be obtained on the screen. Thus, we can estimate the focal of a concave mirror by obtaining a real image of a distant object at its focus.

• Hold the given concave mirror with the help of its stand without obstructing its aperture.
• Face the mirror towards the window of your laboratory from which a distant object like a tree is visible.
• Place the screen in front of the concave mirror.
• Select the right position of the screen by moving it back and forth so that a sharp, clear and inverted image of the distant tree is formed on it. We can get a clear and bright image if the distant object is illuminated with sunlight and the screen is placed in the shade. We can also get an image of the sun if sunlight is made to fall directly on the concave mirror.
• Now, mark the position of the centre of the stands holding the mirror and the screen.
• Measure the horizontal distance between the concave mirror and the screen using a metre scale.
• Note the observations in the observation table.
• Repeat the experiment two more times by obtaining the image of two different distant objects. Measure the distances between the concave mirror and the screen in each case.
• Record the observations in table.
• Calculate the mean value of the focal length.

Observations

Calculation

Mean focal length f = (f1 + f2 + f3 / 3) cm.

The approximate focal length of the given concave mirror is .............. cm.

Precautions

• The distant object taken for seeing the image should be clearly visible.
• Image obtained on the screen should be sharp and distinct.
• Mirror surface should be clean.
• No obstacle should be in the path of rays of light from the distant object, incident on the concave mirror.
• Sunlight should never be focussed with a concave mirror on the sheet of paper. The heat produced due to the concentration of sunlight ignites the paper. The image of the sun should be focussed only on the screen.

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• 1 To determine the focal length of a concave mirror by obtaining image of a distant object.

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• 10 Science Lab Manual
• Focal Length Of Concave Mirror And Convex Lens Solution

Focal Length Of Concave Mirror And Convex Lens

Class 10 th science lab manual cbse solution.

• AIMTo Determine the focal length of the concave mirror by obtaining the image of a distant…
• AIMTo determine the focal length of a convex lens by obtaining the image of a distant…

What is a spherical mirror, give its type ?

What is an aperture of the spherical mirror?

what is centre and radius of curvature?

Define principal focus and axis?

• Where is the brightest and sharpest image is formed by a concave mirror ?…

Comment on the image formed by a concave mirror?

Why concave mirror is called a converging mirror?

Give two uses of the convex mirror?

What is the difference between a real and virtual image?

State law of reflection of light ?

What is a lens?

What are different types of lens?

Write the lens formula?

Write the mirror formula?

When does a lens behave like a plane glass plate?

What is the way to increase the magnification power of a lens ?

• Comment on the magnification sign for the concave and convex lens?…

What is the power of the lens?

What is a factor the power of lens?

• If we cover one half of the lens while focusing on a distant object, in what ways will it…

Lab Experiment 10a

AIM To Determine the focal length of the concave mirror by obtaining the image of a distant object.

MATERIALS REQUIRED

A concave mirror, a metre scale, a mirror holder.

1. A spherical mirror whose reflecting surface is curved inwards, i.e., faces towards the centre of the sphere is called a concave mirror.

2. It obeys the law of the reflection of light like a plane mirror.

3. Thus, the parallel beam of a ray of light coming from a distant object, such as the sun or a building can be considered as parallel.

4. When the parallel ray is incident on the reflecting surface of a mirror, then after reflection, the rays converge at a point, and this point is called principal focus of the concave mirror as shown in the figure.

5. since the image formed by the mirror is real so that it can be obtained on the screen.

6. The distance between the pole O and principal focus F of a concave mirror is called the focal length of the mirror. It is equal to half the radius of curvature of the mirror.

1. Choose a distant object like a tree or the sun to at as an object for our experiment.

2. Mount the concave mirror in a mirror holder.

3. Adjust the concave mirror in such a way that the rays of light coming from the tree fall directly on its reflecting surface.

4. Measure the horizontal distance between the wall and the centre of a concave mirror with the help of a meter scale.

5. Repeat the experiment by selecting the different distant objects at different distances to measure the focal length of the concave mirror.

OBSERVATION AND CALCULATION

The approximate focal length of the given concave mirror is 18 cm as determined by the above method.

According to the sign conventions, the focal length of a concave mirror is negative. Therefore, f = - 18 cm

PRECAUTIONS

1. The mirror should be placed vertically in the lens holder.

2. There should not be any obstacle in the path of rays of light incident on the concave mirror

3. The meter scale must be correctly positioned between the wall and centre of the concave mirror.

Lab Experiment 10b

AIM To determine the focal length of a convex lens by obtaining the image of a distant object.

A convex lens, a lens holder, a white screen such as a wall or white painted board, meter scale, distant object.

1. Choose any distant object like a tree or the sun as an object for the experiment.

2. Held the lens vertically inside a lens holder and lens must be kept in vertical position during the experiment.

3. Place a screen on the other side of the lens.

4. Obtain the image of tree/building on a white screen or wall. Move the lens to get a sharp image.

5. Using a meter scale, measure the distance from the lens to the screen.

6. Repeat this procedure by changing the object.

The approximate focal length of a convex lens = 13.25 cm.

1. The lens should be kept vertical inside the lens holder.

2. While measuring the distance, the distance between the sharp image and the centre of the lens is to be measured.

3. The lens the screen must be at the same level.

Viva Questions

The spherical mirror is a part of a hollow sphere with one side having silver/mercury coating, further coated with paint to protect it from damage.

It is of two types :

1. Concave mirror: Silvered at the outer surface so that reflection takes place from the inner surface(concave).

2. Convex mirror: Silver at inner surface so that reflection takes place from the outer surface(convex).

It is the width of the reflecting surface from which reflection takes place.

, The centre of curvature, is the geometrical centre of the hollow sphere “c” from which the mirror is formed.

The radius of curvature is the radius “R” of the hollow sphere whose part is a spherical mirror.

The Principal axis is the straight line joining the pole and centre of curvature.

The principal focus is the point “f” on the parallel axis where a parallel beam of light actually meets after reflection.

Where is the brightest and sharpest image is formed by a concave mirror ?

The brightest and the sharpest image for the concave mirror is formed at the focus.

1. Virtual, erect and enlarged image of an object is formed behind the concave mirror.

2. The real and inverted image is formed in front of the mirror.

A concave mirror is called a converging mirror because it converges the parallel beam of light ray after reflection at a point.

The uses of concave mirror are:

1. They are reflectors in the torch to reflect the light rays.

2. It is used as a shaving mirror to get an erect and enlarged image of the face.

A real image is obtained on the screen, they are formed by the interaction of a ray of light after reflection.

A virtual image is formed when the ray of light appear to meet after the reflection; Image cannot be taken on the screen.

1. The angle incidence is equal to the angle of reflection.

2. The incident ray, the reflected ray and the normal at the point of incidence,all lie on the same plane.

A homogenous transparent material or medium bounded by two surfaces at different or same radii of curvature is called a lens.

1. Double convex lens: if both the refracting surface is convex, it is thicker at the middle and thinner at the edges.

2. Double concave lens: If both the refracting surface of the lens is concave, it is thinner at middle and thicker at the edges.

Lens formula:

Where f is the focal length of the lens

u is the distance between the object, and the lens

v is the distance between the image and the lens.

The mirror formula:

Where f is the focal length of the mirror

v is the distance between the image and lens

u is the distance between the object and the mirror.

The lens will behave like a plane glass only when it is kept in a medium whose refractive index is equal to that of the lens .

The magnification power of the lens can be increased by using a number of lenses.

Comment on the magnification sign for the concave and convex lens?

1. Convex lens, magnification is positive for virtual image and negative for real image.

2. Concave lens, magnification is always positive because virtual image is always formed.

The power lens is the degree of convergence or divergence of the light ray incident on any refracting surface of the lens.

the magnification power of the lens is inversely proportional to the focal length of the lens.

If we cover one half of the lens while focusing on a distant object, in what ways will it affect the image formed?

A full-size image is formed, but only the intensity or brightness of the image will reduce.

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• Open access
• Published: 12 September 2024

Cylindrical compression of thin wires by irradiation with a Joule-class short-pulse laser

• Alejandro Laso Garcia   ORCID: orcid.org/0000-0002-7671-0901 1   na1 ,
• Long Yang   ORCID: orcid.org/0009-0000-9207-6026 1   na1 ,
• Victorien Bouffetier 2 ,
• Karen Appel   ORCID: orcid.org/0000-0002-2902-2102 2 ,
• Carsten Baehtz   ORCID: orcid.org/0000-0003-1480-511X 1 ,
• Johannes Hagemann   ORCID: orcid.org/0000-0003-2768-9496 3 ,
• Hauke Höppner   ORCID: orcid.org/0009-0000-1929-5097 1 ,
• Oliver Humphries   ORCID: orcid.org/0000-0001-6748-0422 2 ,
• Thomas Kluge   ORCID: orcid.org/0000-0003-4861-5584 1 ,
• Mikhail Mishchenko   ORCID: orcid.org/0000-0002-2362-9392 2 ,
• Motoaki Nakatsutsumi   ORCID: orcid.org/0000-0003-0868-4745 2 ,
• Alexander Pelka 1 ,
• Thomas R. Preston   ORCID: orcid.org/0000-0003-1228-2263 2 ,
• Lisa Randolph   ORCID: orcid.org/0000-0001-9587-404X 2 ,
• Ulf Zastrau   ORCID: orcid.org/0000-0002-3575-4449 2 ,
• Thomas E. Cowan   ORCID: orcid.org/0000-0002-5845-000X 1 , 4 ,
• Lingen Huang   ORCID: orcid.org/0000-0003-1184-2097 1 &
• Toma Toncian   ORCID: orcid.org/0000-0001-7986-3631 1  

Nature Communications volume  15 , Article number:  7896 ( 2024 ) Cite this article

Metrics details

• Astrophysical plasmas
• Imaging techniques
• Laser-produced plasmas
• Phase transitions and critical phenomena

Equation of state measurements at Jovian or stellar conditions are currently conducted by dynamic shock compression driven by multi-kilojoule multi-beam nanosecond-duration lasers. These experiments require precise design of the target and specific tailoring of the spatial and temporal laser profiles to reach the highest pressures. At the same time, the studies are limited by the low repetition rate of the lasers. Here, we show that by the irradiation of a thin wire with single-beam Joule-class short-pulse laser, a converging cylindrical shock is generated compressing the wire material to conditions relevant to the above applications. The shockwave was observed using Phase Contrast Imaging employing a hard X-ray Free Electron Laser with unprecedented temporal and spatial sensitivity. The data collected for Cu wires is in agreement with hydrodynamic simulations of an ablative shock launched by highly impulsive and transient resistive heating of the wire surface. The subsequent cylindrical shockwave travels toward the wire axis and is predicted to reach a compression factor of 9 and pressures above 800 Mbar. Simulations for astrophysical relevant materials underline the potential of this compression technique as a new tool for high energy density studies at high repetition rates.

Introduction

Dynamic shock compression serves as a crucial tool for creating warm and hot dense matter under extreme conditions that exist throughout the universe, such as the interior of planets, supernovae, and astrophysical jets. To generate these conditions in the laboratory, a wide array of techniques are employed, such as gas guns 1 , pulse power systems 2 , 3 , and nanosecond high-energy laser pulses 4 , 5 . Two types of shock geometries are commonly employed: planar shocks, which are prevalent in most cases, and converging shocks. Converging shocks are particularly valuable as they deliver energy to a small volume, resulting in the compression of material to exceedingly high densities and pressures. The generation of converging shocks requires precise design and facilities enabling laser irradiation with multiple beams such as OMEGA, NIF, or LMJ 6 , 7 , 8 , 9 , 10 , 11 .

X-ray Free Electron Lasers (XFEL) provide a novel platform for studying compression and shock physics. The high number of X-ray photons per pulse, low bandwidth, short temporal pulse length and high coherence make XFELs a great tool to study ultra-fast structural dynamics via a combination of techniques: X-ray diffraction, small-angle and wide-angle X-ray scattering, phase contrast imaging, X-ray absorption and emission spectroscopy, etc. The combination of the XFEL beams with high-power optical laser drivers has enabled precision measurements of extreme states of matter. The generation of high-pressure states at these facilities has been restricted to the use of high-energy (60 J) nanosecond pulse duration lasers and recently upgraded to 100 J. With these drivers, scientists have been able to study the equation of state and phase transitions of materials 12 , 13 , 14 , as well as generating conditions relevant to Earth’s mantle 15 , and large planet interiors 16 , 17 , 18 , 19 . However, the phase-space coverage is constrained in pressure range to a few Mbar due to the limited laser energy. The extension of the experimental capabilities is currently discussed by upgrade roadmaps involving coupling multi-kilojoule ns-lasers at existing XFEL instruments, like the Matter at Extreme Conditions 20 with the MEC-U upgrade at LCLS and the HED/HiBEF and with the HiBEF 2.0 upgrade at EuXFEL.

On the other hand, instruments at XFELs are also equipped with short-pulse lasers delivering Joule-level energies, pulse duration of tens of femtoseconds, and reaching intensities up to 10 20 W/cm 2   21 , 22 when focused on a sample. The interaction of such lasers with the matter generates a blast wave following the strong localized heating in the focal spot of the laser 23 , 24 , 25 , 26 , 27 , 28 , and secondary radiation absorbed heterogeneously by the sample can drive hydrodynamic motion 29 , 30 , 31 . While these past experimental studies have focused on the rarefaction subsequent to the shock propagation following this blast wave, we experimentally demonstrate that by irradiating a thin wire with a short-pulse laser, conditions are met where a cylindrical shock is generated at the surface that propagates toward the wire axis. At the convergence point, this shock achieves a high compression factor and pressure. We attribute the generation of the radial compression wave to an ablative shock created by transient resistive heating of a thin surface layer of the wire. We perform particle-in-cell (PIC) simulations, shedding light on the conditions driving the shock and hydrodynamic simulations recovering the experimentally measured compression wave evolution. As an outlook we investigate the potential of this compression scheme for different materials relevant in the astrophysical context, showing that Jovian and white dwarf conditions could be reached, enabling complementary studies to those performed at kJ-class facilities.

Experimental setup for imaging the convergent shocks

The experiment was performed at the European X-ray Free Electron Laser facility using the ReLaX laser as a relativistic plasma driver operating at the HED-HiBEF instrument 32 . A schematic of the experiment is shown in Fig.  1 a. The ReLaX laser was used at 100 TW level, delivering laser pulses with an energy of 3 J on target and sub 30 fs full-width half-maximum (FWHM) pulse duration. The laser was focused employing an F/2 off-axis parabola to a spot size of approximately 4 μm FWHM resulting in an average intensity of 10 20 W/cm 2 . The 8.2 keV X-rays generated by the SASE2 undulator were used to illuminate a square region (250 μm) 2 around the ReLaX focal spot. This plane was imaged and magnified by a compound refractive lens 33 stack consisting of 10 beryllium lenses with a focal length of approximately 53 cm to an imaging X-ray detector located 3.3 m away. The detector was a GAGG scintillator imaged to an Andor Zyla CMOS camera via a ×7.5 objective. The detector pixel pitch is 6.5 μm, and after accounting the total magnification factor, it results in an equivalent pixel size on target of 150 nm/pixel. The imaging system was tested using resolution test targets (NTT-XRESO 50HC) to resolve 500 nm structures. It has to be noted that the resolution was limited for this experiment by the chromaticity of the SASE X-ray beam with a bandwidth of approximately 20 eV. The temporal evolution was recorded by variation of the pump-probe delay with a precision of 200 fs (RMS) given by the chosen temporal synchronization scheme (locked to the accelerator’s radio frequency). The raw images are flat-fielded using the free-beam X-ray intensity distribution (without a target) and accounting for the instrument backgrounds. The main experimental results are summarized in Fig.  1 b, showing Phase Contrast Images (PCI) of the wire, for different time delays ranging from 100 ps to 1 ns after laser irradiation. It is worth mentioning the used X-ray pulse duration ≤50 fs is much shorter than the typical few picosecond duration of laser-driven X-ray backlighters used conventionally in all-optical setups 34 that are also typically limited to 10s μm spatial resolution due to the source proprieties. In the PCI data, the 25 μm diameter wires are oriented vertically with the optical laser propagating from the left side and focused to the left edge of the wire in the vertical center of the illuminated area. Besides the attenuation and phase contrast generated by the wire, the evolution of two distinct structures can be measured. During the first 300 ps, a spherical shock originating from the focal spot volume is observed, similar to ref. 28 . At 300 ps after the laser irradiation, this shock has already propagated through the wire. At the same time, one can follow a second nearly cylindrical shock moving radially inward toward the wire axis originating from both the left and right edges of the wire. The velocity of this shock decreases with increasing distance from the ReLaX focus. At 500–1000 ps, a convergence of the shock close to the axis of the wire can be seen. Simultaneous with the inward radial motion, a radial expansion resulting in a smoothing of the wire edge is observed. This effect is attributed to wire plasma expansion. To quantify the evolution associated with the radially converging structure, we have analyzed lineouts at 42 and 100 μm from the focal position of the laser.

a Experimental setup of the PCI configuration used for imaging the compressed wire. The whole setup until the last 50 cm before the detector is placed in vacuum conditions, minimizing air scattering. A slit system (not shown) is used to limit the X-ray illumination to a field of view at the sample position to 250 × 250 μm and minimize fringe scattering by the CRLs (300 μm diameter). b X-ray PCI data measured at delays from 100 to 1000 ps after the irradiation of a 25 μm Cu wire by a 3 J, 30 fs laser pulse. The color scale gives the change in transmission compared to free-beam propagation. c Zoom into the red highlighted area of ( b ) for improved visibility of the converging shock.

We calculate the velocity of the shock by measuring the distance traveled by the shockwave between the time delays of 300 ps and 500 ps. These time delays lie within a constant shock velocity region, avoiding the initial deceleration as well as the final acceleration, as predicted by simulations and explained in the next section. An average velocity of 14.3 ± 1.3 km/s is observed 42 μm close to the laser focus and of 10.5 ± 1.3 km/s at 100 μm. Additionally, we evaluated the shock front speed emerging from the laser focus. Here, the velocity of the front decreases from 180 km/s at 20 ps to 50 km/s at 300 ps at the time of the shock release and is in agreement with ref. 28 .

Origin of the shocks unraveled by numerical simulations

In this section the origin of the observed cylindrical converging shocks is investigated. It is instructive to start by looking at the details of the laser-wire interaction. Particle in cell simulations are commonly used for the simulation of laser-matter interaction as predictive tools, giving detailed insights into plasma evolution at the time of laser interaction and shortly after (at ps timescales). Performing a simulation with a wire as target, one would observe processes similar to refs. 35 , 36 : first, the generation of a highly energetic electron population by the direct interaction of the impulsive laser pulse with the wire material that will propagate longitudinally through the wire, leading, for example, to generation of a plasma sheath with strong electric fields that accelerate ions 37 . Simultaneously, part of the hot electron population will move at close to the speed of light transversely away from the focal spot. The hot electrons are electrostatically trapped close to the wire surface, and in their wake, the wire surface is ionized. To achieve charge balance a return current along the surface is established reaching current densities of 10 13 A/cm 2 . These currents encompass the whole wire surface up to a skin depth. Proton imaging techniques (employing laser-accelerated protons) have been successfully applied to measuring the sheath fields and thus the dynamics of the associated hot electron transport and subsequent return current 35 , 38 , 39 , 40 .

While the lifetime of such currents was estimated by Quinn et al. to be 20 ps 35 , one order of magnitude longer than the ps duration of the laser pulse used, recent theoretical work has been focused on laser pulses matched to our experiment of several tens fs 41 , work that follows a multi-scale approach. PIC simulations are used in the first step to calculate the current density profile along an extended wire. With a model detailed by equation ( 1 ), a temperature distribution is evaluated from the current density within the skin depth at the wire surface. Finally, this temperature distribution is used as initial conditions for hydrodynamic simulations to predict the long-term shock formation, propagation and density compression. To confirm this scenario, we have performed 2D PIC simulations of the laser interaction with a 10 μm diameter Cu wire target. We observe the formation of a return peak current of 2.8 × 10 13 A/cm 2 , with a lifetime of 100fs (Fig.  2 a shows j y and associated magnetic field for t = 38 fs after the peak of the interaction, and Fig.  2 b the time evolution peak of j y at a distance of 8 μm away from the focus). The surface return current has two effects on the wire target which can lead to compression: the magnetic compression and Joule heating and the associated ablation. While the return current magnitude scales inverse of the wire radius 41 , for a 25 μm diameter copper wire, the peak of the surface return current density is predicted to be in the range of 0.4–1.1 × 10 13 A/cm 2 , with a current strength decaying further away from the laser focus. At the time when the return current is maximal, the thermal pressure is evaluated to be 8 times higher than the magnetic pressure of 250 Mbar. The resulting plasma β  ≈ 8 confirms the kinetic nature of our shock formation. The electron temperature distribution to be used as initial condition for the hydro simulations is estimated using the electron energy equation 41 , 42 ,

where \({K}_{{T}_{e}}\) is the thermal conductivity of cold electrons, \({\sigma }_{{T}_{e}}\) is the electric conductivity, and j e ( r ) is the surface return current distribution in the radial direction from PIC (Fig.  2 c). The equation’s right side accounts for heat diffusion in the radial direction and the Joule heating due to the surface return current. The electron resistivity model and the heat diffusion coefficient are functions of temperature and density and can be extracted from the SESAME equation of state 43 . The calculated electron temperature with this equation peaks at 140–320 eV for the return current range of 0.4–1.1 × 10 13 A/cm 2 and is distributed within a 0.1 μm skin depth layer. This temperature distribution is used as the initial condition for hydrodynamic simulation to investigate plasma evolution until 1 ns. These simulations with the FLASH code 44 , 45 solve in a 1D cylindrical symmetry the one fluid, two-species (ion and electron) and two-temperatures hydrodynamic equations with copper SESAME equation of state 43 . The ion temperature and electron temperature are assumed to be equal due to the high collision rates between these.

a , b and c correspond to the 2D PIC simulations using a copper 10 μm diameter wire; d , e and f correspond to the 1D hydro simulations using the SESAME equation of state. a Snapshot of the current density parallel to the wire axis and the associated magnetic field strength 38 fs after peak of the interaction, b time evolution of peak current density 8 μm away from the laser focus and c distribution of the current density, magnetic field, bulk temperature for the surface layer for t = 38 fs. Hydrodynamic temporal evolution of compression factor ( d ), pressure ( e ) and temperature ( f ) for a shock driven by a surface temperature of 250 eV.

Figure  2 d–f shows the result of the simulation of the shock dynamic for a 25 μm copper wire under the experimental laser irradiation conditions, giving the temporal and spatial evolution of compression (density normalized to initial density), pressure and temperature. The initial condition used is a peak temperature of 250 eV with an exponential decay depth and a decay constant τ  = 0.067 μm. The shock is formed within the first 5 ps due to the ablation pressure. It starts at the surface with a compression factor of 2.8 with respect to cold copper and a peak pressure of 111 Mbar. It travels toward the wire axis with a starting velocity of 44 km/s. At a time of 200 ps, the shockwave has decelerated to 15 km/s, and the temperature of the shock front reaches 22 eV, while the pressure decreases to 12 Mbar. The peak compression factor at this point is 2. Between 200 ps and 650 ps, the shock propagates with constant conditions, and the shock front moves from 8.1 μm away from the wire axis down to 1.1 μm. After this point, the shock gains velocity until it converges at the wire axis, reaching compression factors of 9, corresponding to densities of 80.6g/cm 3 , a temperature of 38 eV and a pressure of 830 Mbar.

Comparison between experimental and synthetic imaging data

Using the radial density profile from the hydrodynamic simulations, the expected X-ray PCI profiles were calculated and compared with experimental results. Using a forward Abel transform of the density profile, the projected mass density is obtained as probed by the X-ray in the experimental geometry. The intensity at the detector plane can be calculated via the Transport of Intensity Equation 46 , as:

where I ( x 1 ,  z  =  z 1 ) is the intensity at the detector located at a propagation distance z 1 , I ( x ,  z  = 0) is the intensity at contact, that is directly at the exit plane of the target and given by the attenuation of the X-rays by the target, k is the X-ray wave-vector, and Φ( x ,  z  = 0) is the phase shift at contact. The PCI data measured with an undriven wire was used to characterize the propagation distance, resulting in an equivalent plane located at a distance z 1  = 6 mm after the target that is imaged by the CRLs onto the detector. Figure  3 a shows the temporal evolution of the forward calculated PCI pattern up to 1 ns for two initial temperatures, 210 eV and 250 eV. The convergence time is 758 ps for the 210 eV simulation and 698 ps for 250 eV. The resulting profiles at time-steps 300, 500 and 700 ps are compared in Fig.  3 b with experimental data 42 μm away from the laser focus. The synthetic profiles reproduce quantitatively and qualitatively the experimentally observed PCI patterns with features such as the position of the inward propagating shock-front and outward beam refraction. In this context, it is important to consider the effect of temperature-induced opacity changes. Using the TOPS/ATOMIC database 47 , it was confirmed that the values for the mass attenuation coefficient for a temperature of 22 eV differ on percent level from those of cold copper material, thus it can be neglected for the rest of the analysis.

a Simulated PCI profiles using the hydrodynamic simulation data for initial temperatures of 210 and 250 eV. b Comparison of experimental and simulated data 300 ps, 500 ps, and 700 ps for 210 eV and 250 eV initial temperature. The matching of the experimentally observed convergence at 700 ps occurs for the 250 eV case.

We have selected the experimental data at 300 ps and 500 ps delay and further analyzed the PCI profiles at 42 μm away from the laser focus to extract the shock parameters. As the shock acceleration is minimal between these delays, the uncertainty in the velocity estimation and its effect on the shock pressure is reduced. The generalized Paganin method 48 was applied to calculate the intensity at the target plane. Here the assumption is that all the intensity variation is due to absorption in the wire. An inverse Abel transform is used to extract the radial mass attenuation and, consequently, the density profile. Finally, the Rankine-Hugoniot equation is used to calculate the shock pressure using the experimental shock velocity and density, obtaining a value \(p=11.{0}_{-2.6}^{+4.0}\) Mbar for 300 ps and \(p=10.{0}_{-2.6}^{+3.9}\) Mbar for 500 ps. The values extracted from hydrodynamic simulations are 10.5 Mbar and 10.6 Mbar, in close agreement with the experimental value. Since an experimental velocity was not available, only the evaluated central density of \(10{4}_{-21}^{+21}\) g/cm 3 for convergence at 700 ps is highlighted in Fig.  4 .

The red dots represent the pressure extracted at delays of 300 ps and 500 ps for copper by this work, the squares and triangle corresponds to previous published results 58 , 59 . The blue area represents the experimental compression factor at convergence within experimental uncertainties. The blue dashed line is states reached for Cu according to hydrodynamic simulations, and the brown and orange lines are Fe and C simulations.

Scaling for various materials

A comparison of this value with previously published results for copper is shown in Fig.  4 , together with the predicted states achieved in our experiment according to the FLASH simulations. Furthermore, Fig.  4 shows simulation predictions for carbon and iron as representative materials in the context of astrophysical research. Using the same return current conditions and target diameter as for the Cu wires, the simulations predict pressures up to 790 Mbar for iron and up to 400 Mbar for carbon. The temperatures, in these cases, range up to 32 eV and up to 16 eV for iron and carbon, respectively, at the time of convergence. The carbon states are comparable to the ones expected in Jovian worlds as well as exoplanets 49 , showing the potential of this platform for planetary interior research. The iron states are in the range of the stellar conditions for white dwarf envelopes, as shown in MJ experiments at NIF 50 . While further investigation is beyond the scope of this paper, the consideration of higher dimensionality of the compression is paramount for a fully quantitative prediction of the compression capabilities. First 2D simulations assuming a 33% drop of the initial temperature between the front and back surface of the wire show that the maximal density would decrease by 25% compared to the case of ideal compression, demonstrating the robustness of the process and potential of this method as a platform for HED studies.

In summary, we have demonstrated the capabilities of a Joule-class laser irradiating thin wire targets to generate extreme pressure states relevant to astrophysical studies. We have shown how the state can be characterized via imaging techniques exploiting the ultra-short duration and high brilliance of an XFEL beam. In particular, converging cylindrical shocks in copper with pressures up to 11 Mbar have been measured, with simulations predicting pressures up to 830 Mbar at convergence, supported by the excellent quantitative agreement between experimental and forward calculated data. This method of shock-generation paves the way for performing astrophysical experiments in the laboratory providing large statistics thanks to the high repetition rate of the lasers (shot per minute) and involving simple and ubiquitous targets.

X-ray setup

The X-ray beam was characterized in energy via the elastic scattering of the beam on a YAG scintillator. The elastic signal was measured via a von Hamos X-ray spectrometer 51 , which was previously calibrated via copper K α emission. The energy was determined to be 8.2 keV. The pulse energy of 600 μJ on average was measured via an X-ray gas monitor (XGM) in the X-ray tunnel. An X-ray lens configuration was chosen that resulted in a pencil-like beam at the target chamber (vacuum in the 10 −5 mbar range). The beam size was measured with a YAG scintillator at the pulse arrival monitor located 9.5 m before the target chamber center (TCC) and at 3.3 m after TCC with the same detector used for PCI measurements. The beam size at TCC was interpolated between those two points. The compound refractive lenses stack consisted of 10 Be lenses, with a radius of curvature of 50 μm manufactured by RXOptics Germany with a web thickness of 50 μm for each lens. The resolution of the system was characterized by imaging of a Siemens star test target, NTT-XRESO-50HC. The resolution target is made of tantalum with a thickness of 500 nm. The imaging of the target is shown in Supplementary Fig.  S1 .

Optical laser setup

The ReLaX laser was used at 100 TW energy level, delivering 3 J of energy on target. The pulse duration was optimized using a self-referencing spectral interferometer WIZZLER 800 by Fastlite and was regularly checked for best compression by a second harmonic generator autocorrelator. The pulse intensity contrast was measured to be within the specs presented by Laso Garcia 21 . The focal spot quality was monitored and optimized by using a ×20 APO PLAN microscope objective. The spatial phase was optimized by an adaptive deformable mirror coupled to a wavefront sensor. The synchronization between optical and X-ray laser was measured by spatial photon arrival monitor techniques 52 .

Data processing

Each of the X-ray images taken was flat-fielded according to the following procedure: first, the detector background is subtracted by subtracting the average of 313 empty frames. Then the image is normalized by the pulse energy measured by the XGM. Next, the scattering pattern generated by the slits on the shot of interest is compared to an ensemble of scattering patterns taken without target. The slit scattering pattern is sensitive to the X-ray intensity and the beam pointing. A chi-square minimization is used to find the best match. The normalized on-shot image is divided by the normalized free-beam best match. Finally, any residual intensity variation due to imperfect match is locally corrected by fitting a third-order polynomial to an area 200 pixels around the wire shadow and dividing by the fit result (shown in Supplementary Fig.  S2 ). The final uncertainty of the flatfield is obtained from the peak-to-valley transmission variation associated with the transmission baseline. This results in an uncertainty of s ( T ) = 0.12.

Particle in cell simulations

The 2D PIC simulations are performed with the PICLS code 53 to evaluate the surface return current. To best match the experimental conditions, we first simulate the interaction of the prepulse of ReLaX laser and Cu targets using the MHD code FLASH, which gives the initial density profile for the PIC simulations. The main pulse of the ReLaX laser is modeled by a Gaussian profile both in the spatial and temporal dimensions with full-width at half-maximum (FWHM) spot size w F W H M = 4 μm and duration τ F W H M = 30 fs respectively, resulting in a peak intensity I 0  = 5 × 10 20 W/cm 2 . In order to resolve the plasma wavelength for a fully ionized Cu plasma, that corresponds to an electron density of 1400 n c , where n c  = 1.74 × 10 21 cm −3 is the plasma critical density, the cell size and time step are set Δx = Δy = 5.3 nm and Δt = 0.0178 fs respectively. The simulation box consists of N x  × N y = 7500 × 5000 cells, corresponding to the real space size of 40 × 27 μm. The PIC simulations use an absorbing boundary condition and include field and direct impact ionization models 54 . In addition, the relativistic binary collisions between charged particles are included. To save computational time, the diameter of the Cu wire is reduced from 25 μm to 10 μm while maintaining the same level of fast electron refluxing within the target 55 .

Hydrodynamic simulations

The FLASH code as version 4.6.2, developed by the University of Rochester, was employed for hydrodynamic simulations. These simulations utilized a 1D cylindrical symmetry geometry. The total simulation box is 50 μm in size, with the wire target material occupying 12.5 μm and the rest containing vacuum. This vacuum is filled with low-density hydrogen at a density of 10 −5  gcm −2 and a temperature of 1 eV. The target material varies depending on the case and includes copper, iron, or carbon. Each target region contains solid targets at their respective solid densities. The initial temperature distribution of the target is determined using the electron energy equation 1. Based on the return current scaling theory, magnetic compression driven by the J  ×  B force is disregarded, setting the initial fluid velocity to zero. The boundary conditions use a reflective boundary condition for the symmetry axis and free space for the vacuum. A self-adaptive mesh grid and derived material properties from the corresponding SESAME equation of state were used.

Synthetic PCI data

The density output of the hydrodynamic simulations is used to calculate the synthetic PCI intensity I ( x ,  z ) with x the transverse distance from the axis of the wire relative to the X-ray propagation direction and z the distance from the wire along the X-ray propagation distance. In the first step, the Abel transform is evaluated

to obtain Γ the total mass density projected along the line of sight x using ρ ( r ) the 1D mass density in cylindrical coordinates. From Γ ( x ), the absorption mass attenuation coefficient μ( x ) and the phase Φ( x ,  z  = 0) are calculated at the exit plane of the wire. The change of PCI intensity after a propagation distance z is given by

with I 0 the original intensity, k the wave number of the X-ray. In the last step, we consider both the experimental resolution and also the bandwidth of the SASE X-ray beam. The limitations of the resolution are simulated by applying a Butterworth filter with a cut-off frequency of 0.00057nm −1 corresponding to the Nyquist frequency of the detector imaging system. The SASE bandwidth effect is calculated by approximating the spectral distribution as a Gaussian with 20 eV FWHM and sampling 40 wavelengths ω . For each of these wavelengths, a corrected z (compared to the 6 mm) is used to obtain I ( x ,  z ( ω )). The final PCI intensity is integrated from the I ( ω ) with weights given by the spectral distribution.

Shock density reconstruction

The procedure to reconstruct the density from experimental PCI profiles is the direct inversion of the procedure to produce the synthetic PCI data. The Paganin method is used to invert the TIE equation and obtain the intensity profile at target contact. The intensity at contact, I ( x ,  y ,  z  = 0) relates to the measured one, I ( x ,  y ,  z ) as

where μ is the mass attenuation coefficient, δ is the real part of the index of refraction, d is the pixel size on target and k x , y are the spatial frequencies at the detector plane, and \({{\mathcal{F}}}\) the Fourier transform. The intensity at contact is then directly related to the mass attenuation coefficient. The attenuation as a function of radius is calculated via the inverse Abel transform of the intensity. Finally, division by the mass attenuation coefficient returns the radial density profile. An estimate of the uncertainty of the Abel reconstruction was obtained by shifting the center of rotation by a distance of 500 nm around the nominal axis to account for the finite resolution. This procedure was done for the experimental intensity profile as well as the upper and lower boundaries. The final peak density and uncertainty are calculated as the average and standard deviation of the peak density of all profiles. The peak density for the 300 ps data point is \(\rho=2{2}_{-5.0}^{+10}\) gcm −3 , for the 500 ps \(\rho=2{0}_{-4.5}^{+8.4}\) gcm −3 and for 700 ps \(\rho=10{4}_{-21}^{+21}\,{{\rm{g}}}{{{\rm{cm}}}}^{-3}\) .

Calculation of the shock pressure

The pressure was extracted from the experimental data via the Rankine-Hugoniot relation. The pressure can be expressed as:

where p is the shock pressure, p 0 is the pressure of the matter in front of the shock, u s is the shock velocity, ρ 0 is the uncompressed density and ρ is the compressed density. In our case, the pressure in front of the shock is ambient pressure. The shock velocity can be extracted from the imaging data by measuring the distance traveled by the shock between two time delays, specifically between 300 ps and 500 ps. The uncertainty in the timing measurement is negligible due to the synchronization of the beams with an RMS  <200 fs. The uncertainty in the position is limited by the imaging resolution of  ≈500 nm. With this approach, the shock velocity results in u s  = 14.3 ± 1.3 km/s (at 42 μm from the interaction point). The density reconstruction has been discussed in the previous section. The uncertainty on the pressure is calculated by propagating the uncertainties of the reconstructed density and shock velocity.

Data availability

Data recorded for the experiment at the European XFEL are available at EuXFEL data repository, HED 4597 56 , after the expiration of the embargo period or upon reasonable request. The simulation data used to generate Figs  2 – 4 are available at the Rossendorf data repository 57 .

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Acknowledgements

We acknowledge the European XFEL in Schenefeld, Germany, for the provision of X-ray free electron laser beam time at the Scientific Instrument HED (High Energy Density Science) and would like to thank the staff for their assistance. The authors are indebted to the HIBEF user consortium for the provision of instrumentation and staff that enabled this experiment. FLASH was developed in part by the DOE NNSA and DOE Office of Science-supported Flash Center for Computational Science at the University of Chicago and the University of Rochester.

Open Access funding enabled and organized by Projekt DEAL.

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These authors contributed equally: Alejandro Laso Garcia, Long Yang.

Authors and Affiliations

Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, Dresden, 01328, Germany

Alejandro Laso Garcia, Long Yang, Carsten Baehtz, Hauke Höppner, Thomas Kluge, Alexander Pelka, Thomas E. Cowan, Lingen Huang & Toma Toncian

European XFEL, Holzkoppel 4, Schenefeld, 22869, Germany

Victorien Bouffetier, Karen Appel, Oliver Humphries, Mikhail Mishchenko, Motoaki Nakatsutsumi, Thomas R. Preston, Lisa Randolph & Ulf Zastrau

Deutsches Elektronen-Synchrotron DESY, Notkestraße 86, Hamburg, 22607, Germany

Johannes Hagemann

Technische Universität Dresden, Dresden, 01062, Germany

Thomas E. Cowan

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Contributions

A.L.G. and T.T. conceptualized the experiment. L.Y. and L.H. developed the theory and performed the PIC and hydrodynamic simulations. L.Y., L.H., A.L.G. and T.T. analyzed the simulations. A.L.G., V.B., K.A., C.B., H.H., O.H., M.M., M.N., A.P., T.R.P., L.R. and T.T. performed the experiment. A.L.G., L.Y., L.H., T.E.C., J.H. and T.T. analyzed the data. A.L.G., L.Y., L.H., T.E.C. and T.T. wrote the original manuscript draft. All authors, including T.K. and U.Z., reviewed and edited the manuscript. L.H. and T.T. supervised the project.

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Correspondence to Lingen Huang or Toma Toncian .

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Laso Garcia, A., Yang, L., Bouffetier, V. et al. Cylindrical compression of thin wires by irradiation with a Joule-class short-pulse laser. Nat Commun 15 , 7896 (2024). https://doi.org/10.1038/s41467-024-52232-6

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To Find the Focal Length of a Convex Mirror Using a Convex Lens

To find the focal length of a convex mirror, using a convex lens.

Apparatus and Materials Required

• An optical bench
• A convex mirror
• Three needles
• A half-metre scale
• A convex lens
• Four uprights with at least two having lateral motion
• A lens holder

A convex lens generates a real image of a subject. A convex mirror is positioned in the way of the light rays between the image and lens such that the light rays, after refraction through the lens, normally strike on the mirror’s surface. The light rays are then reflected back, retracing their trajectory and generating an image of the object.

The focal length of the mirror is calculated as,

where f is the focal length mirror and R is the radius of curvature.

• Calculate the probable focal length of the convex lens by focussing the image of a distant subject (example: a tree) on a clear screen and calculating the length between the image and the lens.
• Fix the object needle, the mirror, and the lens on the optical bench as shown in the above figure. Then adjust their height in a way that the needle’s tip, the pole of the mirror, and the lens’s optical centre lie at the same horizontal alignment.
• Place the object pin between 2F and F. Tune the mirror and the needle in such a way that there is no significant parallax between the object needle’s tip and its inverted image produced at O.

4. Measure the positions of the lens (L), the mirror (M), and the object (O).

5. Take out the mirror and place another needle, C, on the same side of the mirror. Tune the needle C in such a way as to remove the parallax between the needle C and the image of object O. Note down the position of C.

6. Repeat procedures 3 to 5 to note down at least five various locations of the lens and the object.

7. Calculate the index correction between the imaging needle and the mirror as described previously.

Observation

• The focal length of the convex lens, F = _____ cm.
• The actual length of the index needle, L = _____ cm.
• Observed length of the index needle = Position of mirror upright – Position of pin upright on the scale = _____ cm

4. Index correction, e = Actual length – observed length = … cm.

Calculations

Determine the mean value of the radius of curvature of the convex mirror, R , and determine its focal length using the following relation

f = R/2 = ____ cm

Sources of Error

1. Parallax cannot be corrected completely.

2. The uprights are not ideally vertical.

3. The pole of the mirror, the tip of the object needle and the optical centre are not placed in a line.

The focal length of the given convex mirror is …… cm. Here f is the mean value of the focal length.

1. What are the main types of mirrors?

Answer: There are two main types of mirrors, and they are as follows:

• Plane Mirror
• Spherical Mirror

2. Is silvering in mirrors generated by applying a silver coating or some other substances?

Answer: In inexpensive mirrors, silvering is generated by the application of mercuric oxide, while the silver coating in excellent quality mirrors is generated by the application of silver nitrate.

3. What kind of mirror is typically used for the dressing desk?

Answer: A plane mirror is employed in a dressing desk as it provides a virtual image of the exact dimension of the object positioned in front of it.

4. What is meant by the index error?

Answer: The variation between the real distance between the point object and the mirror’s pole and the recorded distance calculated on the optical bench is known as the index error. It is also called the bench error.

Index Correction = Actual Distance – Observed Distance

Index Error = Observed Distances – Actual Distance

5. What is meant by focal length?

Answer: Focal length is the length between the optical centre and the principal focus of a lens.

6. Is a convex mirror a transparent or opaque optical tool?

Answer: A convex mirror is an opaque optical tool.

7. What happens to the incident light ray when it falls on a convex lens?

Answer: A convex lens coincides with the incident light rays approaching the principal axis.

8. Which lens is called a diverging lens?

Answer: A concave lens is called a diverging lens.

9. When a convex lens is paired with a concave lens, what will the image’s resolution be?

Answer: A sharper image is produced when a convex lens is combined with a concave lens.

10. What is the principle behind the working of a lens?

Answer: Refraction is the principle behind the working of a lens.

11. What is the type of lens found in the human eye?

Answer: Biconvex lens is the lens found in the human eye.

Watch the video and revise all the important concepts in the chapter Light Reflection and Refraction Class 10

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